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A neural network can begin as a flat sheet and learn the shape of hidden data A self-organizing map turns learning into geometry. Each data point pulls one winning neuron toward it, but nearby neurons move too, and so the whole lattice bends without losing its neighborhood structure. The... show more

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A Neural Sheet Discovers the Shape of Data A self-organizing map (SOM) starts as a flat sheet of neurons with no knowledge of the data around it. For every input sample, it finds the best-matching neuron, c = argminᵢ ‖x - wᵢ‖, and updates that neuron together with its neighbours, wᵢ ← wᵢ + η(t)hᶜᵢ(t)(x - wᵢ). As the neighbourhood radius shrinks during training, the sheet gradually bends and folds onto the hidden geometry of the dataset while preserving its neighbourhood structure. Developed by Finnish computer scientist Teuvo Kohonen, the Self-Organizing Map remains one of the most elegant examples of competitive learning, where order emerges from thousands of simple local updates. #MachineLearning #ArtificialIntelligence #SelfOrganizingMap #NeuralNetworks #DataVisualization #Mathematics #ComputerScience #Finland #TeuvoKohonen

A Neural Sheet Discovers the Shape of Data A self-organizing map (SOM) starts as a flat sheet of neurons with no knowledge of the data around it. For every input sample, it finds the best-matching neuron, c = argminᵢ ‖x - wᵢ‖, and updates that neuron together with its neighbours, wᵢ ← wᵢ + η(t)hᶜᵢ(t)(x - wᵢ). As the neighbourhood radius shrinks during training, the sheet gradually bends and folds onto the hidden geometry of the dataset while preserving its neighbourhood structure. Developed by Finnish computer scientist Teuvo Kohonen, the Self-Organizing Map remains one of the most elegant examples of competitive learning, where order emerges from thousands of simple local updates. #MachineLearning #ArtificialIntelligence #SelfOrganizingMap #NeuralNetworks #DataVisualization #Mathematics #ComputerScience #Finland #TeuvoKohonen

Mathelirium

175,749 görüntüleme • 1 ay önce

We often hear that machine learning models "learn patterns in data". But what does that actually look like in geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a self-organizing map...a simple unsupervised neural model...learn the shape of a two-dimensional dataset arranged in a spiral arm. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud...it knows nothing about the structure underneath. Learning is a repeated game: pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

We often hear that machine learning models "learn patterns in data". But what does that actually look like in geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a self-organizing map...a simple unsupervised neural model...learn the shape of a two-dimensional dataset arranged in a spiral arm. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud...it knows nothing about the structure underneath. Learning is a repeated game: pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

Mathelirium

139,682 görüntüleme • 8 ay önce

We often hear that Machine Learning models learn patterns in data. But what does that actually look like in Geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a Self-Organizing Map (SOM), a simple unsupervised neural model, learn the shape of a 3D datasets l, one static and the other dynamic. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud. It knows nothing about the structure underneath. Learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. Python code is available for Subscribers. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

We often hear that Machine Learning models learn patterns in data. But what does that actually look like in Geometry? If you dropped a little elastic mesh into a cloud of points and let it learn, how would it fold itself to match the shape of the data? In this scene we watch a Self-Organizing Map (SOM), a simple unsupervised neural model, learn the shape of a 3D datasets l, one static and the other dynamic. On top of this, we lay down a square grid of neurons whose weights live in the same plane. At the start, this grid is just a flat net floating across the cloud. It knows nothing about the structure underneath. Learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, and then nudge that neuron and its neighbours toward the point. Do this again and again, while slowly shrinking how far the neighbourhood influence spreads. Python code is available for Subscribers. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML

