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When I first took ordinary differential equations, the pattern was always the same. Week 1 turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations… and by Week 2 or 3 the course has quietly degenerated into hammer-picking. Spot the form, apply...

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Mathelirium

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We just watched Professor Arthur Mattuck kick off MIT’s ODE course with the one interpretation that most differential equations classes somehow postpone: An ordinary differential equation isn’t primarily a method hunt. It’s a geometric rule. You write dy/dx = f(x,y), and that right-hand side is literally telling you the slope your solution curve must have at each point (x,y). So I built this animation as a visual companion to that first lecture. It draws the direction field (little line elements whose slope is f(x,y)) and then shows integral curves sliding through it...curves that are tangent to the field everywhere they go. Two quick examples from the animation: For dy/dx = −x/y, the slope field steers you onto circles x² + y² = R². You also see a subtle point that gets missed when everything is taught as y(x)...even when the curve exists smoothly, the graph y(x) may only exist on a limited interval (|x|<R for the upper semicircle). For dy/dx = 1 + x − y, the isoclines (curves where the slope is constant) make the global behavior obvious...trajectories get funneled into a corridor and become asymptotic to the special solution y=x. You learn qualitative behavior without solving it the traditional way. #DifferentialEquations #ODEs #MITOCW #VectorFields #MathAnimation #Mathematics
0:30

We just watched Professor Arthur Mattuck kick off MIT’s ODE course with the one interpretation that most differential equations classes somehow postpone: An ordinary differential equation isn’t primarily a method hunt. It’s a geometric rule. You write dy/dx = f(x,y), and that right-hand side is literally telling you the slope your solution curve must have at each point (x,y). So I built this animation as a visual companion to that first lecture. It draws the direction field (little line elements whose slope is f(x,y)) and then shows integral curves sliding through it...curves that are tangent to the field everywhere they go. Two quick examples from the animation: For dy/dx = −x/y, the slope field steers you onto circles x² + y² = R². You also see a subtle point that gets missed when everything is taught as y(x)...even when the curve exists smoothly, the graph y(x) may only exist on a limited interval (|x|<R for the upper semicircle). For dy/dx = 1 + x − y, the isoclines (curves where the slope is constant) make the global behavior obvious...trajectories get funneled into a corridor and become asymptotic to the special solution y=x. You learn qualitative behavior without solving it the traditional way. #DifferentialEquations #ODEs #MITOCW #VectorFields #MathAnimation #Mathematics

Mathelirium

22,367 Aufrufe • vor 5 Monaten

Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics
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Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

106,981 Aufrufe • vor 6 Monaten

In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics
0:30

In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

23,254 Aufrufe • vor 1 Monat

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics
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Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Mathelirium

30,340 Aufrufe • vor 3 Monaten

Do you actually know what convex optimization is in the geometric, guarantee-theoretic sense or have you only met it through solvers and loss curves? Convexity is rare comfort in optimization...there are no spurious local minima, no surprise traps, and inequalities you can use like tools instead of prayers. So, what is this convexity? Let x = (x₁, x₂) and let f(x) be convex. Plot the surface z = f(x). Pick a contact point x₀. The local slope is the gradient p = ∇f(x₀). That p is exactly the data that defines the supporting plane: z = f(x₀) + p · (x − x₀). Thus, f is said to be convex because for every x, f(x) ≥ f(x₀) + p · (x − x₀). So the plane at x₀ can slide under the surface, but it never slices through it. Not near the point...everywhere. Now for here is the interesting part: The slope becomes a coordinate system! Rewrite the same plane as z = p · x − b, where b is the offset. Because the plane passes through (x₀, f(x₀)), the offset is forced to be b = p · x₀ − f(x₀). And that number isn’t just geometry trivia. It’s the convex conjugate: f*(p) = sup over x ( p · x − f(x) ). At a differentiable contact point, the supporting plane touches f tightly enough that the supremum is achieved at x₀, giving the identity f*(p) = p · x₀ − f(x₀) when p = ∇f(x₀). So one moving contact point gives two linked readouts: primal position x₀ dual position (slope) p = ∇f(x₀) dual offset f*(p) One surface. Two worlds. #ConvexOptimization #Optimization #MachineLearning #SignalProcessing #AppliedMath #Engineering
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Do you actually know what convex optimization is in the geometric, guarantee-theoretic sense or have you only met it through solvers and loss curves? Convexity is rare comfort in optimization...there are no spurious local minima, no surprise traps, and inequalities you can use like tools instead of prayers. So, what is this convexity? Let x = (x₁, x₂) and let f(x) be convex. Plot the surface z = f(x). Pick a contact point x₀. The local slope is the gradient p = ∇f(x₀). That p is exactly the data that defines the supporting plane: z = f(x₀) + p · (x − x₀). Thus, f is said to be convex because for every x, f(x) ≥ f(x₀) + p · (x − x₀). So the plane at x₀ can slide under the surface, but it never slices through it. Not near the point...everywhere. Now for here is the interesting part: The slope becomes a coordinate system! Rewrite the same plane as z = p · x − b, where b is the offset. Because the plane passes through (x₀, f(x₀)), the offset is forced to be b = p · x₀ − f(x₀). And that number isn’t just geometry trivia. It’s the convex conjugate: f*(p) = sup over x ( p · x − f(x) ). At a differentiable contact point, the supporting plane touches f tightly enough that the supremum is achieved at x₀, giving the identity f*(p) = p · x₀ − f(x₀) when p = ∇f(x₀). So one moving contact point gives two linked readouts: primal position x₀ dual position (slope) p = ∇f(x₀) dual offset f*(p) One surface. Two worlds. #ConvexOptimization #Optimization #MachineLearning #SignalProcessing #AppliedMath #Engineering

