发表机构
University of Cambridge; Shanghai Normal University; YMSC, Tsinghua University（剑桥大学; 上海师范大学; 清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对物理信息PDE求解器表示结构固定的问题，提出ReCAP，通过局部细化、分裂、剪枝和合并动态重塑容量，在5个3D/4D基准上相对最强竞争降低误差10.7%-27.5%。
AI 中文摘要
物理信息神经偏微分方程求解器通过调整参数以满足控制方程，但其表示结构在训练过程中通常保持固定。这种刚性结构与在空间和时空上具有强烈异质性复杂度的偏微分方程解匹配不佳，导致在物理困难处容量不足，在简单处容量冗余。我们引入了物理信息神经可塑性，这是一种在优化过程中表示本身根据未解析物理进行重塑的范式。我们通过面向偏微分方程的表示容量自适应（ReCAP）实例化了这一原则，这是一种高斯局部化求解器，通过局部细化、残差导向分裂、基于门控的剪枝和功能感知合并来动态重新分配容量。ReCAP使用责任加权误差指标和残差能量的几何形状来确定在何处以及如何细化。为限制分裂引入的扰动，我们引入了安静子代细化，通过传输父代表示同时控制瞬时功能扰动来初始化新组件。我们进一步建立了条件后验可靠性和结构稳定性保证，将局部物理残差与解误差和稳定细化联系起来。在五个具有挑战性的3D和4D偏微分方程基准测试中，与11个物理信息求解器相比，ReCAP在每个问题上都实现了最低的相对$L^2$误差，相对于最强竞争结果将误差降低了$10.7\%$至$27.5\%$。这些结果表明，物理信息求解器不仅需要学习其参数——它们还可以学习其表示容量应如何组织。
英文摘要
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.
