Geometry influences PDE learning accuracy and efficiency across domains
Geometry-physics confounding impairs PDE learning across varying domains
Machine Learning
Summary
When trying to teach computers how physical systems change over different shapes or spaces, variations in shapes can confuse the learning process. The authors found that these shape changes affect both how the data looks and the mathematical rules behind it, mixing up geometry with true physical behavior. They created a method that clearly separates the effects of geometry from the underlying physics, which helps computers learn more accurately and use less data. This approach improved predictions and helped identify the correct physical equations much better than before.
What this means in practice
- •For computational engineers: Improve simulation accuracy and efficiency for systems whose shapes or boundaries change over time by explicitly accounting for geometry effects in PDE learning.
- •For environmental modelers: Better identify governing equations in evolving natural domains like coastlines or weather patterns by separating geometric influences from physical dynamics.
Authors
Yinghao Cheng, Gengxiang Chen, Xu Liu, Qinglu Meng, Yixin Jing, Xiangguo Tang, Wenping Mou, Lihui Wang, Yingguang Li
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.