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.

翻译：标准物理信息神经网络在建模具有急剧模态转换（如分岔）的参数化动力学系统时常常面临挑战。在这些场景中，从参数到解的连续映射可能导致谱偏差或"模态坍缩"，即网络将不同的物理行为进行平均。我们提出了一种拓扑感知物理信息神经网络，旨在通过监督度量正则化构建潜在空间来缓解这一挑战。与直接将物理参数映射到解的标准参数化物理信息神经网络不同，我们的方法将求解器条件置于一个经过优化的潜在状态上，该状态旨在反映模态间基于度量的分离，其物理残差降低了约49%（0.082对比0.160）。我们采用基于阶段的交替优化训练策略来训练该架构，以管理度量目标与物理目标之间的梯度冲突。在Duffing振子上的初步实验表明，虽然标准基线方法遭受谱偏差问题，且高容量超网络存在过拟合（记忆数据但违反物理规律），我们的方法实现了稳定收敛，其梯度方差比多输出Sobolev误差基线低2.18倍，且参数数量比基于超网络的替代方案少5倍。