Mathelirium

76,194 görüntüleme • 5 ay önce

A Self-Organizing Map Learns The Shape of Data

A Self-Organizing Map Learns The Shape of Data

Mathelirium

44,616 görüntüleme • 6 ay önce

We often hear that Machine Learning models learn patterns in data. But what does that look like in geometry? Picture dropping an elastic mesh into a cloud of points and letting it adapt. How would it bend, stretch, and settle so it matches the shape hiding in the data? In this scene we watch a self-organizing map(SOM)...a classic unsupervised neural model...learn a 2D dataset arranged like a spiral arm. Over the points we place a square grid of neurons whose weights live in the same plane. At the start it’s just a flat net drifting across the cloud. No clue what the structure is. Fir the SOM, learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, then nudge that neuron and its neighbours toward the point. Repeat, repeat, repeat...while gradually shrinking how wide the neighbourhood influence spreads. The result is satisfying...the grid stops being a grid and turns into a coordinate sheet wrapped onto the spiral. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML #SelfOrganizingMap #Topology

We often hear that Machine Learning models learn patterns in data. But what does that look like in geometry? Picture dropping an elastic mesh into a cloud of points and letting it adapt. How would it bend, stretch, and settle so it matches the shape hiding in the data? In this scene we watch a self-organizing map(SOM)...a classic unsupervised neural model...learn a 2D dataset arranged like a spiral arm. Over the points we place a square grid of neurons whose weights live in the same plane. At the start it’s just a flat net drifting across the cloud. No clue what the structure is. Fir the SOM, learning is a repeated game: Pick a random data point, find the neuron whose weight is closest, then nudge that neuron and its neighbours toward the point. Repeat, repeat, repeat...while gradually shrinking how wide the neighbourhood influence spreads. The result is satisfying...the grid stops being a grid and turns into a coordinate sheet wrapped onto the spiral. #MachineLearning #ManifoldLearning #UnsupervisedLearning #NeuralMaps #GeometricML #SelfOrganizingMap #Topology

Mathelirium

23,021 görüntüleme • 6 ay önce

A Neural Network Can Grow New Neurons Where It Is Confused? In 1994, Bernd Fritzke published A Growing Neural Gas Network Learns Topologies. He introduced a network that starts small, follows incoming data, and inserts new neurons where its error is highest. In the animation, the fog is the drifting data. The glowing nodes are neurons. The fibers are learned connections. The network grows into a living skeleton of the manifold.

A Neural Network Can Grow New Neurons Where It Is Confused? In 1994, Bernd Fritzke published A Growing Neural Gas Network Learns Topologies. He introduced a network that starts small, follows incoming data, and inserts new neurons where its error is highest. In the animation, the fog is the drifting data. The glowing nodes are neurons. The fibers are learned connections. The network grows into a living skeleton of the manifold.

Mathelirium

38,306 görüntüleme • 3 ay önce

The Non-Traversable Bridge Einstein Found Inside Gravity Einstein’s equations not only describe gravity as a force but as the shape of Spacetime as well. In 1935, Einstein and Rosen found that the Schwarzschild solution could be written in a way that reveals something astonishing: Two separate exterior regions of Spacetime joined by a bridge-like throat Rᵤᵥ = 0 This became known as the Einstein-Rosen Bridge. It is often called a Wormhole, but the classical Einstein-Rosen bridge is not a science-fiction tunnel. In the standard Schwarzschild geometry, it is non-traversable. The throat does not remain open long enough for ordinary travel from one region to the other. Its real power is mathematical. It shows that General Relativity is not merely about objects moving through space. The theory allows Spacetime to have global structure such as different regions, horizons, extensions, and hidden connections that are not visible from one local patch alone. The Einstein-Rosen bridge is one of the earliest glimpses of that deeper idea. Gravity is geometry. And sometimes, geometry has two worlds joined by a throat. #Astrophysics #GeneralRelativity #Einstein #BlackHole #Spacetime #Cosmology #Gravity #Wormhole #Mathematics #Physics

