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 auto-encoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train a neural network to map the latent variables representing the boundary data to the solution of the PDE. Finally, we solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method in the linear setting and establish optimal error estimates in the $H^1$ and $L^2$-norms. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.

翻译：我们考虑一个反问题，其涉及在边界条件未知的情况下重构非线性偏微分方程（PDE）的解。我们并非直接获得边界数据，而是拥有一个包含典型解边界观测值（集体数据）的大型数据集，以及一个特定实现的体测量数据。为利用此集体数据，我们首先通过线性展开中的本征正交分解（POD）对边界数据进行压缩。接着，我们利用自编码器识别展开系数中可能存在的非线性低维结构，从而在更低维的隐空间中对数据集进行参数化。随后，我们训练一个神经网络，将表示边界数据的隐变量映射到PDE的解。最后，我们通过在隐空间上优化数据拟合项来求解该反问题。我们分析了线性情形下的底层稳定化有限元方法，并在$H^1$和$L^2$范数下建立了最优误差估计。随后对非线性问题进行了数值研究，验证了我们方法的有效性。