Mathelirium

38,506 Aufrufe • vor 6 Monaten

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics
0:46

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

Mathelirium

66,204 Aufrufe • vor 5 Monaten

Lecture 1 on Physics-Informed Neural Networks: A Mini-Series Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:00

Lecture 1 on Physics-Informed Neural Networks: A Mini-Series Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

47,308 Aufrufe • vor 5 Monaten

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics
0:46

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

Mathelirium

133,454 Aufrufe • vor 4 Monaten

What if Your Neural Network Was Forced to Obey Physics? Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:25

What if Your Neural Network Was Forced to Obey Physics? Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

17,285 Aufrufe • vor 1 Monat

What are Physics-Informed Neural Networks (PINNs) Physics-Informed Neural Networks (PINNs) are neural nets trained to satisfy a differential equation. The trick is simple. You bake the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work. Meshes. Stencils. Stability tuning. And once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain. Most people first meet PINNs in the least helpful way. A pretty solution surface, almost no clarity on what was enforced to make it appear. In this series we keep the enforcement visible. We pick a PDE, represent the unknown solution as a flexible function, measure how badly that function violates the equation across the domain, and train it to reduce that mismatch at the points we sample. A normal neural net learns from labels. You give it inputs and target outputs. A PINN learns from an equation. You give it inputs (x,t), and it gets penalized whenever its output fails the PDE. Smaller mismatch means smaller loss. Bigger mismatch means bigger loss. That’s all “punish” and “reward” mean here. The network isn’t replacing physics. It’s just a flexible function that we force to obey the same calculus you’d demand from any candidate solution. The math breakdown: We start with a PDE on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we’d have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or from an initial condition. The training objective is a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is how we get the derivatives inside rᵩ. We don’t approximate them with finite differences. We compute them with automatic differentiation: ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … Then we differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. That’s the whole idea. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold, a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t). Each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large, with color showing the sign. As training runs, those threads go slack across the domain, not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:00

What are Physics-Informed Neural Networks (PINNs) Physics-Informed Neural Networks (PINNs) are neural nets trained to satisfy a differential equation. The trick is simple. You bake the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work. Meshes. Stencils. Stability tuning. And once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain. Most people first meet PINNs in the least helpful way. A pretty solution surface, almost no clarity on what was enforced to make it appear. In this series we keep the enforcement visible. We pick a PDE, represent the unknown solution as a flexible function, measure how badly that function violates the equation across the domain, and train it to reduce that mismatch at the points we sample. A normal neural net learns from labels. You give it inputs and target outputs. A PINN learns from an equation. You give it inputs (x,t), and it gets penalized whenever its output fails the PDE. Smaller mismatch means smaller loss. Bigger mismatch means bigger loss. That’s all “punish” and “reward” mean here. The network isn’t replacing physics. It’s just a flexible function that we force to obey the same calculus you’d demand from any candidate solution. The math breakdown: We start with a PDE on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we’d have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or from an initial condition. The training objective is a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is how we get the derivatives inside rᵩ. We don’t approximate them with finite differences. We compute them with automatic differentiation: ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … Then we differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. That’s the whole idea. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold, a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t). Each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large, with color showing the sign. As training runs, those threads go slack across the domain, not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

44,675 Aufrufe • vor 4 Monaten

Sometimes the thing that hurts you, isn’t the moment itself, it’s the fact that you keep reliving it. Not once. Not twice. But in patterns that look different outside and feel identical inside. You start wondering why it keeps happening. Why the same fears return. Why the same version of you shows up even when you swore you were done with him. Most people think life moves in a straight line. But the Vishnu Purana says something completely different: “Krita, Treta, Dvapara and Kali, these ages turn like a wheel.” A wheel. Not a timeline. A rotation that keeps circling back until something in you changes. Centuries later, Nietzsche said almost the same thing, his idea of “eternal recurrence.” If you had to live this day again and again, would you be proud of it, or terrified that you’re still choosing the same life? And even modern physics keeps stumbling toward the same shape, cyclic universe models, Big Bangs followed by cosmic “bounces,” universe after universe rising from the ashes of the last. Once you see this, your own life starts making a different kind of sense. You stop blaming fate and start noticing your habits. You stop praying for a new life while living the old one on autopilot. Because you’re not trapped in a cycle. You’re participating in one. So ask yourself, What keeps returning because you haven’t faced it yet? What pattern keeps knocking like a teacher you ignore? What moment repeats because you haven’t shown up differently? In a universe where time moves in circles, even the smallest choice isn’t small. It becomes a pattern. A future. A repetition, or a release. Maybe this is the moment where the wheel comes back around and you finally choose a different version of you. Maybe this is your cycle break.
2:55

