PDE
Symplectic Geometry and Structure-Preserving Neural Networks
Learn physics-informed neural networks that preserve energy and symplectic structure. Covers HNN, LNN, SympNet, symplectic integrators, and four classical experiments.
PDE and Machine Learning (8): Reaction-Diffusion Systems and Graph Neural Networks
Deep GNNs collapse because they are diffusion equations on graphs. Turing's reaction-diffusion theory tells us how to fix it -- and closes the eight-chapter PDE+ML series.
PDE and Machine Learning (7): Diffusion Models and Score Matching
Diffusion models are PDE solvers in disguise. We derive the heat equation, Fokker-Planck, score matching, DDPM, and DDIM from a unified PDE perspective and visualise every step.
PDE and Machine Learning (6): Continuous Normalizing Flows and Neural ODE
How do you turn a Gaussian into a complex data distribution? This article derives Neural ODEs, the adjoint method, continuous normalizing flows (FFJORD), and Flow Matching from the underlying ODE/PDE theory, and shows …
PDE and Machine Learning (5): Symplectic Geometry and Structure-Preserving Networks
Standard neural networks violate conservation laws. This article derives Hamiltonian mechanics, symplectic integrators, HNNs, LNNs, and SympNets from the geometry of phase space.
PDE and Machine Learning (4): Variational Inference and the Fokker-Planck Equation
Variational inference and Langevin MCMC are two faces of the same Fokker-Planck PDE. We derive the equivalence, build SVGD as an interacting-particle approximation, and quantify convergence under log-Sobolev …
PDE and Machine Learning (3): Variational Principles and Optimization
What is the essence of neural-network training? When we run gradient descent in a high-dimensional parameter space, is there a deeper continuous-time dynamics at work? As the network width tends to …
PDE and Machine Learning (2) — Neural Operator Theory
A classical PDE solver — finite difference, finite element, spectral — is a function: feed it one initial condition and one set of coefficients, get back one solution. A PINN is the same kind of …
PDE and Machine Learning (1): Physics-Informed Neural Networks
From finite differences to PINNs: automatic differentiation, PDE residual losses, NTK-based training pathologies, Burgers inverse problems, and a side-by-side comparison with FEM and neural operators. Seven figures …