A novel artificial neural network method is proposed for solving Cauchy inverse problems. Using multiple-layers network as an approximation we present a non-mesh discretization to solve the problems. The existence and convergence are shown to establish the well-posedness of neural network approximations for the Cauchy inverse problems. Numerical results on 2D to 8D cases show that compared to finite element method, the neural network approach easier extends to high dimensional case. The stability and accuracy of the proposed network approach are investigated by the experiments with noisy boundary and irregular computational domain. Our studies conclude that the neural network method alleviates the influence of noise and it is observed that networks with wider and deeper hidden layers could lead to better approximation.

提出了一种新的人工神经网络方法来解决柯西逆问题。使用多层网络作为近似值，我们提出了一种非网格离散化来解决这些问题。证明存在性和收敛性来建立柯西逆问题的神经网络近似的适定性。2D 到 8D 案例的数值结果表明，与有限元方法相比，神经网络方法更容易扩展到高维案例。通过噪声边界和不规则计算域的实验研究了所提出的网络方法的稳定性和准确性。我们的研究得出结论，神经网络方法减轻了噪声的影响，并且观察到具有更宽和更深隐藏层的网络可以导致更好的近似。