Sometimes the thing that hurts you, isn’t the moment itself, it’s the fact that you keep reliving it. Not once. Not twice. But in patterns that look different outside and feel identical inside. You start wondering why it keeps happening. Why the same fears return. Why the same version of you shows up even when you swore you were done with him. Most people think life moves in a straight line. But the Vishnu Purana says something completely different: “Krita, Treta, Dvapara and Kali, these ages turn like a wheel.” A wheel. Not a timeline. A rotation that keeps circling back until something in you changes. Centuries later, Nietzsche said almost the same thing, his idea of “eternal recurrence.” If you had to live this day again and again, would you be proud of it, or terrified that you’re still choosing the same life? And even modern physics keeps stumbling toward the same shape, cyclic universe models, Big Bangs followed by cosmic “bounces,” universe after universe rising from the ashes of the last. Once you see this, your own life starts making a different kind of sense. You stop blaming fate and start noticing your habits. You stop praying for a new life while living the old one on autopilot. Because you’re not trapped in a cycle. You’re participating in one. So ask yourself, What keeps returning because you haven’t faced it yet? What pattern keeps knocking like a teacher you ignore? What moment repeats because you haven’t shown up differently? In a universe where time moves in circles, even the smallest choice isn’t small. It becomes a pattern. A future. A repetition, or a release. Maybe this is the moment where the wheel comes back around and you finally choose a different version of you. Maybe this is your cycle break.

Wisdom Walk

11,296 Aufrufe • vor 6 Monaten

Paul Verhoeven on whether the events that take place in "Total Recall" (1990) are real or just a dream: "It is both. To be honest, that’s what I want. I made the movie in a way that it would be true on both levels, and I spent a lot of time to get that. If you want a scientific explanation, you know, of course, in quantum mechanics there is a very interesting principle, the principle of uncertainty, Heisenberg’s principle. If you have a big object and if you try to measure the place of the object and the velocity of the object at the same time, the more precisely you measure velocity the less precise place gets. So that’s the principle. That means, of course, that there are different realities possible at the same moment. What I wanted to do in 'Total Recall' is to do a movie where both levels are true. I mean for me, of course, the film anyhow has to do with two realities, one being the reality of going as a secret agent to Mars and discovering that there is a problem, and solving the problem, which is starting the nuclear reactor and helping the guerrillas and destroying Cohaagen. The second level of the movie, of course, is that from the moment that he goes into the Rekall chair ‘til the end it’s a dream, and I tried to make that second level work throughout the whole movie. So there’s the dream level which starts when he gets into the chair and the thing is in his neck, and that would go throughout the whole movie, so in the next scene where they say, Oh, there’s a problem, there’s a big glitch here, that would be already the dream, of course. That’s where the dream starts. And the next scene where they are fighting and stuff would be part of his dream, convincing him that it is real, because there is a glitch but that would be part of the program. It would be built into the program to make him accept the fact that it’s real, but it’s a dream. If you look at the movie, if you haven’t seen it, or for the second time, you’ll see that the whole program that’s set up at the beginning when he goes to the Rekall office and he talks to this guy who sells him the program on Mars, you’ll see that he gets everything that he wants: he gets the trip to Mars, he gets the girl, the exotic girl, he ki!!s the bad guys, and he saves the entire planet. That’s what he does. And that’s basically the dream. Even halfway through the movie, you may remember, this other guy comes in, Dr. Edgemar, and tells him that he’s in a dream, that he’s still in the Rekall chair, and then Arnold says, “If I’m there, I can ki!! you.” And he puts a gun to his head and the guy says, “Sure, no problem for me, big problem for you, because you will be psychotic from now on because the walls of reality will fall apart. One moment you will be the savior of the rebel cause, the next moment you’ll be Cohaagen’s bosom buddy, but in the end—you will even have these strange fantasies about alien civilizations—but at the end you will be lobotomized.” And then if you see the movie, you realize that all these things happen. I mean he is lobotomized at the end. That’s why at the last shot, when they are so happy and kissing each other, it slowly fades to white, which for me meant, “OK, there he goes. That’s the end-that’s the dream—they lobotomized him.” And all the other things happened, he finds the alien civilization, he rescues the planet, he finds the good girl, he k!!ls the bad guys, but it’s a dream. Now, of course you can see it as a reality, too. So at the end of the movie, getting to white means either it’s a happy ending or he loses his brains . . . which is probably also a happy ending, I don’t know. That was basically what l wanted—that at the end there would be two possibilities, and they would be both true—for me they are both true—it’s not either one or the other. It’s not that either it’s a dream or it is a reality. It is a dream and it is a reality. And I think they’re both there." (Paul Verhoeven's interview with Chris Shea & Wade Jennings, 1992) P.S: On this day, 36 years ago, "Total Recall" (1990) premiered in Los Angeles, California, USA.
1:25

