Structure-Preserving Learning Improves Geometry Generalization in Neural PDEs

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-NeWF): 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 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.

Simulations are widely used to predict fluid flow, heat transfer, and mechanical stress in engineering designs, but they can be too slow to evaluate every possible shape. Machine learning can accelerate these predictions, yet many existing methods struggle when asked to evaluate geometries that differ from those seen during training. We introduce Geo-NeW, a learning-based simulator that incorporates the geometry of each design into a compact physical model and then solves that model to produce a prediction. By building conservation laws and boundary conditions directly into the prediction process, Geo-NeW is better able to remain accurate on new shapes. Across several two-dimensional steady-state simulation tasks, it matches or improves standard prediction accuracy and gives substantially lower error on geometries outside the training distribution. This capability is important for engineering design, where the goal is often to evaluate new candidate geometries rather than reproduce previously observed ones.