正在加载视频...
视频加载失败
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... show more
0 条评论
暂无评论
原始帖子的评论将显示在这里
![When the Lorenz Attractor Becomes a Cosmos Many of us have seen plots of the famous Lorenz system that famous butterfly-shaped chaos: ẋ = σ(y - x) ẏ = x(ρ - z) - y ż = xy - βz But here, we went one layer deeper. At every point (x, y, z), we computed the local Jacobian matrix, J(x, y, z) = [ -σ σ 0 ] [ ρ - z -1 - x ] [ y x - β ] Then we found its eigenvalues λ and plotted them in the complex plane. Re(λ) tells us whether nearby motion is locally growing or decaying. Im(λ) tells us whether that local motion is rotating. That gives us the spectral flow of the Lorenz system and not the attractor itself, but the hidden local linear geometry driving its instability. When I saw that flow, it looked strangely cosmic with two mirrored swarms curling like particles trapped in a mathematical whirlpool. So I mapped that spectral flow onto our living fabric, not as Einstein Spacetime this time, but as a creative geometric stage for chaos itself. #LorenzSystem #ChaosTheory #DynamicalSystems #Eigenvalues #SpectralFlow #ComplexPlane #NonlinearDynamics #Mathematics #Physics](https://image.24vids.com/tw-2072014826516476128/amplify_video_thumb/2072014299434762240/img/g6jhjcX-oWz0uxxz.jpg)
![Lecture 2 of our Physics-Informed Neural Networks mini-series. In Lecture 1 we made the idea visible...a neural network isn’t predicting a PDE solution, it is the candidate function uᵩ(x,t), and the PDE residual rᵩ(x,t) is the leash that keeps it honest. Now the natural question follows: How can a neural network be punished for breaking a PDE when nobody ever handed it the true solution, and the equation itself contains derivatives like uᵩₜₜ and uᵩₓₓ? Here’s the satisfying answer: A PINN doesn’t need the true answer to be corrected. It only needs a way to measure how wrong it is according to the PDE! The network outputs uᵩ(x,t). A software called "autodiff" is used to compute the derivatives (uᵩₓ, uᵩₜ, uᵩₓₓ, …) exactly by applying the chain rule through the network. Those derivatives get dropped into the PDE to produce rᵩ(x,t). If rᵩ is big at some point, the loss spikes there, and gradient descent pushes the parameters so that rᵩ shrinks. The math breakdown We want a function u(x,t) that satisfies a PDE on a domain Ω. In this lecture we keep a concrete nonlinear example in mind, the damped sine-Gordon equation uₜₜ(x,t) + γ uₜ(x,t) − c² uₓₓ(x,t) + sin(u(x,t)) = 0. A PINN replaces the unknown function u with a neural network uᵩ(x,t), where ᵩ means all the network parameters (weights and biases). Now we build the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t) + sin(uᵩ(x,t)). If uᵩ were a true solution, rᵩ would be 0 everywhere. So we sample points (xⱼ,tⱼ) inside the domain. These are collocation points. At each one we evaluate rᵩ, and we define a physics loss L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|². This is the punishment mechanism. (Punish just means: if |rᵩ| is big, L_phys is big; training updates ᵩ to make L_phys smaller. Reward means the loss drops, so those parameter changes are kept.) The key question was where the derivatives come from. Since uᵩ is built out of differentiable operations, we can compute uᵩₜ(x,t), uᵩₜₜ(x,t), uᵩₓ(x,t), uᵩₓₓ(x,t), at any input (x,t) we choose. Imagine a simple differentiable model written as a sum of nonlinear features uᵩ(x,t) = Σₖ vₖ σ( wₖx x + wₖt t + bₖ ) + b₀. Then the derivatives are just chain rule uᵩₓ(x,t) = Σₖ vₖ σ′(·) wₖx uᵩₓₓ(x,t) = Σₖ vₖ σ″(·) (wₖx)² uᵩₜ(x,t) = Σₖ vₖ σ′(·) wₖt uᵩₜₜ(x,t) = Σₖ vₖ σ″(·) (wₖt)². So rᵩ(x,t) is an explicit computable number at every (x,t). For the damped sine-Gordon example, it’s the same story, just with one extra nonlinear term: rᵩ(x,t) = [uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t)] + sin(uᵩ(x,t)). A real PINN is a deeper composition of these same building blocks, but it’s still just a chain rule, and autodiff is the machinery that does that bookkeeping reliably for big graphs. Then we train by gradient descent on the total loss. Even if we use only physics for the moment, the update is conceptually just ᵩ ← ᵩ − η ∇ᵩ L_phys(ᵩ), with learning rate η. In practice we also include initial/boundary conditions or data, because PDEs aren’t uniquely determined without them L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ), where L_bc/ic(ᵩ) enforces things like uᵩ(x,0) ≈ u₀(x) and uᵩₜ(x,0) ≈ v₀(x), or boundary conditions at x = ±L. So Lecture 2’s punchline is simple: the PDE becomes a training signal. We keep differentiating uᵩ, measuring rᵩ, and updating ᵩ until the residual goes quiet across Ω. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #AutoDiff #Backpropagation #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics](https://image.24vids.com/tw-2009748938266685482/amplify_video_thumb/2009748619386101760/img/q0u_ZxmvYMtDl2eQ.jpg)