Paul Verhoeven on whether the events that take place in "Total Recall" (1990) are real or just a dream: "It is both. To be honest, that’s what I want. I made the movie in a way that it would be true on both levels, and I spent a lot of time to get that. If you want a scientific explanation, you know, of course, in quantum mechanics there is a very interesting principle, the principle of uncertainty, Heisenberg’s principle. If you have a big object and if you try to measure the place of the object and the velocity of the object at the same time, the more precisely you measure velocity the less precise place gets. So that’s the principle. That means, of course, that there are different realities possible at the same moment. What I wanted to do in 'Total Recall' is to do a movie where both levels are true. I mean for me, of course, the film anyhow has to do with two realities, one being the reality of going as a secret agent to Mars and discovering that there is a problem, and solving the problem, which is starting the nuclear reactor and helping the guerrillas and destroying Cohaagen. The second level of the movie, of course, is that from the moment that he goes into the Rekall chair ‘til the end it’s a dream, and I tried to make that second level work throughout the whole movie. So there’s the dream level which starts when he gets into the chair and the thing is in his neck, and that would go throughout the whole movie, so in the next scene where they say, Oh, there’s a problem, there’s a big glitch here, that would be already the dream, of course. That’s where the dream starts. And the next scene where they are fighting and stuff would be part of his dream, convincing him that it is real, because there is a glitch but that would be part of the program. It would be built into the program to make him accept the fact that it’s real, but it’s a dream. If you look at the movie, if you haven’t seen it, or for the second time, you’ll see that the whole program that’s set up at the beginning when he goes to the Rekall office and he talks to this guy who sells him the program on Mars, you’ll see that he gets everything that he wants: he gets the trip to Mars, he gets the girl, the exotic girl, he ki!!s the bad guys, and he saves the entire planet. That’s what he does. And that’s basically the dream. Even halfway through the movie, you may remember, this other guy comes in, Dr. Edgemar, and tells him that he’s in a dream, that he’s still in the Rekall chair, and then Arnold says, “If I’m there, I can ki!! you.” And he puts a gun to his head and the guy says, “Sure, no problem for me, big problem for you, because you will be psychotic from now on because the walls of reality will fall apart. One moment you will be the savior of the rebel cause, the next moment you’ll be Cohaagen’s bosom buddy, but in the end—you will even have these strange fantasies about alien civilizations—but at the end you will be lobotomized.” And then if you see the movie, you realize that all these things happen. I mean he is lobotomized at the end. That’s why at the last shot, when they are so happy and kissing each other, it slowly fades to white, which for me meant, “OK, there he goes. That’s the end-that’s the dream—they lobotomized him.” And all the other things happened, he finds the alien civilization, he rescues the planet, he finds the good girl, he k!!ls the bad guys, but it’s a dream. Now, of course you can see it as a reality, too. So at the end of the movie, getting to white means either it’s a happy ending or he loses his brains . . . which is probably also a happy ending, I don’t know. That was basically what l wanted—that at the end there would be two possibilities, and they would be both true—for me they are both true—it’s not either one or the other. It’s not that either it’s a dream or it is a reality. It is a dream and it is a reality. And I think they’re both there." (Paul Verhoeven's interview with Chris Shea & Wade Jennings, 1992) P.S: On this day, 36 years ago, "Total Recall" (1990) premiered in Los Angeles, California, USA.

DepressedBergman

66,465 Aufrufe • vor 1 Monat

Mark Cuban says Elon f#cked up Twitter "First of all, Twitter X is such a cesspool now. It's a shithole. Instagram and Tik Tok are still chill where you can mess with people and have fun, but you just can't do it on Twitter" "The way it works is called the nearest tree algorithm. If you are a big user on X and you have a lot of followers and those followers engage with you, then not only does your stuff get elevated but all the people you engage with get elevated" "So when Elon interacts with some of these crazy people, their stuff gets elevated and the people that they talk to gets elevated"
1:27

Mark Cuban says Elon f#cked up Twitter "First of all, Twitter X is such a cesspool now. It's a shithole. Instagram and Tik Tok are still chill where you can mess with people and have fun, but you just can't do it on Twitter" "The way it works is called the nearest tree algorithm. If you are a big user on X and you have a lot of followers and those followers engage with you, then not only does your stuff get elevated but all the people you engage with get elevated" "So when Elon interacts with some of these crazy people, their stuff gets elevated and the people that they talk to gets elevated"

dank

1,938,339 Aufrufe • vor 2 Monaten

How do we make invisible progress visible? "If you look at technologies that are really good at building habits, they’re great at visualization. You’re playing a video game; you have a score in the top corner of the screen. It’s increasing all the time as you go through the level. Every time you pick up a weapon or a ruby or a gem or whatever, there’s a little jingle or a chime; there’s some kind of power-up. Even the pitter-patter of footsteps as you run through the level is a signal that you’re progressing. So you have all these visual and audio signals of progress; the feedback is immediate. And if you compare that experience of just playing this level for 15 minutes to what it’s like to sit down and try to write a blog post or record a podcast interview or do anything in the normal world, it’s very different. The feedback comes much slower in real life. My parents like to swim. And one of the challenges of swimming is that your body looks exactly the same when you get out of the water as it did when you jumped in. If they’re doing it for the visual results, it takes two years for your body to change in the mirror. And so they have a little habit tracker where they just put an X down for each day they do their workout, and then at the end of the month, they count up how many workouts they did and compare that number to the month before. Putting that X down at the end of the workout is such a small little thing, but it’s one little visual signal that we showed up and we did the right thing today, that I did what I was supposed to do. It’s a way of visualizing your progress. Finding ways to visualize your progress as you go through your habits is important because a hallmark of almost any compounding process is that the greatest returns are delayed." (James Clear on The Knowledge Project)
1:47