The Non-Traversable Bridge Einstein Found Inside Gravity Einstein’s equations not only describe gravity as a force but as the shape of Spacetime as well. In 1935, Einstein and Rosen found that the Schwarzschild solution could be written in a way that reveals something astonishing: Two separate exterior regions of Spacetime joined by a bridge-like throat Rᵤᵥ = 0 This became known as the Einstein-Rosen Bridge. It is often called a Wormhole, but the classical Einstein-Rosen bridge is not a science-fiction tunnel. In the standard Schwarzschild geometry, it is non-traversable. The throat does not remain open long enough for ordinary travel from one region to the other. Its real power is mathematical. It shows that General Relativity is not merely about objects moving through space. The theory allows Spacetime to have global structure such as different regions, horizons, extensions, and hidden connections that are not visible from one local patch alone. The Einstein-Rosen bridge is one of the earliest glimpses of that deeper idea. Gravity is geometry. And sometimes, geometry has two worlds joined by a throat. #Astrophysics #GeneralRelativity #Einstein #BlackHole #Spacetime #Cosmology #Gravity #Wormhole #Mathematics #Physics

Mathelirium

91,291 görüntüleme • 1 ay önce

A hydraulic press can take a totally flat metal sheet and form it into a precise final shape in seconds. It looks simple, but that kind of accuracy is the difference between a hobby shop and real manufacturing.

A hydraulic press can take a totally flat metal sheet and form it into a precise final shape in seconds. It looks simple, but that kind of accuracy is the difference between a hobby shop and real manufacturing.

Tansu Yegen

21,357 görüntüleme • 3 ay önce

Lockheed Martin can grow the skin of an aircraft that contains sensors & stores energy, without any wires. Data flows through the structure. They have new materials that change shape on command, like a muscular material. They can make autonomous aircraft that change shape, and make swarms of small ones that learn and interact with a larger vehicle. #ufoX #uapX #ufotwitter #uaptwitter

Lockheed Martin can grow the skin of an aircraft that contains sensors & stores energy, without any wires. Data flows through the structure. They have new materials that change shape on command, like a muscular material. They can make autonomous aircraft that change shape, and make swarms of small ones that learn and interact with a larger vehicle. #ufoX #uapX #ufotwitter #uaptwitter

Chris

567,517 görüntüleme • 2 yıl önce

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.

Mathelirium

54,730 görüntüleme • 4 ay önce

What Shape Does Chaos Take Before It Looks Random? Take one particle moving in the Hénon-Heiles Hamiltonian H = ½(pₓ² + pᵧ²) + ½(x² + y²) + λ(x²y - ⅓y³) Its motion is forced by Hamilton’s equations ẋ = pₓ ẏ = pᵧ ṗₓ = -x - 2λxy ṗᵧ = -y - λ(x² - y²) The saddle energy is Eₛ = 1/(6λ²) Below that energy, the particle remains trapped. But a tiny packet of nearby initial conditions stretches into filaments, folds through the potential, and begins to look almost random. However, it is not random. Hamiltonian flow preserves the hidden phase-space structure. What looks like chaos in ordinary space is still a deterministic loom of energy conserved, equations obeyed, geometry stretched without being erased. #HamiltonianMechanics #ChaosTheory #PhaseSpace #DynamicalSystems #MathematicalPhysics #NonlinearDynamics #HenonHeiles #SymplecticGeometry #ScienceAnimation

What Shape Does Chaos Take Before It Looks Random? Take one particle moving in the Hénon-Heiles Hamiltonian H = ½(pₓ² + pᵧ²) + ½(x² + y²) + λ(x²y - ⅓y³) Its motion is forced by Hamilton’s equations ẋ = pₓ ẏ = pᵧ ṗₓ = -x - 2λxy ṗᵧ = -y - λ(x² - y²) The saddle energy is Eₛ = 1/(6λ²) Below that energy, the particle remains trapped. But a tiny packet of nearby initial conditions stretches into filaments, folds through the potential, and begins to look almost random. However, it is not random. Hamiltonian flow preserves the hidden phase-space structure. What looks like chaos in ordinary space is still a deterministic loom of energy conserved, equations obeyed, geometry stretched without being erased. #HamiltonianMechanics #ChaosTheory #PhaseSpace #DynamicalSystems #MathematicalPhysics #NonlinearDynamics #HenonHeiles #SymplecticGeometry #ScienceAnimation

Mathelirium

26,893 görüntüleme • 1 ay önce

A Physics-Informed Neural Network (PINN) is trying to learn a solution to the Klein-Gordon PDE PINNs are neural nets trained to satisfy a partial differential equation. They use a simple trick of baking 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 such as meshes, stencils, and stability tuning. 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.

