On the recovery of two function-valued coefficients in the Helmholtz equation for inverse scattering problems via neural networks On the recovery of two function-valued coefficients in the Helmholtz

• Published in: Advances in computational mathematics Vol. 51; no. 1
• Main Author: Zhou, Zehui
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2025
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Visualization; Approximation; Two function-valued coefficients; Inverse scattering problem; Neural network; Generalization
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-025-10225-z

Recently, deep neural networks (DNNs) have become powerful tools for solving inverse scattering problems. However, the approximation and generalization rates of DNNs for solving these problems remain largely under-explored. In this work, we introduce two types of combined DNNs (uncompressed and compressed) to reconstruct two function-valued coefficients in the Helmholtz equation for inverse scattering problems from the scattering data at two different frequencies. An analysis of the approximation and generalization capabilities of the proposed neural networks for simulating the regularized pseudo-inverses of the linearized forward operators in direct scattering problems is provided. The results show that, with sufficient training data and parameters, the proposed neural networks can effectively approximate the inverse process with desirable generalization. Preliminary numerical results show the feasibility of the proposed neural networks for recovering two types of isotropic inhomogeneous media. Furthermore, the trained neural network is capable of reconstructing the isotropic representation of certain types of anisotropic media.