How do we make invisible progress visible? "If you look at technologies that are really good at building habits, they’re great at visualization. You’re playing a video game; you have a score in the top corner of the screen. It’s increasing all the time as you go through the level. Every time you pick up a weapon or a ruby or a gem or whatever, there’s a little jingle or a chime; there’s some kind of power-up. Even the pitter-patter of footsteps as you run through the level is a signal that you’re progressing. So you have all these visual and audio signals of progress; the feedback is immediate. And if you compare that experience of just playing this level for 15 minutes to what it’s like to sit down and try to write a blog post or record a podcast interview or do anything in the normal world, it’s very different. The feedback comes much slower in real life. My parents like to swim. And one of the challenges of swimming is that your body looks exactly the same when you get out of the water as it did when you jumped in. If they’re doing it for the visual results, it takes two years for your body to change in the mirror. And so they have a little habit tracker where they just put an X down for each day they do their workout, and then at the end of the month, they count up how many workouts they did and compare that number to the month before. Putting that X down at the end of the workout is such a small little thing, but it’s one little visual signal that we showed up and we did the right thing today, that I did what I was supposed to do. It’s a way of visualizing your progress. Finding ways to visualize your progress as you go through your habits is important because a hallmark of almost any compounding process is that the greatest returns are delayed." (James Clear on The Knowledge Project)

Shane Parrish

24,833 Aufrufe • vor 5 Monaten

While revising group theory for a graduate abstract algebra course, I went looking for a second voice, because algebra has this specific kind of pain…you can follow every line of a proof and still lose the idea. That’s how I landed on Benedict H. Gross’s Harvard Extension Math 122 lectures. The difference is immediate. He isn’t reading theorems off a page and transcribing them onto the board like most Algebra lecturers. He rebuilds the argument from the objects upward, and you can feel the mechanism engage: definitions, examples, the one obstruction you forgot to check. And the weird part is you don’t feel the urge to take notes. You just listen. The lecture has that story-logic where each sentence sets up the next, and by the time he writes the key line, you already know why it has to be true. It’s elegant in the literal sense.🫡 Teaching isn’t for everyone. There are a lot of smart people who simply can’t teach! Real teaching is a separate craft…timing, taste in examples, and an instinct for where students will get lost while still nodding. That’s why, if you scan your whole education, you can usually name only a small handful of people who actually moved the needle. Gross is in that small group. In this segment, Benedict Gross is basically saying: “You don’t guess a group law on cosets. You define it by copying (‘transporting’) a group law from somewhere it already exists, using a bijection.” par transport de structure, in one chain: Start with a homomorphism f : G → G′ and set H = ker(f). Cosets are exactly fibers: aH = bH ⇔ f(a) = f(b). Define a bijection φ : G/H → Im(f) by φ(aH) = f(a). Pull the multiplication back from Im(f) to G/H by defining (aH) ⋆ (bH) = φ⁻¹( φ(aH) · φ(bH) ). Then the computation forces the familiar rule: (aH) ⋆ (bH) = φ⁻¹(f(a)f(b)) = φ⁻¹(f(ab)) = (ab)H. So the quotient product isn’t a guess. It’s the pulled-back product that makes φ a homomorphism by construction. #AbstractAlgebra #GroupTheory #QuotientGroups #IsomorphismTheorem #MathLectures #Mathematics
7:57

While revising group theory for a graduate abstract algebra course, I went looking for a second voice, because algebra has this specific kind of pain…you can follow every line of a proof and still lose the idea. That’s how I landed on Benedict H. Gross’s Harvard Extension Math 122 lectures. The difference is immediate. He isn’t reading theorems off a page and transcribing them onto the board like most Algebra lecturers. He rebuilds the argument from the objects upward, and you can feel the mechanism engage: definitions, examples, the one obstruction you forgot to check. And the weird part is you don’t feel the urge to take notes. You just listen. The lecture has that story-logic where each sentence sets up the next, and by the time he writes the key line, you already know why it has to be true. It’s elegant in the literal sense.🫡 Teaching isn’t for everyone. There are a lot of smart people who simply can’t teach! Real teaching is a separate craft…timing, taste in examples, and an instinct for where students will get lost while still nodding. That’s why, if you scan your whole education, you can usually name only a small handful of people who actually moved the needle. Gross is in that small group. In this segment, Benedict Gross is basically saying: “You don’t guess a group law on cosets. You define it by copying (‘transporting’) a group law from somewhere it already exists, using a bijection.” par transport de structure, in one chain: Start with a homomorphism f : G → G′ and set H = ker(f). Cosets are exactly fibers: aH = bH ⇔ f(a) = f(b). Define a bijection φ : G/H → Im(f) by φ(aH) = f(a). Pull the multiplication back from Im(f) to G/H by defining (aH) ⋆ (bH) = φ⁻¹( φ(aH) · φ(bH) ). Then the computation forces the familiar rule: (aH) ⋆ (bH) = φ⁻¹(f(a)f(b)) = φ⁻¹(f(ab)) = (ab)H. So the quotient product isn’t a guess. It’s the pulled-back product that makes φ a homomorphism by construction. #AbstractAlgebra #GroupTheory #QuotientGroups #IsomorphismTheorem #MathLectures #Mathematics

