Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates

TL;DR

This paper introduces novel neural network layers that incorporate physical laws and symmetries on staggered grids, significantly improving the accuracy and generalization of PDE surrogates for fluid dynamics problems.

Contribution

It systematically investigates the combined effect of physical and symmetry constraints on neural PDE surrogates, introducing new layers compatible with staggered grids.

Findings

Symmetries and physical constraints improve surrogate accuracy.

Combining both constraints yields the best performance.

Surrogates with constraints generalize better to new conditions.

Abstract

Neural PDE surrogates can improve the cost-accuracy tradeoff of classical solvers, but often generalize poorly to new initial conditions and accumulate errors over time. Physical and symmetry constraints have shown promise in closing this performance gap, but existing techniques for imposing these inductive biases are incompatible with the staggered grids commonly used in computational fluid dynamics. Here we introduce novel input and output layers that respect physical laws and symmetries on the staggered grids, and for the first time systematically investigate how these constraints, individually and in combination, affect the accuracy of PDE surrogates. We focus on two challenging problems: shallow water equations with closed boundaries and decaying incompressible turbulence. Compared to strong baselines, symmetries and physical constraints consistently improve performance across tasks, architectures, autoregressive prediction steps, accuracy measures, and network sizes. Symmetries are more effective than physical constraints, but surrogates with both performed best, even compared to baselines with data augmentation or pushforward training, while themselves benefiting from the pushforward trick. Doubly-constrained surrogates also generalize better to initial conditions and durations beyond the range of the training data, and more accurately predict real-world ocean currents.