用数据参数化与学习有限元算子，解决非线性PDE反问题。
Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators
- 通过自编码器将边界数据压缩到低维隐空间进行参数化。
- 训练算子网络映射边界系数到有限元解，实现逆问题求解。
- 结合数据拟合优化，适用于有集体观测数据的复杂反问题。
我们研究一个涉及重构具有未知边界条件的非线性偏微分方程（PDE）解的反问题。已知大量典型解的边界观测数据（集体数据）及某一特定解的体测量值。为利用集体数据，首先通过本征正交分解（POD）在线性展开中压缩边界数据；接着利用自编码器识别展开系数中的非线性低维结构，实现数据在低维隐空间中的参数化；随后训练算子网络，将表示边界数据的展开系数映射至有限元（FE）解；最后将自编码器解码器与算子网络连接，通过在隐空间中优化数据拟合项来求解逆问题。我们分析了线性情形下的稳定有限元方法，并建立了$H^1$-范数下的最优误差估计。非线性问题通过数值实验验证了该方法的有效性。
原文摘要 · Abstract (English)
We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of boundary observations for typical solutions (collective data) and a bulk measurement of a specific realization. To leverage this collective data, we first compress the boundary data using proper orthogonal decomposition (POD) in a linear expansion. Next, we identify a possible nonlinear low-dimensional structure in the expansion coefficients using an autoencoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train an operator network to map the expansion coefficients representing the boundary data to the finite element (FE) solution of the PDE. Finally, we connect the autoencoder's decoder to the operator network which enables us to solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method (FEM) in the linear setting and establish an optimal error estimate in the $H^1$-norm. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.
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