通过度量正则化提升多模式物理神经网络的稳定性与精度
Supervised Metric Regularization Through Alternating Optimization for Multi-Regime Physics-Informed Neural Networks
- 用隐空间度量约束优化物理参数到解的映射关系
- 在杜芬振子上实现49%更低的物理残差（0.082 vs 0.160）
- 适合处理含突变相变的复杂动力系统建模
标准物理信息神经网络（PINNs）在建模具有尖锐相变的参数化动力系统时常面临挑战，如分岔现象。此时，从参数到解的连续映射可能导致谱偏差或“模式坍缩”，使网络平均不同物理行为。本文提出拓扑感知PINN（TAPINN），通过监督度量正则化构建隐空间结构以缓解该问题。与直接将物理参数映射到解的标准参数化PINN不同，本方法将求解器条件化于一个经优化的隐状态，该状态反映不同相之间基于度量的分离特性。在杜芬振子上的初步实验表明，尽管标准基线存在谱偏差，高容量超网络会过拟合（记忆数据但违反物理规律），本方法仍能实现稳定收敛，梯度方差比多输出索博列夫误差基线低2.18倍，且参数量仅为超网络方案的1/5。
原文摘要 · Abstract (English)
Standard Physics-Informed Neural Networks (PINNs) often face challenges when modeling parameterized dynamical systems with sharp regime transitions, such as bifurcations. In these scenarios, the continuous mapping from parameters to solutions can result in spectral bias or "mode collapse", where the network averages distinct physical behaviors. We propose a Topology-Aware PINN (TAPINN) that aims to mitigate this challenge by structuring the latent space via Supervised Metric Regularization. Unlike standard parametric PINNs that map physical parameters directly to solutions, our method conditions the solver on a latent state optimized to reflect the metric-based separation between regimes, showing ~49% lower physics residual (0.082 vs. 0.160). We train this architecture using a phase-based Alternating Optimization (AO) schedule to manage gradient conflicts between the metric and physics objectives. Preliminary experiments on the Duffing Oscillator demonstrate that while standard baselines suffer from spectral bias and high-capacity Hypernetworks overfit (memorizing data while violating physics), our approach achieves stable convergence with 2.18x lower gradient variance than a multi-output Sobolev Error baseline, and 5x fewer parameters than a hypernetwork-based alternative.
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