PDE

Jul 28, 2025 Standalone 26 min read

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.

Aug 14, 2024 PDE and Machine Learning 44 min read

PDE and ML (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.

Jul 30, 2024 PDE and Machine Learning 38 min read

PDE and ML (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.

Jul 15, 2024 PDE and Machine Learning 38 min read

PDE and ML (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 …

Jun 30, 2024 PDE and Machine Learning 48 min read

PDE and ML (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.

Jun 15, 2024 PDE and Machine Learning 40 min read

PDE and ML (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 …

May 31, 2024 PDE and Machine Learning 54 min read

PDE and ML (3): Variational Principles and Optimization

Calculus of variations to Wasserstein gradient flow and the mean-field limit of neural networks.

May 16, 2024 PDE and Machine Learning 46 min read

PDE and ML (2): Neural Operator Theory

DeepONet vs FNO from a function-space view: resolution invariance, error bounds, failure modes.

May 1, 2024 PDE and Machine Learning 44 min read

PDE and ML (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 …

Oct 23, 2021 Functional Analysis 70 min read

Functional Analysis (12): Functional Analysis in Action — PDE and Quantum Mechanics

Lax-Milgram for elliptic PDE, variational methods, quantum observables as self-adjoint operators, and Stone's theorem — where the abstract theory meets concrete applications.

Oct 19, 2021 Functional Analysis 76 min read

Functional Analysis (10): Semigroups of Operators — Evolution Equations in Infinite Dimensions

C₀-semigroups provide the abstract framework for evolution equations — the Hille-Yosida theorem characterizes which operators generate well-posed dynamics.