Convergent Operator-Splitting Scheme for Viscosity Solutions: A Foundation for Learning Domain-to-Solution Maps

2025/05/27 by Po‐Yi Wu, Wu, Po-Yi
Engineering · Mathematics · Physics and Astronomy · #35D40 #49L25 #65M12 #65M60 #68T07 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2505.20618

openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work introduces and rigorously analyzes a novel operator-splitting finite element scheme for approximating viscosity solutions of a broad class of constrained second-order partial differential equations. By decoupling the primary PDE evolution from the enforcement of constraints, the proposed method combines a stabilized finite element method for spatial discretization with an efficient semi-implicit time-stepping strategy. The cornerstone of our analysis is a proof that the scheme satisfies a discrete comparison principle. We demonstrate that under a mild time-step restriction and with appropriate stabilization, the discrete operator yields an M-matrix, which is sufficient to guarantee the scheme's monotonicity and consequent L^∞-stability. These properties -- consistency, stability, and monotonicity -- are shown to be sufficient to prove convergence of the numerical approximation to the unique viscosity solution within the celebrated Barles--Souganidis framework. For solutions with enhanced regularity, we further establish an optimal-order error estimate of O(Δt + h²). The rigorously established stability of the scheme provides a blueprint for a novel Physics-Constrained Neural Operator (PCNO) architecture. We prove that by emulating the scheme's structure, the PCNO can provably break the curse of dimensionality for the challenging class of domain-to-solution mapping problems with complex topological variations, a problem for which standard learning approaches often fail. Numerical experiments for both a Hamilton-Jacobi equation with state constraints and a controlled reaction-diffusion system are presented to validate the theoretical findings and demonstrate the scheme's effectiveness.

Citations

• Stochastic approach for elliptic problems in perforated domains
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