Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves

2026-10-07

AI Summary

Physics-informed neural networks (PINNs) with fixed architectures often suffer from capacity mismatch when solving complex partial differential equations (PDEs). This work proposes ReCAP, a paradigm that dynamically reshapes network representations during optimization to adaptively match the underlying physical complexity. The method introduces a silent-subnet refinement mechanism for low-perturbation splitting and establishes reliability and stability guarantees via residual-error conditioned posteriors. Furthermore, it integrates Gaussian localization, residual-guided splitting, gated pruning, and function-aware merging modules. Experimental results demonstrate that ReCAP achieves state-of-the-art accuracy across five 3D and 4D benchmarks, reducing relative L2 errors by 10.7%–27.5% compared to the strongest baseline.

Abstract

Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly matched to PDE solutions with strongly heterogeneous complexity across space and space--time, leaving capacity insufficient where the physics is difficult and redundant where it is simple. We introduce physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics. We instantiate this principle with Representation Capacity Adaptation for PDEs (ReCAP), a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging. ReCAP uses responsibility-weighted error indicators and the geometry of residual energy to determine where and how to refine. To limit the disturbance introduced by splitting, we introduce quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation. We further establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Across five challenging 3D and 4D PDE benchmarks against 11 physics-informed solvers, ReCAP achieves the lowest relative $L^2$ error on every problem, reducing error by $10.7\%$--$27.5\%$ relative to the strongest competing result. These results suggest that physics-informed solvers need not merely learn their parameters---they can learn how their representational capacity should be organized.

Problem

• Physics-Informed Neural Networks
• PDE Solvers
• Neural Plasticity
• Representation Capacity
• Heterogeneous Complexity

Innovation

• Physics-Informed Neural Networks
• Neural Plasticity
• Representation Capacity Adaptation
• PDE Solvers
• Quiet-Child Refinement