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

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.

Figures (4)

• Figure 1: The framework of the proposed PICS solver. PICS organizes PDE solving as a closed solver loop rather than a residual-fitting pipeline. Starting from a gate-structured admissible state, the solver induces the case-level physical fields, retains only the closure-essential differential coordinates through a restricted jet prolongation, evaluates a normalized residual section and its composite residual geometry, extracts a certificate field that identifies fieldwise and global high-risk regions, transports the empirical training measure toward those regions, and then updates the parameter state on the transported measure. This common loop serves as the solver backbone for all benchmark systems studied in Figs. \ref{['F2']}--\ref{['F4']} and Tables \ref{['T1']}--\ref{['T2']}.
• Figure 2: The first benchmark realization of the proposed PICS solver. The top trajectory panel records the evolution of the residual-energy and certificate-governed solver state during training. The field-reconstruction panels compare the predicted $p$, $u$, $v$, $\phi$, and $T$ fields produced by PICS, PINN, DGM, and DRM against the analytic reference under the first benchmark configuration. The corresponding maximum-error maps show that PICS yields more balanced cross-field recovery and contracts the dominant error hotspots toward the interface-active region, whereas the baseline solvers exhibit broader spreading, mild amplitude drift, or stronger interface-localized distortion.
• Figure 3: The second benchmark realization of the proposed PICS solver under altered closure structure. The top trajectory panel shows the evolution of the residual energy and the certificate-guided solver state under the Case 2 thermo-viscous and Leray-regularized closure. The field panels compare the predicted $p$, $u$, $v$, $\phi$, and $T$ fields from PICS, PINN, DGM, and DRM with the analytic reference for the second benchmark configuration. The maximum-error maps show that PICS preserves more stable cross-field coherence and keeps the dominant error concentrations weaker and more localized.
• Figure 4: The third benchmark realization of the proposed PICS solver under the hardest closure configuration. The top trajectory panel shows the evolution of the residual energy and the certificate-guided solver state for the most demanding benchmark in this work. The field panels compare the predicted $p$, $u$, $v$, $\phi$, and $T$ fields from PICS, PINN, DGM, and DRM with the analytic reference for the pressure-regularized and screened electro-thermal closure of Case 3. The maximum-error maps show that PICS maintains more balanced cross-field recovery and confines the dominant error hotspots more effectively.