Graph Neural PDE Solvers with Conservation and Similarity-Equivariance

TL;DR

FluxGNN addresses the challenge of generalizing PDE surrogates across diverse spatial domains by enforcing conservation laws and similarity symmetries within a graph neural PDE solver. It roots its architecture in locally conservative GNNs that mirror finite-volume flux exchanges, with flux functions that are permutation-invariant in edge signals and compatible with similarity-equivariant building blocks. The approach yields high out-of-domain generalization and maintains physical conservation, demonstrated on convection–diffusion and Navier–Stokes with mixtures, outperforming several baselines while remaining integrable with existing numerical methods. This work advances reliable physics-informed surrogates for complex flow and transport problems, enabling efficient, domain-agnostic predictions with strong physical fidelity.

Abstract

Utilizing machine learning to address partial differential equations (PDEs) presents significant challenges due to the diversity of spatial domains and their corresponding state configurations, which complicates the task of encompassing all potential scenarios through data-driven methodologies alone. Moreover, there are legitimate concerns regarding the generalization and reliability of such approaches, as they often overlook inherent physical constraints. In response to these challenges, this study introduces a novel machine-learning architecture that is highly generalizable and adheres to conservation laws and physical symmetries, thereby ensuring greater reliability. The foundation of this architecture is graph neural networks (GNNs), which are adept at accommodating a variety of shapes and forms. Additionally, we explore the parallels between GNNs and traditional numerical solvers, facilitating a seamless integration of conservative principles and symmetries into machine learning models. Our findings from experiments demonstrate that the model's inclusion of physical laws significantly enhances its generalizability, i.e., no significant accuracy degradation for unseen spatial domains while other models degrade. The code is available at https://github.com/yellowshippo/fluxgnn-icml2024.

Figures (18)

• Figure 1: Overview of a typical graph neural network (GNN), finite volume method (FVM), and proposed model, FluxGNN. Our method combines GNN and FVM, realizing high expressibility from GNN and generalizability from FVM.
• Figure 2: Overview of the FluxGNN model for the convection--diffusion equation with an encoder ${\mathcal{F}}_\mathrm{encode}$, locally conservative GNN ${\mathcal{F}}_\mathrm{L}$, and decoder ${\mathcal{F}}_\mathrm{decode}$. The model outputs a time series, and the loss is computed using all steps of the model's output.
• Figure 3: Comparison of the initial condition, ground truth, prediction of FVM, and prediction of FluxGNN taken from a sample in the test dataset at time $t = 1.0$.
• Figure 4: Visual comparison of the velocity field between the ground truth, MP-PDE, PENN, and FluxGNN.
• Figure 5: Speed--accuracy tradeoff obtained through hyperparameter studies for MP-PDE, PENN, and FluxGNN (proposed method), with error bars corresponding to the standard error of the mean. Light and dark colors correspond to the evaluation of the test and taller datasets, respectively. Lines represent Pareto fronts, with arrows denoting shifts of the fronts caused by changes in the dataset considered for the evaluation. The results of FVM are excluded because they are far from the Pareto front and are shown in \ref{['fig:mixture_tradeoff_detail']}.

Theorems & Definitions (10)

• Theorem 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• proof
• Remark 3.1
• proof
• proof
• proof
• Remark 3.2