A method for computing inverse parametric PDE problems with random-weight neural networks

Abstract

We present a method for computing the inverse parameters and the solution field to inverse parametric partial differential equations (PDE) based on randomized neural networks. This extends the local extreme learning machine technique originally developed for forward PDEs to inverse problems. We develop three algorithms for training the neural network to solve the inverse PDE problem. The first algorithm (termed NLLSQ) determines the inverse parameters and the trainable network parameters all together by the nonlinear least squares method with perturbations (NLLSQ-perturb). The second algorithm (termed VarPro-F1) eliminates the inverse parameters from the overall problem by variable projection to attain a reduced problem about the trainable network parameters only. It solves the reduced problem first by the NLLSQ-perturb algorithm for the trainable network parameters, and then computes the inverse parameters by the linear least squares method. The third algorithm (termed VarPro-F2) eliminates the trainable network parameters from the overall problem by variable projection to attain a reduced problem about the inverse parameters only. It solves the reduced problem for the inverse parameters first, and then computes the trainable network parameters afterwards. VarPro-F1 and VarPro-F2 are reciprocal to each other in some sense. The presented method produces accurate results for inverse PDE problems, as shown by the numerical examples herein. For noise-free data, the errors of the inverse parameters and the solution field decrease exponentially as the number of collocation points or the number of trainable network parameters increases, and can reach a level close to the machine accuracy. For noisy data, the accuracy degrades compared with the case of noise-free data, but the method remains quite accurate. The presented method has been compared with the physics-informed neural network method.