A Physics-Informed Neural Network (PINN) is trying to learn a solution to the Klein-Gordon PDE PINNs are neural nets trained to satisfy a partial differential equation. They use a simple trick of baking 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 such as meshes, stencils, and stability tuning. 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.

Mathelirium

56,467 görüntüleme • 4 ay önce

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

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 görüntüleme • 2 ay önce

Zandile Dabula (Gudluza) your remarks show a serious misunderstanding of what it means to protect the sovereignty of a nation. Sovereignty is not declared through words, it is built, defended, and sustained through the understanding of the frontline state. The frontline state is not just a point on a map. It represents a nation’s strength, its readiness to defend its people, and its commitment to guard political independence. It is where sovereignty is proven, not proclaimed. To speak of sovereignty while ignoring the role of the frontline state is to miss the essence of leadership and defense. Sovereignty without awareness of the frontline is a hollow idea, one without grounding or protection. True sovereignty exists when a nation knows its line of defense and is prepared to stand firm behind it. Anything less is a performance, not protection. A true leader must carry the weight of their history with pride, understanding that the struggles, sacrifices, and victories of the past shape the decisions of the present.

Zandile Dabula (Gudluza) your remarks show a serious misunderstanding of what it means to protect the sovereignty of a nation. Sovereignty is not declared through words, it is built, defended, and sustained through the understanding of the frontline state. The frontline state is not just a point on a map. It represents a nation’s strength, its readiness to defend its people, and its commitment to guard political independence. It is where sovereignty is proven, not proclaimed. To speak of sovereignty while ignoring the role of the frontline state is to miss the essence of leadership and defense. Sovereignty without awareness of the frontline is a hollow idea, one without grounding or protection. True sovereignty exists when a nation knows its line of defense and is prepared to stand firm behind it. Anything less is a performance, not protection. A true leader must carry the weight of their history with pride, understanding that the struggles, sacrifices, and victories of the past shape the decisions of the present.

𝔹𝕝𝕖𝕤𝕤𝕚𝕟𝕘 𝕋𝕤𝕙𝕖𝕘𝕠 𝕆𝕣𝕒𝕥𝕚𝕝𝕨𝕖

31,407 görüntüleme • 9 ay önce

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

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 görüntüleme • 7 ay önce

Can Spacetime Move Faster Than Light? In 1994, Miguel Alcubierre proposed a strange solution to Einstein’s field equations: A Spacetime geometry where a compact region could be transported without the object inside locally breaking the speed of light. The idea is not that a ship accelerates through space faster than light. The local interior remains comparatively calm. The violent part is the surrounding geometry. Space contracts in front of the bubble, expands behind it, and forms a moving transport shell that carries the interior region forward. A simplified form of the Alcubierre metric is ds² = −dt² + [dx − vₛ f(rₛ)dt]² + dy² + dz² where f(rₛ) shapes the warp bubble and vₛ describes the motion of its centre. The physics is still highly speculative, especially because the original construction requires exotic stress-energy, but the mathematical idea is powerful. #GeneralRelativity #AlcubierreDrive #WarpDrive #Spacetime #EinsteinEquations #Relativity #Physics #Astrophysics #Cosmology

Can Spacetime Move Faster Than Light? In 1994, Miguel Alcubierre proposed a strange solution to Einstein’s field equations: A Spacetime geometry where a compact region could be transported without the object inside locally breaking the speed of light. The idea is not that a ship accelerates through space faster than light. The local interior remains comparatively calm. The violent part is the surrounding geometry. Space contracts in front of the bubble, expands behind it, and forms a moving transport shell that carries the interior region forward. A simplified form of the Alcubierre metric is ds² = −dt² + [dx − vₛ f(rₛ)dt]² + dy² + dz² where f(rₛ) shapes the warp bubble and vₛ describes the motion of its centre. The physics is still highly speculative, especially because the original construction requires exotic stress-energy, but the mathematical idea is powerful. #GeneralRelativity #AlcubierreDrive #WarpDrive #Spacetime #EinsteinEquations #Relativity #Physics #Astrophysics #Cosmology

