Geometry-physics confounding impairs PDE learning across varying domains

2026-09-29

AI Summary

This study addresses the entanglement between geometric variations and physical properties, which leads to low data efficiency in cross-domain PDE learning and biased equation discovery. We identify, for the first time, a unified failure mechanism arising from this geometry-physics confounding and propose a decoupling framework that separates known geometric effects from intrinsic physical laws by explicitly modeling the geometry-to-operator transformation. By integrating geometry-induced coefficient fields with a candidate library incorporating geometric terms, the framework enables both forward operator learning and equation discovery. Extensive evaluations across five benchmarks demonstrate that our approach significantly improves prediction accuracy and data efficiency, faithfully recovers generating equations, and reduces residual errors by two orders of magnitude.

Abstract

Learning partial differential equation (PDE) dynamics across varying domains is central to predictive modelling and data-driven discovery of governing equations. However, geometric variation alters both field representation and the governing differential operators, confounding geometric effects with intrinsic physical properties in the observed dynamics. This work identifies geometry-physics confounding as a unified failure mechanism for PDE learning across varying domains. In forward operator learning, this confounding increases the burden of inferring geometry-dependent operator changes from finite data, reducing data efficiency and generalisation. In equation discovery, omitting geometry-induced operators misspecifies the candidate library, leading to biased parameters, missed governing terms and spurious terms. We propose a de-confounding framework that makes the known geometry-to-operator transformation explicit. Geometry-induced coefficient fields improve prediction and data efficiency across five operator-learning benchmarks, while geometry-complete candidate libraries recover the generating equations and reduce held-out PDE residuals by more than two orders of magnitude in both evolving-domain systems. By separating known geometric action from intrinsic physics, the proposed framework supports more reliable and data-efficient PDE learning across scientific and engineering problems with varying geometries.

Problem

• Partial Differential Equations
• Geometry-physics confounding
• Operator learning
• Equation discovery
• Varying domains

Innovation

• Geometry-physics confounding
• Partial differential equations
• Operator learning
• Equation discovery
• De-confounding framework