Mathelirium

34,256 Aufrufe • vor 5 Monaten

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics
1:00

Quantum mechanics has a reputation for being mystical mainly because people skip the rules and jump to interpretations. In this lecture series, we’re doing the opposite. We start from the rules, follow the algebra, and let the picture be the calculation. Classical Probability Theory combines alternatives by adding their probabilities. Quantum Theory combines them one step earlier…add complex amplitudes first, then square at the end. That swap in order is everything. Expand |a₁ + a₂|² and you don’t just get |a₁|² + |a₂|²…you get a cross-term, 2 Re(a₁ a₂*). Its sign is set by phase, so the same two contributions can reinforce or cancel. Interference is just the algebra of squaring a sum. In the 3D render, the surface height is proportional to |a(x)| (so peaks become bright bands after squaring), while the surface skin is colored by the local phase arg(a(x)). As the phase knob φ(t) is swept on path 2, the cross-term oscillates, and you literally watch the interference ridges slide across the screen. We model a detector screen with coordinates x in R² (think x = (x,y)). A quantum state assigns a complex amplitude a(x). The rule for outcomes is p(x) = |a(x)|² Now the key situation: two coherent alternatives contribute to the same outcome x. Let their amplitudes be a₁(x) and a₂(x). Quantum says a(x) = a₁(x) + a₂(x) So the probability density becomes p(x) = |a₁(x) + a₂(x)|² Expand it (this is the whole episode): p(x) = (a₁ + a₂)(a₁* + a₂*) = |a₁|² + |a₂|² + a₁ a₂* + a₁* a₂ = |a₁|² + |a₂|² + 2 Re(a₁ a₂*) That last term is the interference term. It can be positive or negative. To see phase explicitly, write each contribution in polar form: a₁(x) = r₁(x) exp(i θ₁(x)) a₂(x) = r₂(x) exp(i θ₂(x)) Then a₁ a₂* = r₁ r₂ exp(i(θ₁ − θ₂)) So the cross-term is 2 Re(a₁ a₂*) = 2 r₁ r₂ cos(θ₁(x) − θ₂(x)) That’s the fringe engine: p(x) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x)) Now the phase knob we animate: Add a controllable phase shift φ to path 2: a₂(x) → a₂(x) exp(i φ) Then Δθ(x) → Δθ(x) − φ, so p(x; φ) = r₁² + r₂² + 2 r₁ r₂ cos(Δθ(x) − φ) As φ changes smoothly, the bright/dark pattern slides continuously. Same setup, same geometry, same magnitudes r₁,r₂, only phase changed. #QuantumMechanics #WaveInterference #ComplexAmplitudes #DoubleSlit #Physics #Mathematics

Mathelirium

81,405 Aufrufe • vor 6 Monaten

🚨ELON MUSK: "Legacy Media is from a time when in order to read the news it had to go through a centralized institution. You don't need that anymore, you have the Internet. That's why X is the future, it's Citizen Journalism. By the people, for the people. Legacy media is controlled by a handful of editors in chiefs, they decide, but on X you get to hear from actual experts in the field."
2:19

🚨ELON MUSK: "Legacy Media is from a time when in order to read the news it had to go through a centralized institution. You don't need that anymore, you have the Internet. That's why X is the future, it's Citizen Journalism. By the people, for the people. Legacy media is controlled by a handful of editors in chiefs, they decide, but on X you get to hear from actual experts in the field."

Autism Capital 🧩

9,996,142 Aufrufe • vor 1 Jahr

Ever Heard of a Metric in Differential Geometry? How About a Geodesic? These two ideas are central to how Mathematics and Physics describe Spacetime. A metric is how a surface measures distance and angle. A geodesic is the path that counts as "straight" after that measuring rule is chosen. Once the metric changes from point to point, straightness is no longer something you judge by eye. On a curved surface, the shortest paths are geodesics. They are chosen by the metric. Take a surface written as r(x,y) = (x, y, f(x,y)) Its metric is gᵢⱼ = ∂ᵢr · ∂ⱼr A geodesic q(t) = (x(t), y(t)) obeys q̈ᵏ + Γᵏᵢⱼ q̇ᶦq̇ʲ = 0 where Γᵏᵢⱼ are the Christoffel symbols of the surface metric. Launch many almost-parallel paths. At the start, they look like a clean beam until curvature takes over. Some paths focus, others separate and some slide around ridges. The surface is essentially the rulebook for motion. The shortest path is local. The geometry remembers everything. #DifferentialGeometry #Geodesics #RiemannianGeometry #MathematicalPhysics #Curvature #TensorCalculus #AppliedMathematics #Geometry #ScienceAnimation
0:33

Ever Heard of a Metric in Differential Geometry? How About a Geodesic? These two ideas are central to how Mathematics and Physics describe Spacetime. A metric is how a surface measures distance and angle. A geodesic is the path that counts as "straight" after that measuring rule is chosen. Once the metric changes from point to point, straightness is no longer something you judge by eye. On a curved surface, the shortest paths are geodesics. They are chosen by the metric. Take a surface written as r(x,y) = (x, y, f(x,y)) Its metric is gᵢⱼ = ∂ᵢr · ∂ⱼr A geodesic q(t) = (x(t), y(t)) obeys q̈ᵏ + Γᵏᵢⱼ q̇ᶦq̇ʲ = 0 where Γᵏᵢⱼ are the Christoffel symbols of the surface metric. Launch many almost-parallel paths. At the start, they look like a clean beam until curvature takes over. Some paths focus, others separate and some slide around ridges. The surface is essentially the rulebook for motion. The shortest path is local. The geometry remembers everything. #DifferentialGeometry #Geodesics #RiemannianGeometry #MathematicalPhysics #Curvature #TensorCalculus #AppliedMathematics #Geometry #ScienceAnimation

