Improved Training of Physics-Informed Neural Networks with Model Ensembles

Learning the solution of partial differential equations (PDEs) with a neural network (known in the literature as a physics-informed neural network, PINN) is an attractive alternative to traditional solvers due to its elegancy, greater flexibility and the ease of incorporating observed data. However, training PINNs is notoriously difficult in practice. One problem is the existence of multiple simple (but wrong) solutions which are attractive for PINNs when the solution interval is too large. In this paper, we propose to expand the solution interval gradually to make the PINN converge to the correct solution. To find a good schedule for the solution interval expansion, we train an ensemble of PINNs. The idea is that all ensemble members converge to the same solution in the vicinity of observed data (e.g., initial conditions) while they may be pulled towards different wrong solutions farther away from the observations. Therefore, we use the ensemble agreement as the criterion for including new points for computing the loss derived from PDEs. We show experimentally that the proposed method can improve the accuracy of the found solution.