A Structure-Preserving Domain Decomposition Method for Data-Driven Modeling

TL;DR

This paper introduces a domain decomposition approach that uses data-driven, structure-preserving finite element discretizations to model complex systems without known governing equations, ensuring stability and scalability.

Contribution

It develops a novel method combining trainable Whitney form elements with mortar methods for data-driven modeling of complex systems without explicit equations.

Findings

Method accurately reproduces high-fidelity data

Ensures stability and well-posedness of the data-driven system

Effective for multiscale problems with complex microstructure

Abstract

We present a domain decomposition strategy for developing structure-preserving finite element discretizations from data when exact governing equations are unknown. On subdomains, trainable Whitney form elements are used to identify structure-preserving models from data, providing a Dirichlet-to-Neumann map which may be used to globally construct a mortar method. The reduced-order local elements may be trained offline to reproduce high-fidelity Dirichlet data in cases where first principles model derivation is either intractable, unknown, or computationally prohibitive. In such cases, particular care must be taken to preserve structure on both local and mortar levels without knowledge of the governing equations, as well as to ensure well-posedness and stability of the resulting monolithic data-driven system. This strategy provides a flexible means of both scaling to large systems and treating complex geometries, and is particularly attractive for multiscale problems with complex microstructure geometry. While consistency is traditionally obtained in finite element methods via quasi-optimality results and the Bramble-Hilbert lemma as the local element diameter $h\rightarrow0$, our analysis establishes notions of accuracy and stability for finite h with accuracy coming from matching data. Numerical experiments and analysis establish properties for $H(\operatorname{div})$ problems in small data limits ($\mathcal{O}(1)$ reference solutions).