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

Shuai Jiang, Jonas Actor, Scott Roberts, Nathaniel Trask

TL;DR

The paper tackles learning data-driven, structure-preserving models when governing equations are unknown but conservation laws are known. It introduces a scalable domain-decomposition framework that trains subdomain Whitney-form elements to realize local Dirichlet-to-Neumann maps and couples them with a non-conforming mortar method that preserves flux conservation and stability. The authors establish both continuous and discrete coercivity and error bounds, and demonstrate through multiple numerical experiments (pure FEM, pure FEEC, and hybrids) that the approach can capture multiscale features and preserve important structural properties, even under subdomain refinement and limited data. This work provides a principled path to scalable, physics-respecting data-driven discretizations suitable for complex geometries and microstructures, with potential impact on multiscale simulations and unexplained-physics modeling.

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).

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

TL;DR

The paper tackles learning data-driven, structure-preserving models when governing equations are unknown but conservation laws are known. It introduces a scalable domain-decomposition framework that trains subdomain Whitney-form elements to realize local Dirichlet-to-Neumann maps and couples them with a non-conforming mortar method that preserves flux conservation and stability. The authors establish both continuous and discrete coercivity and error bounds, and demonstrate through multiple numerical experiments (pure FEM, pure FEEC, and hybrids) that the approach can capture multiscale features and preserve important structural properties, even under subdomain refinement and limited data. This work provides a principled path to scalable, physics-respecting data-driven discretizations suitable for complex geometries and microstructures, with potential impact on multiscale simulations and unexplained-physics modeling.

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 , our analysis establishes notions of accuracy and stability for finite h with accuracy coming from matching data. Numerical experiments and analysis establish properties for problems in small data limits ( reference solutions).
Paper Structure (22 sections, 8 theorems, 88 equations, 18 figures, 6 tables)

This paper contains 22 sections, 8 theorems, 88 equations, 18 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Relation to previous work
  3. Data-driven DEC/FEEC and Dirichlet-to-Neumann maps
  4. Structure-preserving ML vs. physics-informed ML
  5. Choice of mortar scheme
  6. Local learning of Whitney form elements
  7. Mortar Method
  8. Stability analysis for continuous case
  9. Discretized Case
  10. Data-Driven Elements with Mortar Method
  11. Neumann Boundary Conditions and Conservation
  12. Numerical Results
  13. Example 1: Pure Finite Elements
  14. Example 2: Pure FEEC and Pure FEM Elements Comparison
  15. Example 3: Hybrid Methods
  16. ...and 7 more sections

Key Result

Lemma 4.1

\newlabellem:cont-subdomains0 For $1 \le i \le n$, let $(\bm{u}_i, p_i, \lambda) \in (L^2(\Omega_i)^2, H^1(\Omega_i), \Lambda_g)$ such that with continuity of state and flux enforced via the boundary condition $p_i|_{\Gamma_i} = \lambda|_{\Gamma_i}$ and weak flux continuity condition Then $\bm{u} = \sum_{i=1}^n \bm{u}_i \in L^2(\Omega)^2, p = \sum_{i=1}^n p_i \in H^1_g(\Omega)$ solves eqn:prima

Figures (18)

  • Figure 1: To construct a PPOU, we first consider an underlying tensor product grid of B-splines with trainable vertex locations. By taking a trainable convex combination of these shape functions, we arrive at more complex geometries. Noting that B-splines form a partition of unity, and that partitions of unity are closed under convex combination, this process provides a trainable partition of unity which may be integrated exactly via a pull-back onto the fine grid. For purposes of illustration, the underlying tensor product is shown to be uniform in the figure, but they are allowed to shift in the general case.
  • Figure 1: Figure of a square domain $\Omega$ divided into four subdomains. The edge $\Gamma_{3,4}$ is denoted explicitly and the highlighted boundary is $\Gamma_1$.
  • Figure 1: Figure illustrating the initial mesh, the corresponding mortar space, and their first refinement for the example in \ref{['sec:example-1']}.
  • Figure 1: Plots illustrating some of the training data used for \ref{['sec:cylinder']} with pressure, flux of $x$ and flux of $y$ in the columns respectively. The data is generated from a low order FEM method with $h = 1/100$. The key differences between each data set is that the boundary conditions are varied so that the element can respond to the different mortars.
  • Figure 2: Sketch of a 4 element mortar $\Lambda_H$ and its two adjacent subdomains. The colors on the subdomains represent the PPOUs constructed as convex combinations of a fine-scale B-splines. We note that while the mortar matches the fine-scale nodes on $\Omega_2$, it is disjoint from $\Omega_1$, requiring analysis of a remap/projection between the two meshes. Because the FEEC fine-scale nodes on $\Omega_i$ evolve during training, they will generally not coincide with mortar nodes.
  • ...and 13 more figures

Theorems & Definitions (18)

  • Remark 3.1
  • Lemma 4.1
  • Proof 1
  • Lemma 4.2
  • Proof 2
  • Lemma 4.3
  • Proof 3
  • Lemma 4.4
  • Lemma 4.5
  • Theorem 4.6
  • ...and 8 more