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

TL;DR

This work introduces a novel family of neural operators based on the 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.

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.

Figures (9)

• Figure 1: Top: Network structure of the Neural Operator. The input is passed as $a$. (1) It is lifted to a higher dimensional space via a lifting operator $\mathcal{P}$. (2) Multiple G-FuNK layers are then applied. (3) The network projects to the output dimensional space via $\mathcal{Q}$. Bottom: Single G-FuNK layer structure. Starting from input $k$, (1) $\mathcal{F}$ denotes the GFT of the input which is then reduced to the $k_{\text{max}}$-lowest modes. (2) The function is expanded onto a set of the eigenvalues raised to different powers in $\mathcal{L}$ (3) A linear transform is applied with learnable parameters in $\mathcal{R}$. (4) $\mathcal{F}^{-1}$ represents the inverse GFT, mapping back to the original domain. (5) The input function $k$ undergoes a linear mapping in $\mathcal{W}$. (6) The outputs of the top and bottom branches are combined and then passed through an activation function $\sigma$. The output of the G-FuNK layer is $z$ and is operated on by the next G-FuNK layer.
• Figure 2: Heat Equation: G-FuNK vs Target. A comparison between the target and the G-FuNK predicted heat equation. On the left is a stream plot example of the primary direction of the diffusion, described as $\Hat{F}^{(1)}$ in (\ref{['heat_fibers_eq_2']}), which was unique for the test trajectory shown on the right. The prediction of G-FuNK is on the bottom and the target is above for three different time points after the initial condition at $t_0$.
• Figure 3: Reaction Diffusion on Random Rectangle: G-FuNK vs Target.A and B are two comparisons of G-FuNK for 2 different test geometries. Both panels use the colorbar on the right which is min-max scaled from -85 to 20 mV and $\Delta t$ is 10 milliseconds.
• Figure 4: Example of a patient-specific fiber field providing anisotropic information for the PDE on the 3D left atrial chamber surface.
• Figure 5: Comparing target and G-FuNK predictions on an out-of-training test geometry. A Left atrial posterior wall view at $2\Delta t$ (20 ms) after initial condition. B Left atrial anterior wall view at $4\Delta t$ (40 ms) after initial condition.