Automated discovery of finite volume schemes using Graph Neural Networks

TL;DR

This paper demonstrates that Graph Neural Networks can be used to automatically discover and generate finite volume numerical schemes for differential equations, including higher-order schemes, through training on simple graph datasets and symbolic regression.

Contribution

The work shows GNNs can extrapolate finite volume schemes beyond training data and autonomously rediscover classical schemes using symbolic regression and physics-informed training.

Findings

GNNs trained on two-node graphs can approximate first-order FV schemes with errors proportional to training loss.

GNNs can rediscover the exact analytical form of FV schemes through symbolic regression.

Unsupervised GNN training with residual loss can recover FV schemes without ground-truth data.

Abstract

Graph Neural Networks (GNNs) have deeply modified the landscape of numerical simulations by demonstrating strong capabilities in approximating solutions of physical systems. However, their ability to extrapolate beyond their training domain (\textit{e.g.} larger or structurally different graphs) remains uncertain. In this work, we establish that GNNs can serve purposes beyond their traditional role, and be exploited to generate numerical schemes, in conjunction with symbolic regression. First, we show numerically and theoretically that a GNN trained on a dataset consisting solely of two-node graphs can extrapolate a first-order Finite Volume (FV) scheme for the heat equation on out-of-distribution, unstructured meshes. Specifically, if a GNN achieves a loss $\varepsilon$ on such a dataset, it implements the FV scheme with an error of $\mathcal{O}(\varepsilon)$. Using symbolic regression, we show that the network effectively rediscovers the exact analytical formulation of the standard first-order FV scheme. We then extend this approach to an unsupervised context: the GNN recovers the first-order FV scheme using only a residual loss similar to Physics-Informed Neural Networks (PINNs) with no access to ground-truth data. Finally, we push the methodology further by considering higher-order schemes: we train (i) a 2-hop and (ii) a 2-layers GNN using the same PINN loss, that autonomously discover (i) a second-order correction term to the initial scheme using a 2-hop stencil, and (ii) the classic second-order midpoint scheme. These findings follows a recent paradigm in scientific computing: GNNs are not only strong approximators, but can be active contributors to the development of novel numerical methods.