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This research introduces "physics-informed neural plasticity," a novel approach where the architecture of PDE solvers dynamically reshapes itself during training to match heterogeneous solution complexity. By implementing mechanisms like residual-directed splitting and function-aware merging, the proposed ReCAP method achieves superior accuracy on 3D and 4D benchmarks compared to fixed-structure competitors. This advancement demonstrates that adaptive representational capacity is crucial for efficiently solving complex physical systems governed by partial differential equations.
Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves
arXiv:2610.09510v1 Announce Type: new 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.
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.
Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2610.09510 [cs.LG]
(or arXiv:2610.09510v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.09510 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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