  Learning the solution of partial differential equations (PDEs) with a neural
network is an attractive alternative to traditional solvers due to its
elegance, greater flexibility and the ease of incorporating observed data.
However, training such physics-informed neural networks (PINNs) is notoriously
difficult in practice since PINNs often converge to wrong solutions. In this
paper, we address this problem by training an ensemble of PINNs. Our approach
is motivated by the observation that individual PINN models find similar
solutions in the vicinity of points with targets (e.g., observed data or
initial conditions) while their solutions may substantially differ farther away
from such points. Therefore, we propose to use the ensemble agreement as the
criterion for gradual expansion of the solution interval, that is including new
points for computing the loss derived from differential equations. Due to the
flexibility of the domain expansion, our algorithm can easily incorporate
measurements in arbitrary locations. In contrast to the existing PINN
algorithms with time-adaptive strategies, the proposed algorithm does not need
a pre-defined schedule of interval expansion and it treats time and space
equally. We experimentally show that the proposed algorithm can stabilize PINN
training and yield performance competitive to the recent variants of PINNs
trained with time adaptation.
