Discretization of parameter identification in PDEs using neural networks. (1st December 2022)

• Record Type: Journal Article
• Authors: Kaltenbacher, Barbara; Nguyen, Tram Thi Ngoc
• Is Part Of: Inverse problems. Volume 38:Number 12(2022)
• Publication Date: 2022-12-01
• Subjects: neural networks -- discretization of regularization -- parameter identification -- nonlinear PDEs -- Tikhonov regularization -- Landweber iteration -- unsupervised learning; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac9c25
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent partial differential equation model. The nonlinearity is approximated by a neural network (NN), and needs to be determined alongside other unknown physical parameters and the unknown state. Hence, it is not possible to construct input–output data pairs to perform a supervised training process. Proposing an all-at-once approach, we bypass the need for training data and recover all the unknowns simultaneously. In the general case, the approximation via a NN can be realized as a discretization scheme, and the training with noisy data can be viewed as an ill-posed inverse problem. Therefore, we study discretization of regularization in terms of Tikhonov and projected Landweber methods for discretization of inverse problems, and prove convergence when the discretization error (network approximation error) and the noise level tend to zero.