Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains

TL;DR

This paper introduces G-FuNK, a neural operator framework that efficiently predicts solutions to nonlinear diffusive PDEs across multiple domains by combining graph-based domain adaptation with Fourier neural operators, enabling accurate and fast simulations.

Contribution

The paper presents a novel neural operator architecture that integrates graph Laplacian-based domain adaptation with Fourier Neural Operators for solving nonlinear PDEs across diverse geometries and parameters.

Findings

Accurately predicts heat and reaction diffusion equations across various geometries.

Achieves low relative errors on unseen domains and fiber fields.

Significantly faster than traditional finite-element methods.

Abstract

Predicting time-dependent dynamics of complex systems governed by non-linear partial differential equations (PDEs) with varying parameters and domains is a challenging task motivated by applications across various fields. We introduce a novel family of neural operators based on our Graph Fourier Neural Kernels, designed to learn solution generators for nonlinear PDEs in which the highest-order term is diffusive, across multiple domains and parameters. G-FuNK combines components that are parameter- and domain-adapted with others that are not. The domain-adapted components are constructed using a weighted graph on the discretized domain, where the graph Laplacian approximates the highest-order diffusive term, ensuring boundary condition compliance and capturing the parameter and domain-specific behavior. Meanwhile, the learned components transfer across domains and parameters using our variant Fourier Neural Operators. This approach naturally embeds geometric and directional information, improving generalization to new test domains without need for retraining the network. To handle temporal dynamics, our method incorporates an integrated ODE solver to predict the evolution of the system. Experiments show G-FuNK's capability to accurately approximate heat, reaction diffusion, and cardiac electrophysiology equations across various geometries and anisotropic diffusivity fields. G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions compared to traditional finite-element solvers.