Enforcing physics-based algebraic constraints for inference of PDE models on unstructured grids

Valerii Iakovlev, Markus Heinonen, Harri Lähdesmäki

Abstract: Data-driven neural network models have recently shown great success in modelling and learning complex PDE systems. Several works have proposed approaches to include specific physics-based constraints to avoid unrealistic modelling outcomes. While previous works focused on specific constraints and uniform spatial grids, we propose a novel approach for enforcing general pointwise, differential and integral constraints on unstructured spatial grids. The method is based on representing a black-box PDE model's output in terms of a function approximation and enforcing constraints directly on that function. We demonstrate applicability of our approach in learning PDE-driven systems and generating spatial fields with GANs, both on free-form spatial and temporal domains, and show how both kinds of models benefit from incorporation of physics-based constraints.