We present a method for computing the inverse parameters and the solution
field to inverse parametric PDEs 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 (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 (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 (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 a sense. The presented method produces accurate results for inverse
PDE problems, as shown by the numerical examples herein. For noise-free data,
the errors for 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.Comment: 40 pages, 8 figures, 34 table