Mathelirium

37,986 görüntüleme • 1 ay önce

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

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

44,806 görüntüleme • 6 ay önce

Swift is a network of thousands of institutions worldwide, making it possible to move value seamlessly around the world. We’re now collaborating across that network to raise the bar for cross-border payments on current and future rails. Discover what’s next and how we’re helping shape it: #CrossBorderPayments #SwiftBlockchainLedger #SwiftScheme

Swift is a network of thousands of institutions worldwide, making it possible to move value seamlessly around the world. We’re now collaborating across that network to raise the bar for cross-border payments on current and future rails. Discover what’s next and how we’re helping shape it: #CrossBorderPayments #SwiftBlockchainLedger #SwiftScheme

Swift

136,741 görüntüleme • 1 ay önce

Geometry 🤝 Topology What you're seeing here is called Hopf Fibration. In pure-math terms, S³ (the unit sphere in 4D) can be decomposed into circles called fibers. The Hopf map sends each of those circles to a single point on S². So you don’t fill S³ with random loops, you fill it with a perfectly organized family of circles, one circle sitting above each point of S². For the animation, we sample thousands of fibers on S³, apply a smooth SO(4) rotation (a real rotation in 4D, not a 3D cheat), then stereographically project into R³ so we can actually draw it. Pure mathematics is beautiful alz #HopfFibration #Topology #Geometry #StereographicProjection #PureMathematics

Geometry 🤝 Topology What you're seeing here is called Hopf Fibration. In pure-math terms, S³ (the unit sphere in 4D) can be decomposed into circles called fibers. The Hopf map sends each of those circles to a single point on S². So you don’t fill S³ with random loops, you fill it with a perfectly organized family of circles, one circle sitting above each point of S². For the animation, we sample thousands of fibers on S³, apply a smooth SO(4) rotation (a real rotation in 4D, not a 3D cheat), then stereographically project into R³ so we can actually draw it. Pure mathematics is beautiful alz #HopfFibration #Topology #Geometry #StereographicProjection #PureMathematics

Mathelirium

15,727 görüntüleme • 6 ay önce

This is one of the more unusual flying robots I have seen. Researchers at the University of Tokyo built DRAGON, a transformable aerial robot that can change its shape while flying. Instead of a rigid frame, it is made of four linked segments. Each segment has two ducted rotors mounted on gimbals, so every link can redirect thrust independently. • The robot controls its full pose in SE(3) while reshaping mid-air • It can form different geometries to stabilize itself or interact with objects • The navigation stack computes the most efficient shape for the current task • It lifts over 3 kg, which is unusually high for this class of system This effectively turns a drone into a flying manipulator. Not a gripper bolted on top, but the airframe itself becomes the manipulator. The authors also mention an interesting idea for range extension: letting the robot walk on the ground when flying is not required. Paper ——- Weekly robotics and AI insights. Subscribe free:

This is one of the more unusual flying robots I have seen. Researchers at the University of Tokyo built DRAGON, a transformable aerial robot that can change its shape while flying. Instead of a rigid frame, it is made of four linked segments. Each segment has two ducted rotors mounted on gimbals, so every link can redirect thrust independently. • The robot controls its full pose in SE(3) while reshaping mid-air • It can form different geometries to stabilize itself or interact with objects • The navigation stack computes the most efficient shape for the current task • It lifts over 3 kg, which is unusually high for this class of system This effectively turns a drone into a flying manipulator. Not a gripper bolted on top, but the airframe itself becomes the manipulator. The authors also mention an interesting idea for range extension: letting the robot walk on the ground when flying is not required. Paper ——- Weekly robotics and AI insights. Subscribe free:

Ilir Aliu

39,213 görüntüleme • 3 gün önce