In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks. The numerical scheme is based on the standard output least-squares formulation where the Galerkin finite element method (FEM) is employed to approximate the state and neural networks (NNs) act as a smoothness prior to approximate the unknown diffusion coefficient. A projection operation is applied to the NN approximation in order to preserve the physical box constraint on the unknown coefficient. The hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias / approximation properties of NNs. We derive \textsl{a priori} error estimates in the standard $L^2(\Omega)$ norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data. The error bounds depend explicitly on the noise level, regularization parameter and discretization parameters (e.g., spatial mesh size, time step size, and depth, upper bound and number of nonzero parameters of NNs). We also provide extensive numerical experiments, indicating that the hybrid method is very robust for large noise when compared with the pure FEM approximation.

翻译：本文研究利用神经网络数值识别椭圆与抛物问题中的扩散系数。该数值方案基于标准输出最小二乘公式，其中采用Galerkin有限元法（FEM）逼近状态，而神经网络（NNs）作为光滑性先验逼近未知扩散系数。为保持未知系数的物理箱型约束，对神经网络近似施加投影操作。该混合方法兼具FEM的严格数学基础与NNs的归纳偏置/逼近特性。我们在一个可对广泛问题数据类验证的正性条件下，推导出数值重构在标准$L^2(\Omega)$范数下的\textsl{先验}误差估计。误差界显式依赖于噪声水平、正则化参数及离散化参数（如空间网格尺寸、时间步长、深度、上界及神经网络非零参数数量）。我们还提供了大量数值实验，表明与纯FEM逼近相比，该混合方法对强噪声具有极强鲁棒性。