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

Shane Loeffler, Zan Ahmad, Syed Yusuf Ali, Carolyna Yamamoto, Dan M. Popescu, Alana Yee, Yash Lal, Natalia Trayanova, Mauro Maggioni

Keywords: Neural Operator, Graph Neural Networks, Graph Fourier Transform, Partial Differential Equations, Operator Learning, Cardiac Electrophysiology

TL;DR: Learning temporal dynamics of diffusive PDEs on multiple anisotropic domains using neural operators that embed Graph Fourier Transforms.

Abstract: Understanding and predicting the time-dependent dynamics of complex systems governed by non-linear partial differential equations (PDEs), with varying parameters and domains, is a difficult problem that is motivated by applications in many fields. We introduce a novel family of neural operators based on a Graph Fourier Neural Kernel (G-FuNK), for learning solution generators of nonlinear PDEs with varying coefficients, across multiple domains, for which the highest-order term in the PDE is diffusive. G-FuNKs are constructed by combining components that are parameter- and domain-adapted, with others that are not. The latter components are learned from training data, using a variation of Fourier Neural Operators, and are transferred directly across parameters and domains. The former, parameter- and domain-adapted components are constructed as soon as a parameter and a domain on which the PDE needs to be solved are given. They are obtained by constructing a weighted graph on the (discretized) domain, with weights chosen so that the Laplacian on that weighted graph approximates the highest order, diffusive term in the generator of the PDE, which is parameter- and domain-specific, and satisfies the boundary conditions. This approach proves to be a natural way to embed geometric and directionally-dependent information about the domains, allowing for improved generalization to new test domains without need for retraining. Finally, we equip G-FuNK with an integrated ordinary differential equation (ODE) solver to enable the temporal evolution of the system's state. Our experiments demonstrate G-FuNK's ability to accurately approximate heat, reaction diffusion, and cardiac electrophysiology equations on multiple geometries and varying anisotropic diffusivity fields. We achieve low relative errors on unseen domains and fiber fields, significantly speeding up prediction capabilities compared to traditional finite-element solvers.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Submission Number: 7131