MultiPINNs: Using an Ensemble of Physics-Informed Neural Networks to Generalize to Unseen Equilibria

Physics-Informed Neural Networks (PINNs) [1], are a flexible family of function approximators which have shown promise as PDE solvers. PINNs can be used to solve boundary value problems, fitting a function on the boundary of a domain (subject to constraints ensuring that predictions satisfy a set of diff. eqns on the interior). We demonstrate that PINNs can reproduce plasma equilibria calculated by TAE's multi-fluid equilibrium code (LR_eqMI) [2]. We train a separate PINN on each equilibrium, showing they can solve our PDEs of interest to comparable levels of numerical error. Once trained, evaluating a PINN is much faster than a run of the equilibrium code. Additionally, we show that once we have a trained ensemble of PINNs, we can generalize to equilibria outside of our training dataset. This ensemble method increases the utility of PINNs, turning them from single-use tools for a particular equilibrium (which find one PDE solution) to a joint solver which finds a solution to an entire class of PDEs. We demonstrate applications of this technique to simulation hyperparameter estimation and surrogate modelling of plasma physics. 

[1]Raissi et al. "Physics informed deep learning (part I)" arXiv preprint arXiv:1711.10561 (2017).

[2] L.Galeotti et al. Phys. Plasmas, 18, 082509 (2011)