Fundamental solution neural networks for solving inverse Cauchy problems for the Laplace and biharmonic equations

This paper proposes a novel fundamental solution neural networks method (FSNNs) to solve inverse Cauchy problems, which combines the method of fundamental solutions (MFS) with the physics-informed neural networks (PINNs). To optimize the distribution of source points, the method starts by partitioning the interval into equal segments, determined by the number of the source points. The coordinate system is situated at the center of the computational domain. The resulting angles as network inputs for the FSNNs and the intermediate variable as outputs, which is subsequently substituted into a length function to obtain the final length. The coordinates of the source points are then determined, and the MFS is employed to approximate the numerical solutions. The loss function is formulated based on the boundary conditions on the accessible boundary, and the training is employed to optimize the network parameters in the FSNNs and source point intensities in the MFS. The introduction of the PINNs overcomes the challenge of source point selection in the MFS and effectively addresses the ill-posedness of inverse problems. In summary, the proposed scheme is a machine learning-based semi-analytical meshless method which is simple, accurate and easily implemented, making it highly suitable for the numerical solution of inverse problems. Four numerical experiments, including the Laplace and biharmonic equations, validate the effectiveness and accuracy of the proposed FSNNs.