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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.21271 |
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| _version_ | 1866911549543677952 |
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| author | Tao, Ze Zhou, Hongfu Liang, Hanbing Liu, Fujun |
| author_facet | Tao, Ze Zhou, Hongfu Liang, Hanbing Liu, Fujun |
| contents | Coupled partial differential equations underpin a wide range of multiphysics systems, yet existing neural PDE solvers still struggle to resolve localized high-risk regions and often fail to preserve structural admissibility across coupled fields. To address these limitations, we propose the Partition-of-unity Information-geometric Certified Solver (PICS), a closed-loop framework that strictly enforces structural admissibility at the level of representation rather than relying on an additional soft penalty. By constructing a gate-structured admissible manifold coupled with a restricted jet prolongation, PICS ensures that geometry-sensitive approximations and closure-essential differential coordinates enter the solver as a strongly enforced, structure-preserving ansatz. Furthermore, the framework integrates entropic tail-risk control and \textit{a posteriori} certificate-driven empirical measure transport, dynamically reallocating training efforts toward uncertified, error-prone transition zones. Evaluated against standard baseline methods across three two-dimensional coupled benchmarks, PICS achieves more consistently accurate and balanced cross-field recovery while retaining practical computational efficiency, thereby providing a rigorous route toward highly reliable multiphysics simulation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21271 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | PICS: A Partition-of-unity Information-geometric Certified Solver for Coupled Partial Differential Equations Tao, Ze Zhou, Hongfu Liang, Hanbing Liu, Fujun Computational Physics Coupled partial differential equations underpin a wide range of multiphysics systems, yet existing neural PDE solvers still struggle to resolve localized high-risk regions and often fail to preserve structural admissibility across coupled fields. To address these limitations, we propose the Partition-of-unity Information-geometric Certified Solver (PICS), a closed-loop framework that strictly enforces structural admissibility at the level of representation rather than relying on an additional soft penalty. By constructing a gate-structured admissible manifold coupled with a restricted jet prolongation, PICS ensures that geometry-sensitive approximations and closure-essential differential coordinates enter the solver as a strongly enforced, structure-preserving ansatz. Furthermore, the framework integrates entropic tail-risk control and \textit{a posteriori} certificate-driven empirical measure transport, dynamically reallocating training efforts toward uncertified, error-prone transition zones. Evaluated against standard baseline methods across three two-dimensional coupled benchmarks, PICS achieves more consistently accurate and balanced cross-field recovery while retaining practical computational efficiency, thereby providing a rigorous route toward highly reliable multiphysics simulation. |
| title | PICS: A Partition-of-unity Information-geometric Certified Solver for Coupled Partial Differential Equations |
| topic | Computational Physics |
| url | https://arxiv.org/abs/2603.21271 |