Improved Training of Physics-Informed Neural Networks with Model Ensembles

Katsiaryna Haitsiukevich, Alexander Ilin

Keywords: Label propagation, Model ensembles, Partial differential equations, Physics-informed neural networks

Abstract: 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 propose a training algorithm that starts approximation of the PDE solution in the neighborhood of initial conditions and gradually expands the solution domain based on agreement of an ensemble of PINNs. PINNs in the ensemble find similar solutions in the vicinity of points with targets (e.g., observed data or initial conditions) while the found solutions may substantially differ farther away from the observations. 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.

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