Structure-Preserving Learning Improves Geometry Generalization in Neural PDEs

2026-02-02

AI Summary

This work addresses the challenge of generalizing partial differential equation (PDE) solvers to unseen geometric domains by proposing Geo-NeW, a novel method that jointly learns differential operators and compatible reduced finite element spaces within the framework of finite element exterior calculus to rigorously preserve physical conservation laws. Geo-NeW introduces geometry-aware neural Whitney forms that embed mesh geometric information into both Transformer encodings and basis function construction, thereby endowing neural PDE solvers with strong structure-preserving inductive biases. Furthermore, it devises a new constitutive model parameterization that guarantees the existence and uniqueness of solutions. Evaluated on multiple steady-state PDE benchmarks, the method achieves state-of-the-art performance and significantly outperforms conventional approaches on out-of-distribution geometries.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• geometry generalization
• neural PDEs
• structure preservation
• out-of-distribution generalization
• physics-informed learning

Innovation

• Structure-Preserving Learning
• Neural PDEs
• Finite Element Exterior Calculus
• Geometry Generalization
• Neural Whitney Forms