发表机构
Cornell University; Siemens Energy; Johns Hopkins University（康奈尔大学; 西门子能源; 约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出三阶段框架，利用不完美神经算子提供结构先验，通过多项式延拓外推解分支并蒸馏至PINN，解决其收敛到错误解盆地的问题，在多个非线性PDE上验证了有效性。
AI 中文摘要
神经算子能够廉价地评估参数化偏微分方程，但在其训练分布之外性能急剧下降。物理信息神经网络避免了对标注数据的依赖，但其优化可能具有“盆地脆弱性”：当控制残差存在多个解时，从头训练的PINN可能收敛到物理上不正确的状态，尽管实现了较小的残差。我们表明这些失败模式可以联合解决：一个不完美的神经算子提供了将PINN置于正确解盆地所需的结构先验，而PDE残差则将解精化到超出算子精度的水平。我们引入了一个三阶段框架，该框架冻结物理信息神经算子的空间基，使用多项式延拓先验将其解分支外推到分布外参数，并将所得场蒸馏到一个全新的PINN中。神经算子无需在目标处准确；它传递解分支信息，而PINN中的PDE残差最小化控制收敛。我们在三个非线性PDE上评估该框架：1D粘性Burgers方程、2D稳态Allen-Cahn方程（接近叉形分岔）以及2D稳态顶盖驱动空腔流。对于Allen-Cahn，平凡解精确满足PDE残差，标准PINN坍缩到平凡的零分支，而从粗略外推算子蒸馏则恢复了与有限差分参考匹配的非平凡分支。对于顶盖驱动空腔，外推到雷诺数Re=3200加速了向正确物理状态的收敛，使用比近期文献基线更少的参数和优化步骤实现了有竞争力的精度。这些结果确立了一个简单原则：神经算子无需准确预测解即可有用；它只需识别正确的盆地，PINN优化即可从中恢复解。
英文摘要
Neural operators evaluate parametric partial differential equations cheaply but degrade sharply outside their training distribution. Physics-informed neural networks avoid dependence on labeled data, yet their optimization can be basin-fragile: when the governing residual admits multiple solutions, a PINN trained from scratch may converge to a physically incorrect state despite achieving a small residual. We show that these failure modes can be addressed jointly: an imperfect NO provides the structural prior needed to place a PINN in the correct solution basin, while the PDE residual refines the solution beyond the operator's accuracy. We introduce a three-stage framework that freezes the spatial basis of a physics-informed NO, extrapolates its solution branch to an out-of-distribution parameter using a polynomial continuation prior, and distills the resulting field into a fresh PINN. The NO need not be accurate at the target; it transfers solution-branch information, while PDE residual minimization in the PINN governs convergence. We evaluate the framework on three nonlinear PDEs: 1D viscous Burgers, 2D steady Allen-Cahn near a pitchfork bifurcation, and 2D steady lid-driven cavity flow. For Allen-Cahn, where the trivial solution satisfies the PDE residual exactly, a standard PINN collapses to the trivial zero branch, whereas distillation from the crude extrapolated operator recovers the non-trivial branch that matches the finite-difference reference. For the lid-driven cavity, extrapolating to a Reynolds number of Re = 3200 accelerates convergence to the correct physical state, achieving competitive accuracy using fewer parameters and optimization steps than recent literature baselines. These results establish a simple principle: an NO need not accurately predict the solution to be useful; it only needs to identify the correct basin from which PINN optimization can recover it.