Mathelirium

22,170 Aufrufe • vor 9 Tagen

I asked Luke Kornet about his memories from stretches where he was a DNP for many games in a row, and what his message is to the young guys who find themselves in that position now: “It's just a lot of peace to come to life of accepting the position that you're in and then just doing the best with that, versus trying to fight it and want to be in a different one.” “You can spend a lot of time worrying about things that you can't really control. But when you just take the opportunity that you have and do the best with that, you can have days like today where a lot of guys came in and were able to contribute. You can have peace about knowing that you did your job well, just as you did the day before, the day after. It might just look a different way every time. I feel like that's what I've been taught through my life. That's the only thing attitude that you can have that you can actually do something with – versus trying to control things that are outside of what you can control.”
2:15

I asked Luke Kornet about his memories from stretches where he was a DNP for many games in a row, and what his message is to the young guys who find themselves in that position now: “It's just a lot of peace to come to life of accepting the position that you're in and then just doing the best with that, versus trying to fight it and want to be in a different one.” “You can spend a lot of time worrying about things that you can't really control. But when you just take the opportunity that you have and do the best with that, you can have days like today where a lot of guys came in and were able to contribute. You can have peace about knowing that you did your job well, just as you did the day before, the day after. It might just look a different way every time. I feel like that's what I've been taught through my life. That's the only thing attitude that you can have that you can actually do something with – versus trying to control things that are outside of what you can control.”

Noa Dalzell 🏀

48,992 Aufrufe • vor 1 Jahr

I Built a 37.0 Profit Factor Bot by Cracking Every TradingView Source Code tradingview is a gold mine hiding in plain sight and i just found the master key to unlock every single secret hidden within its community scripts. most traders spend their entire lives staring at candles and hoping for a miracle while the actual alpha is buried in the open source code that nobody bothers to look at. i used to be that guy who sat there getting liquidated at three in the morning because i thought i could outplay the market with my gut feeling and some drawings on a screen. it turns out that the game is completely rigged against you if you are trading manually but there is a specific way to flip the script. i am going to show you how to stop guessing and start knowing exactly what works across every possible market condition before you ever risk a single dollar. i spent years losing money and thousands on developers because i thought i was not smart enough to code the systems myself but i was wrong. the first step to cracking the market is realizing that every indicator on the super charts has a source code section that is completely open to the public. you can literally scroll through the community scripts and pull the exact logic for thousands of different strategies that people claim are the holy grail of trading. but the secret is not just having the code because most of these indicators are actually garbage that will blow your account up in a week. this is where the real loop opens because you need a way to test these ideas across twenty five different data sets in seconds rather than months. i use a custom setup with ai agents specifically a sub agent i call the backtest architect to handle the heavy lifting of turning pine script into python code. the goal is to create a factory where you can feed in a raw indicator and get back a full report on its expectancy and profit factor without lifting a finger. most people find one strategy and marry it for life but a real data dog knows that you have to iterate to success or you will get left behind. i am running eighty one different backtests right now because i know that ninety percent of what i find will be trash but that remaining ten percent is where the wealth is made. the backtest architect knows exactly how to structure the folders and data paths so that we are testing everything from the base indicator to complex versions with filters. you might think that popular tools like fibonacci or order blocks are the way to go because everyone on social media talks about them like they are law. but when i actually ran the numbers through the machine the results were embarrassing and most of those strategies just resulted in negative expectancy. it is a dangerous trap to follow the crowd into a trade just because some guru said a certain level was important when the data shows it is a coin flip at best. the dynamic swing indicator was one of the few that actually held its weight during the recent massive testing sessions we ran. it was pulling in profit factors of over thirty seven with annualized returns that look too good to be true until you see the trade list. we combined it with filters like the adx and the money flow index to see if we could refine the signals and the results were absolutely staggering. when you have a system that can run through forty data sets while you are drinking tea you realize that manual trading is a form of self harm. i realized this after spending hundreds of thousands on apps and devs only to find out that i could just learn to build these bots myself live on the internet. the speed of iteration is the only thing that matters in this game because the faster you can fail the faster you can find the one strategy that actually prints. one of the biggest hurdles i faced was thinking that i needed to be a math genius or a senior engineer to automate my trading systems. the truth is that code is the great equalizer because it allows a regular person to compete with massive hedge funds by using the same logic and speed. i decided to learn everything in public because i wanted people to see the process of losing money with liquidations and then finally finding a path to automation. the reality of the market is that it moves in cycles and what worked yesterday will almost certainly fail tomorrow unless you are constantly testing. that is why i built the agents to automatically look through the results folder and rank the top performers based on a composite score. it takes all the emotion out of the process because i am no longer looking for a reason to enter a trade i am just looking at a csv file that tells me the truth. if you are still drawing lines on a chart and hoping for the best you are basically playing a game of chance against a high speed casino. the transition from a manual trader to a systems builder is the single most important pivot you will ever make in your life. it is not about being right or wrong it is about having a positive expectancy that has been proven across thousands of trades and multiple years of history. i had to fix a few errors in the short selling logic where the agents were getting confused between maximum and minimum values for take profit levels. these tiny bugs are the difference between a winning system and a blown account so you have to be willing to dive into the code and refine the machine. but once the system is tuned and the sub agents are running it becomes a beautiful workflow that functions entirely without your input. we are currently moving through the editors picks and the trending indicators one by one because i want to have a database of every single strategy on the platform. being a data dog means you never stop searching for that edge and you never settle for a strategy that just looks okay on a single chart. you have to demand excellence from your code because the market will not give you a single inch of mercy if you are lazy with your research. the ultimate goal is to have fully automated systems trading for you so you can focus on scaling rather than staring at a screen for ten hours a day. i am already up to over eighty backtests in this single session and i plan on hitting hundreds more by the end of the week. once you realize that you can crack the code of any indicator you see on the internet you will never look at a chart the same way again. this is the power of using agents to bridge the gap between a raw idea and a finished trading bot that actually works in the real world. i am done with getting liquidated and i am done with the stress of over trading because the code handles everything with cold precision. the path to success is paved with data and if you are not willing to automate your process you are just waiting for your next liquidation to happen
49:49

