Hybrid neural-network FEM approximation of diffusion coefficient in elliptic and parabolic Problems
Hybrid neural-network FEM approximation of diffusion coefficient in elliptic and parabolic Problems
复制标题
椭圆和抛物线问题中扩散系数的混合神经网络 FEM 近似
DOI：
10.1093/imanum/drad073
复制
发表时间：
2023
影响因子：
2.1
通讯作者：
Cen S
中科院分区：
文献类型：
--
作者：
Cen S
在这项工作中，我们研究了使用神经网络（NN）对椭圆和抛物线问题中扩散系数的数值识别。该数值方案基于标准输出最小二乘公式，其中采用伽辽金有限元法 (FEM) 来近似状态，并且神经网络充当近似未知扩散系数之前的平滑度。对 NN 近似应用投影运算，以保留对未知系数的物理框约束。混合方法既具有有限元的严格数学基础，又具有神经网络的归纳偏差/近似特性。我们在可对大量问题数据进行验证的正性条件下，以数值重建的标准范数导出先验误差估计。误差界限明确取决于噪声水平、正则化参数和离散化参数（例如，空间网格大小、时间步长大小和深度、神经网络非零参数的上限和数量）。我们还提供了广泛的数值实验，表明与纯 FEM 近似相比，混合方法对于大噪声非常稳健。
In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks (NNs). 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 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 derivea priorierror estimates in the standardnorm 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.