Structure-Preserving Learning Improves Geometry Generalization in Neural PDEs

TL;DR

This paper introduces Geo-NeW, a structure-preserving neural finite element method that improves the generalization of neural PDE solvers to unseen geometries by integrating geometry explicitly into the learning process.

Contribution

The paper presents Geo-NeW, a novel data-driven finite element approach that jointly learns differential operators and finite element spaces, ensuring physical law preservation and enhanced geometric generalization.

Findings

Achieves state-of-the-art results on steady-state PDE benchmarks.

Significantly outperforms traditional methods on out-of-distribution geometries.

Ensures solution existence and uniqueness through a new parameterization.

Abstract

We aim to develop physics foundation models for science and engineering that provide real-time solutions to Partial Differential Equations (PDEs) which preserve structure and accuracy under adaptation to unseen geometries. To this end, we introduce General-Geometry Neural Whitney Forms (Geo-NeW): a data-driven finite element method. We jointly learn a differential operator and compatible reduced finite element spaces defined on the underlying geometry. The resulting model is solved to generate predictions, while exactly preserving physical conservation laws through Finite Element Exterior Calculus. Geometry enters the model as a discretized mesh both through a transformer-based encoding and as the basis for the learned finite element spaces. This explicitly connects the underlying geometry and imposed boundary conditions to the solution, providing a powerful inductive bias for learning neural PDEs, which we demonstrate improves generalization to unseen domains. We provide a novel parameterization of the constitutive model ensuring the existence and uniqueness of the solution. Our approach demonstrates state-of-the-art performance on several steady-state PDE benchmarks, and provides a significant improvement over conventional baselines on out-of-distribution geometries.