![CSS Trick 🧲 You can create magnetic links with the power of custom properties and some JavaScript 💪 a { translate: calc(clamp(-1, var(--x), 1) * var(--pad-x)) ...; transition: translate var(--s, 1s) var(--ease, var(--elastic)); } a:hover { --s: 0s; } The trick here is to pad out the list items wrapping your links and use that as a translation limit 🛑 Start by using some JavaScript to calculate a value between -1 and 1 for both the x/y axis on pointermove for each list item, not the link! 🔗 If your pointer was at the center of the item, you'd get [0,0]. If it was in the top right, you'd get [1,-1] ☝️ It's worth checking out the JavaScript snippet to see how the mapping function works. Essentially, you create a function that when given a value between two bounds, will give you a mapped value back 🤙 const mapX = mapRange( item.offsetWidth * -0.5, item.offsetWidth * 0.5, 1, -1 ) Then, on pointermove, you plug the pointer position in to get the value back out and pass that into your CSS const x = mapX(item.centerX - event.x) document.documentElement.style.setProperty(--x, x) When the pointer leaves the list item, you make sure to reset these values back to 0 ✨ Once CSS has your values, it's the trick of updating the translation of each part You know that in each axis, you only want to translate the link by the padding amount li a { translate: calc(clamp(-1, var(--x), 1) * var(--pad-x)) calc(clamp(-1, var(--y), 1) * var(--pad-y)); transition: translate var(--speed, 1s) var(--ease, var(--elastic)); } This will translate the link within the list item by the desired amount. The cool part here is that you can set an offset for the text inside the link and have that move at a different rate ⭐️ By only updating the --pad-x/y custom properties for the inside the link, you can control how much it moves nav a span { --pad-x: 0.25rem; --pad-y: 0.25rem; } And the last piece, how do you update the behavior for transition speeds? And so it springs back like that? Again, use custom properties ✨ a:hover { --s: 0s; } a { transition: translate var(--s, 1s) var(--ease, var(--elastic)); } By default, a link will use --elastic easing via linear() and have a transition-duration of 1s. When a link is hovered that speed becomes 0s because you want the link to magnetise to your pointer. How about that little gap between when your pointer enters the item but hasn't hovered the link? Set a different transition so it transitions to being hovered 🫶 nav li:hover a { --ease: ease-out; --speed: 0.1s; } That's kinda it! 🙌 Use JavaScript (~40 loc) to get the information and then let CSS do all the lifting for you 💪 Any questions or suggestions, let me know 🙏 If you want a walkthrough video, also let me know please 🙏 CodePen.IO link below 👇](https://image.24vids.com/tw-1745716136824672271/ext_tw_video_thumb/1745612660081881088/pu/img/SsGz-uqmfdr1KZ7H.jpg)
![Lecture 3 of our Quantum Mechanics series. Lecture 2 gave us the one clean privilege quantum theory offers: treat ψ(x,t) as the state and ρ(x,t) = |ψ(x,t)|² as probability, because Schrödinger evolution forces ρ to obey a continuity equation. Lecture 3 is what that continuity equation is really telling you. If ρ behaves like a fluid, then the only question that matters is: What is the velocity field? Write ψ(x,t) = r(x,t) exp(i θ(x,t)). The magnitude r sets how much probability is sitting there. The phase θ sets where it tries to go. When you unpack the current j = Im(ψ* ∇ψ), it collapses to j = (ρ/m) ∇θ, which means the flow lines you draw are literally contours of phase geometry. Then the constraint that makes the picture bite: ψ has to be single-valued, so θ can’t wind by an arbitrary amount. Around any closed loop the total phase change must be 2π n, with n an integer. That’s why vortices aren’t features you add...they’re defects the math permits, in quantized units. In the render you see both layers at once...the 3D surface shows |ψ| breathing while the phase skin slides, and the 2D panel exposes the engine...current lines steering around discrete vortex charges. The math breakdown We write the state as a complex field ψ(x,t) on the plane (x in R²). The Born rule defines the probability density ρ(x,t) = |ψ(x,t)|² Schrödinger evolution (ħ = 1 units) is i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Now derive conservation of probability. Start with ρ = ψ*ψ: ∂ρ/∂t = ψ* (∂ψ/∂t) + ψ (∂ψ*/∂t) Use Schrödinger and its complex conjugate: ∂ψ/∂t = (1/i) [ −(1/2m) ∇²ψ + Vψ ] ∂ψ*/∂t = (−1/i) [ −(1/2m) ∇²ψ* + Vψ* ] Substitute. The V terms cancel, and the remaining terms rearrange into the continuity equation ∂ρ/∂t + ∇·j = 0 with probability current j = (1/2mi) ( ψ* ∇ψ − ψ ∇ψ* ) = (1/m) Im(ψ* ∇ψ) So "probability density" really behaves like a conserved fluid density with flux j. Now expose the phase mechanism. Write ψ in polar form ψ(x,t) = r(x,t) exp(i θ(x,t)) Compute the gradient ∇ψ = exp(iθ) (∇r + i r ∇θ) Then ψ* ∇ψ = r (∇r + i r ∇θ) Taking the imaginary part gives Im(ψ* ∇ψ) = r² ∇θ = ρ ∇θ So the current becomes j = (ρ/m) ∇θ That’s the steering-wheel statement: Phase gradient sets the flow direction and speed (modulated by density and m). Finally, quantized vortices. Because ψ must be single-valued, going around any closed loop must return the same complex value. That forces the phase winding to be an integer multiple of 2π: ∮ ∇θ · dl = 2π n with n in Z n is the vortex charge. Vortex cores sit where ρ ≈ 0 (phase is undefined), and the current streamlines circulate around them. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #Vortices #TopologicalDefects #ComplexAnalysis #MathematicalPhysics #Mathematics #Physics](https://image.24vids.com/tw-2006665244211380338/amplify_video_thumb/2006664944415035392/img/qEDUF2G3erxPqtSk.jpg)
![Quantum Mechanics Series Lecture 4 Lecture 1 established that ρ(x,t) = |ψ(x,t)|² behaves like a conserved probability density. Lecture 2 showed what drives that flow. We also saw that writing ψ = r exp(iθ) makes the probability current proportional to the phase gradient, making it clear that phase geometry literally steers the motion. Lecture 3 then showed that the centroid of that flow can move almost classically when the packet is tight and the external potential is smooth. However, that raises yet another question. If the centroid can look classical, why does the full wave still spread, bend, split, and interfere in ways no classical particle cloud would? This is because the wave is not driven only by the external potential. It is also driven by its own curvature. Write ψ(x,t) = r(x,t) exp(iθ(x,t)) with ρ = r². Then Schrödinger’s equation gives two coupled real equations. One is the continuity equation you already know. The other looks like a Hamilton-Jacobi equation, but with one extra term: Q = −(1/2m) ∇²r / r This is the so-called Quantum Potential. It depends entirely on how the amplitude bends across space. So, the wave is being shaped not only by V(x,t), but also by the geometry of its own envelope. In the animation, the upper surface is still |ψ| and its skin is still colored by arg(ψ). The glowing threads still trace the probability current. But now a second membrane hangs underneath. That lower membrane encodes the quantum potential Q itself. The porcelain bead marks the quantum centroid. The amber bead follows a classical centroid under the same external V. When those paths separate, the lower membrane tells you why. The difference is not magic but the extra term classical mechanics does not have. The math breakdown: Start from Schrödinger evolution in units with ħ = 1: i ∂ψ/∂t = [ −(1/2m) ∇² + V(x,t) ] ψ Write the state in polar form: ψ = r exp(iθ) Then ρ = |ψ|² = r² From the imaginary part, you recover probability conservation: ∂ρ/∂t + ∇·j = 0 with j = (1/m) Im(ψ* ∇ψ) = (ρ/m) ∇θ So the local velocity field is v = j / ρ = ∇θ / m Now take the real part of Schrödinger’s equation. That gives ∂θ/∂t + |∇θ|² / (2m) + V + Q = 0 where Q = −(1/2m) ∇²r / r This is the classical Hamilton-Jacobi equation with one extra term. That extra term is what makes quantum motion locally different from classical motion. Take a gradient of that phase equation and use v = ∇θ / m. Then the flow obeys an Euler-like equation: ∂v/∂t + (v·∇)v = −(1/m) ∇(V + Q) In other words, there are really two forces in the problem. One comes from the external potential V. The other comes from the wave’s own curvature through Q. That is why Ehrenfest is only approximate. The centroid can still satisfy d⟨x⟩/dt = ⟨p⟩/m d⟨p⟩/dt = −⟨∇V⟩ but the internal shape of the packet evolves under the combined influence of V and Q. When the packet stays broad and smooth, Q is gentle and the motion looks more classical. When the packet develops sharp curvature or interference structure, Q becomes strong and the classical picture breaks down. That is what this scene is designed to show live. #QuantumMechanics #Wavefunction #SchrodingerEquation #BornRule #ProbabilityCurrent #ContinuityEquation #Phase #EhrenfestTheorem #QuantumPotential #Madelung #HamiltonJacobi #MathematicalPhysics #Mathematics #Physics](https://image.24vids.com/tw-2044118124148973661/amplify_video_thumb/2044117901251072000/img/IXBMEKzfYfpY_xGi.jpg)