Structure-Preserving Neural Surrogates with Tractable Uncertainty Quantification

2026-06-10

AI Summary

This work addresses the absence of theoretically grounded, real-time solvers for partial differential equations in current scientific machine learning, which hinders reliable simulation validation. The authors propose a physics-informed, data-driven reduced-order model that preserves conservation structures by leveraging exterior calculus to encode topological properties and Gaussian processes to model state-to-flux relationships, yielding a Dirichlet-to-Neumann map with closed-form posterior uncertainty. A novel interface between mixed finite element spaces and Gaussian process regression is established, recasting training as an optimal recovery problem subject to conservation constraints, augmented by an efficient Schur complement strategy. Within a reproducing kernel Hilbert space framework, rigorous posterior error bounds are derived. Numerical experiments demonstrate that the method enables real-time, high-fidelity estimation of boundary fluxes, with its posterior distribution effectively replacing conventional error estimators while offering both interpretability and principled uncertainty quantification.

Abstract

Recent advances in scientific machine learning provide a means of near-real-time solution to partial differential equations (PDEs), but lack the theoretical underpinnings of conventional simulators that support contemporary verification and validation. In this work, we construct data-driven reduced-order models that serve as structure-preserving, real-time surrogates. Remarkably, the exterior calculus that imposes physical conservation structure also exposes topological structure that we use to build a Gaussian process (GP) representation of uncertainty in state-flux relationships, ultimately yielding a Dirichlet-to-Neumann map for quantities of interest with closed-form expressions for posterior uncertainty. We specifically propose structure-preserving $H(\mathrm{div})$--$L^2$ subspaces of conventional Raviart--Thomas and $dgP_0$ elements prescribed by a lightweight transformer. Reduced-order dynamics consistent with this subspace are learned by posing a conservation law in which a GP describes the fluxes between volumes. This work hinges on a novel interface between mixed FEM spaces and GP regression; when training is posed as the optimal recovery problem (ORP), the resulting GP regression can be written as an optimization problem with equality constraints that impose a conservation structure, amenable to a fast Schur-complement training strategy. The trained model can then be solved in real time with closed-form estimators for boundary fluxes driven by prescribed Dirichlet data. The paper includes RKHS posterior error bounds for linear functionals to support uncertainty quantification, as well as numerical experiments demonstrating the accuracy of the posterior distribution as a surrogate for error estimation.

Problem

• structure-preserving
• uncertainty quantification
• reduced-order models
• partial differential equations
• Gaussian process

Innovation

• structure-preserving surrogate
• Gaussian process regression
• mixed finite element methods
• uncertainty quantification
• optimal recovery problem