Fundamental solution neural networks for inverse acoustic Cauchy problems in complex domains

This paper presents a fundamental solution neural networks (FSNNs) method, driven by physics-informed neural networks (PINNs), for solving inverse Cauchy problems in acoustic analysis in both two- and three-dimensional domains. Such problems aim to reconstruct unknown boundary data from partially prescribed boundary conditions and are typically ill-posed and highly sensitive to perturbations. For inverse problems, the conventional MFS typically relies on the empirical selection of source points, which often leads to numerical instability. Although various strategies have been proposed to address this, these approaches tend to either increase computational complexity or compromise overall efficiency. To address the limitations of the MFS, PINNs are incorporated into the MFS framework. In this approach, the unknown quantities are parameterized by a neural network, and a physics-informed loss function is constructed based on the measurable boundary conditions. Guided by physical constraints, the model can adaptively adjust the source point distribution, thereby improving the stability and accuracy of the solution without manual intervention. The proposed method preserves the mesh-free nature, ease of implementation, and semi-analytical characteristics of the MFS, while significantly enhancing its capability for solving inverse problems. Numerical results demonstrate that the proposed method achieves high accuracy, stability, and robustness for problems in complex geometries.