Convergent Operator-Splitting Scheme for Viscosity Solutions: A Foundation for Learning Domain-to-Solution Maps
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909666519285760 |
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| author | Wu, Po-Yi |
| author_facet | Wu, Po-Yi |
| contents | 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 $L^\infty$-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 $O(Δt + h^2)$.
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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_20618 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergent Operator-Splitting Scheme for Viscosity Solutions: A Foundation for Learning Domain-to-Solution Maps Wu, Po-Yi Numerical Analysis 65M60, 35D40, 49L25, 65M12, 68T07 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 $L^\infty$-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 $O(Δt + h^2)$. 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. |
| title | Convergent Operator-Splitting Scheme for Viscosity Solutions: A Foundation for Learning Domain-to-Solution Maps |
| topic | Numerical Analysis 65M60, 35D40, 49L25, 65M12, 68T07 |
| url | https://arxiv.org/abs/2505.20618 |