Hard-constraining Neumann boundary conditions in physics-informed neural networks via Fourier feature embeddings

Christopher Straub, Philipp Brendel, Vlad Medvedev, Andreas Rosskopf

Track: New scientific result

Keywords: Physics-informed neural networks, hard-constraint, Neumann boundary condition, Fourier feature embedding

TL;DR: A new method to hard-constrain Neumann boundary conditions into PINNs based on Fourier feature embeddings.

Abstract: We present a novel approach to hard-constrain Neumann boundary conditions in physics-informed neural networks (PINNs) using Fourier feature embeddings. Neumann boundary conditions are used to described critical processes in various application, yet they are more challenging to hard-constrain in PINNs than Dirichlet conditions. Our method employs specific Fourier feature embeddings to directly incorporate Neumann boundary conditions into the neural network's architecture instead of learning them. The embedding can be naturally extended by high frequency modes to better capture high frequency phenomena. We demonstrate the efficacy of our approach through experiments on a diffusion problem, for which our method outperforms existing hard-constraining methods and classical PINNs, particularly in multiscale and high frequency scenarios.

Supplementary: https://github.com/FhG-IISB-MKI/HC_Neumann_Experiments

Submission Number: 5