Physics-informed neural solvers reshape themselves to solve complex problems
Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves
Machine Learning
Summary
Solving complex physics problems with neural networks can be difficult when different parts of the problem need different levels of detail. The authors introduce a way for these neural solvers to change their own structure during the learning process, focusing their power where it's needed most. This approach, called ReCAP, uses measurements of error and physics clues to add or remove parts of the model dynamically. Their method improved accuracy on several challenging physics problems compared to other techniques. This shows that neural solvers can become smarter by adjusting not just their settings, but also how they represent the solutions.
What this means in practice
- •For simulation engineers: Build adaptive PDE solvers that redistribute computational effort according to local physics complexity for better accuracy in simulations.
- •For computational fluid dynamics developers: Integrate dynamic neural capacity adjustment into solvers to improve prediction fidelity in complex, heterogeneous flow problems.
Authors
Chun-Wun Cheng, Bingcheng Hu, Angelica I. Aviles-Rivero
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.