用微扰法+一次迁移学习，快速求解非线性偏微分方程。
One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs
- 将非线性偏微分方程拆解为线性子问题，用多头物理信息神经网络求解。
- 新问题实例适应时间低于0.2秒，误差约1e-3，媲美经典求解器。
- 适合需快速响应的非线性偏微分方程场景，如实时仿真与参数敏感分析。
我们提出一种结合微扰理论与一次迁移学习的框架，用于求解带有多项式项的非线性偏微分方程（PDE）。通过将非线性PDE分解为一系列线性子问题，利用多头物理信息神经网络（Multi-Head PINN）高效求解。一旦线性算子的隐式表示被学习，即可在无需重新训练的情况下，以闭式形式获得不同扰动、源项或边界/初值条件下新PDE实例的解。我们在KPP-Fisher方程和波动方程上验证该方法，在不到0.2秒内完成新问题适应，误差维持在1e-3量级；其精度与经典求解器相当，但转移速度显著更快。敏感性分析表明，误差随ε和多项式阶数呈可预测增长，明确了该方法的有效适用范围。主要贡献包括：(i) 将一次迁移学习从非线性常微分方程拓展至偏微分方程；(ii) 推导出新实例适应的闭式解；(iii) 在典型非线性偏微分方程上验证了高精度与高效性。最后，展望了向导数相关非线性及高维偏微分方程的扩展。
原文摘要 · Abstract (English)
We propose a framework for solving nonlinear partial differential equations (PDEs) by combining perturbation theory with one-shot transfer learning in Physics-Informed Neural Networks (PINNs). Nonlinear PDEs with polynomial terms are decomposed into a sequence of linear subproblems, which are efficiently solved using a Multi-Head PINN. Once the latent representation of the linear operator is learned, solutions to new PDE instances with varying perturbations, forcing terms, or boundary/initial conditions can be obtained in closed form without retraining. We validate the method on KPP-Fisher and wave equations, achieving errors on the order of 1e-3 while adapting to new problem instances in under 0.2 seconds; comparable accuracy to classical solvers but with faster transfer. Sensitivity analyses show predictable error growth with epsilon and polynomial degree, clarifying the method's effective regime. Our contributions are: (i) extending one-shot transfer learning from nonlinear ODEs to PDEs, (ii) deriving a closed-form solution for adapting to new PDE instances, and (iii) demonstrating accuracy and efficiency on canonical nonlinear PDEs. We conclude by outlining extensions to derivative-dependent nonlinearities and higher-dimensional PDEs.
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