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#

I found that Adam variants perform the best when beta_1=beta_2 in my work on optimizers for physics-informed neural networks (PINNs). And I'm glad to see that people have also found this secret sauce for Adam, on a wider domain. Setting beta_1 equal to beta_2 aligns the momentum tracking history length, stabilizing the scale. (especially when the loss surface is changing a lot) It also puts less penalization on the gradient norm, making the gradients larger, thus converging faster. ⏩Table 4 in this PINN literature: ⏩Newer literatures:

🚀Thrilled to present PINNacle at #NeurIPS2024#! 🌍 📍Poster Location: West Ballroom, #5307#, 11am 🔑Highlights: - A comprehensive benchmark for Physics-Informed Neural Networks (PINNs). - Evaluates 10+ SOTA PINN methods on 20+ challenging PDE problems and more! Let's connect! #AI#

PINNACLE FINANCIAL PARTNERS $PNFP Q3’26 EARNINGS HIGHLIGHTS 🔹 Adj. Revenue: $1.32B (Est. $1.28B) 🟢 🔹 GAAP Revenue: $1.31B 🔹 Adj. EPS: $2.69 (Est. $2.63) 🟢; +19% YoY 🔹 GAAP EPS: $2.40; +10% YoY 🔹 Net Interest Income: $986M; +148% YoY Other Q3 Metrics: 🔹 Net Income: $379M; +119% YoY 🔹 Net Interest Margin: 3.40% 🔹 Non-Interest Revenue: $326M; +122% YoY 🔹 Adj. Non-Interest Revenue: $323M 🔹 Non-Interest Expense: $749M; +147% YoY 🔹 Adj. Non-Interest Expense: $690M 🔹 Efficiency Ratio-TE: 56.6% 🔹 Adj. Tangible Efficiency Ratio: 48.9% 🔹 Provision for Credit Losses: $86M 🔹 Net Charge-Offs: $50M 🔹 Net Charge-Off Ratio: 0.23% 🔹 NPA Ratio: 0.50% 🔹 Period-End Loans: $89.8B 🔹 Period-End Deposits: $103.0B; +125% YoY 🔹 CET1 Ratio: 10.01% 🔹 Book Value/Share: $91.52 🔹 Tangible Book Value/Share: $61.86 Financials: 🔹 Total Assets: $130.69B 🔹 Net Income Available to Common: $364M 🔹 Adjusted Net Income to Common: $409M Capital Return: 🔹 Dividend: $0.50/share Key Updates: 🔹 Adj. Net Income to Common: $409M vs $364M GAAP 🔹 Merger-Related Expense: $56M 🔹 YoY significantly impacted by the merger 🔹 Added 96 Experienced Revenue Producers Comments: 🔸 “The merger changed our size, not how we perform.”

Pennsylvania Governor Josh Shapiro launched a federal political action committee to support Democrats running in the state’s most competitive US House races next month, stoking further questions about his 2028 presidential ambitions

Pennsylvania reports its 6th measles death, in a Franklin County resident The 39-year-old woman had underlying health conditions, including a previous measles infection. Read more: 📷: CDC/ Dr. Erskine Palmer