线性化物理信息神经网络（Linearized PINN）与预训练非线性层
Linearized PINN with pretrained nonlinear layers
- University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
- Pacific Northwest National Laboratory（太平洋西北国家实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出线性化PINN（lPINN），离线预训练连续神经基函数，在线仅优化线性层，在多个方程上降低误差并大幅缩短推理时间。
AI中文摘要：
我们提出了一种线性化物理信息神经网络（lPINN），这是一种用于正问题和反问题微分方程的降阶神经基函数方法。在离线阶段，lPINN从一组数值解中学习与算子兼容的连续神经基函数。这些基函数通过自动微分可微，并使用解数据以及导数信息或物理残差进行预训练。对于每个新的问题实例，基函数被冻结，通过最小化控制方程残差以及适用的初始、边界、正则化和观测项来获得解。与代理模型和算子学习方法不同，训练数据在离线阶段定义试验空间，而实例特定的解通过强制执行控制物理在线计算。相对于原始PINN，lPINN在离线阶段预训练非线性隐藏层表示，并仅在线推断最终线性层。我们在对流扩散方程、Burgers方程和非线性摆方程的正问题和反问题上评估了lPINN。与原始PINN相比，lPINN实现了更低的解和参数误差，同时将在线推断时间减少了大约一个到三个数量级以上，最大的收益通常出现在残差或测量数据有限的情况下。跨分辨率实验表明，学习到的连续表示可以在更细的网格上评估，无需重新训练，且精度几乎不变。
英文摘要：
We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.
