PICS: A Partition-of-unity Information-geometric Certified Solver for Coupled Partial Differential Equations

TL;DR

PICS is a novel neural PDE solver that enforces structural admissibility and improves accuracy in coupled multiphysics systems by integrating geometric, probabilistic, and certification techniques.

Contribution

It introduces a structure-preserving, certified framework for coupled PDEs that outperforms standard methods in accuracy and reliability.

Findings

PICS achieves more accurate cross-field recovery.

It maintains structural admissibility without soft penalties.

The method is computationally efficient and reliable.

Abstract

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.