I Built a 37.0 Profit Factor Bot by Cracking Every TradingView Source Code tradingview is a gold mine hiding in plain sight and i just found the master key to unlock every single secret hidden within its community scripts. most traders spend their entire lives staring at candles and hoping for a miracle while the actual alpha is buried in the open source code that nobody bothers to look at. i used to be that guy who sat there getting liquidated at three in the morning because i thought i could outplay the market with my gut feeling and some drawings on a screen. it turns out that the game is completely rigged against you if you are trading manually but there is a specific way to flip the script. i am going to show you how to stop guessing and start knowing exactly what works across every possible market condition before you ever risk a single dollar. i spent years losing money and thousands on developers because i thought i was not smart enough to code the systems myself but i was wrong. the first step to cracking the market is realizing that every indicator on the super charts has a source code section that is completely open to the public. you can literally scroll through the community scripts and pull the exact logic for thousands of different strategies that people claim are the holy grail of trading. but the secret is not just having the code because most of these indicators are actually garbage that will blow your account up in a week. this is where the real loop opens because you need a way to test these ideas across twenty five different data sets in seconds rather than months. i use a custom setup with ai agents specifically a sub agent i call the backtest architect to handle the heavy lifting of turning pine script into python code. the goal is to create a factory where you can feed in a raw indicator and get back a full report on its expectancy and profit factor without lifting a finger. most people find one strategy and marry it for life but a real data dog knows that you have to iterate to success or you will get left behind. i am running eighty one different backtests right now because i know that ninety percent of what i find will be trash but that remaining ten percent is where the wealth is made. the backtest architect knows exactly how to structure the folders and data paths so that we are testing everything from the base indicator to complex versions with filters. you might think that popular tools like fibonacci or order blocks are the way to go because everyone on social media talks about them like they are law. but when i actually ran the numbers through the machine the results were embarrassing and most of those strategies just resulted in negative expectancy. it is a dangerous trap to follow the crowd into a trade just because some guru said a certain level was important when the data shows it is a coin flip at best. the dynamic swing indicator was one of the few that actually held its weight during the recent massive testing sessions we ran. it was pulling in profit factors of over thirty seven with annualized returns that look too good to be true until you see the trade list. we combined it with filters like the adx and the money flow index to see if we could refine the signals and the results were absolutely staggering. when you have a system that can run through forty data sets while you are drinking tea you realize that manual trading is a form of self harm. i realized this after spending hundreds of thousands on apps and devs only to find out that i could just learn to build these bots myself live on the internet. the speed of iteration is the only thing that matters in this game because the faster you can fail the faster you can find the one strategy that actually prints. one of the biggest hurdles i faced was thinking that i needed to be a math genius or a senior engineer to automate my trading systems. the truth is that code is the great equalizer because it allows a regular person to compete with massive hedge funds by using the same logic and speed. i decided to learn everything in public because i wanted people to see the process of losing money with liquidations and then finally finding a path to automation. the reality of the market is that it moves in cycles and what worked yesterday will almost certainly fail tomorrow unless you are constantly testing. that is why i built the agents to automatically look through the results folder and rank the top performers based on a composite score. it takes all the emotion out of the process because i am no longer looking for a reason to enter a trade i am just looking at a csv file that tells me the truth. if you are still drawing lines on a chart and hoping for the best you are basically playing a game of chance against a high speed casino. the transition from a manual trader to a systems builder is the single most important pivot you will ever make in your life. it is not about being right or wrong it is about having a positive expectancy that has been proven across thousands of trades and multiple years of history. i had to fix a few errors in the short selling logic where the agents were getting confused between maximum and minimum values for take profit levels. these tiny bugs are the difference between a winning system and a blown account so you have to be willing to dive into the code and refine the machine. but once the system is tuned and the sub agents are running it becomes a beautiful workflow that functions entirely without your input. we are currently moving through the editors picks and the trending indicators one by one because i want to have a database of every single strategy on the platform. being a data dog means you never stop searching for that edge and you never settle for a strategy that just looks okay on a single chart. you have to demand excellence from your code because the market will not give you a single inch of mercy if you are lazy with your research. the ultimate goal is to have fully automated systems trading for you so you can focus on scaling rather than staring at a screen for ten hours a day. i am already up to over eighty backtests in this single session and i plan on hitting hundreds more by the end of the week. once you realize that you can crack the code of any indicator you see on the internet you will never look at a chart the same way again. this is the power of using agents to bridge the gap between a raw idea and a finished trading bot that actually works in the real world. i am done with getting liquidated and i am done with the stress of over trading because the code handles everything with cold precision. the path to success is paved with data and if you are not willing to automate your process you are just waiting for your next liquidation to happen

Moon Dev

25,967 Aufrufe • vor 3 Monaten