Physics-Informed Neural Network Diffusion Equation (PINNDE)

Building on the PINNDE proof-of-concept from GSoC 2025, I propose two directions: (1) systematic improvements to the PINN architecture and training — Fourier feature embeddings, GELU activations, and adaptive collocation — informed by my test task results where GELU+Fourier features reduced physics loss by 10x over the baseline tanh network; and (2) a flow matching reformulation that replaces the O(NM) Monte Carlo score estimate with a simulation-free O(N) training objective, producing straighter transport paths that are easier for the PINN to learn. I will implement quantitative evaluation metrics (MMD, Sliced Wasserstein), benchmark both approaches on the existing 3D toy distributions, and apply the best pipeline to Dataset 1 of the Fast Calorimeter Simulation Challenge 2022. Deliverables: improved PINN architectures with benchmarks, a flow-matching PINNDE training mode, a reusable evaluation module, and results on the Calorimeter Challenge.