  Physics-informed neural networks (PINNs) are emerging as popular mesh-free
solvers for partial differential equations (PDEs). Recent extensions decompose
the domain, apply different PINNs to solve the problem in each subdomain, and
stitch the subdomains at the interface. Thereby, they can further alleviate the
problem complexity, reduce the computational cost, and allow parallelization.
However, the performance of multi-domain PINNs is sensitive to the choice of
the interface conditions. While quite a few conditions have been proposed,
there is no suggestion about how to select the conditions according to specific
problems. To address this gap, we propose META Learning of Interface Conditions
(METALIC), a simple, efficient yet powerful approach to dynamically determine
appropriate interface conditions for solving a family of parametric PDEs.
Specifically, we develop two contextual multi-arm bandit (MAB) models. The
first one applies to the entire training course, and online updates a Gaussian
process (GP) reward that given the PDE parameters and interface conditions
predicts the performance. We prove a sub-linear regret bound for both UCB and
Thompson sampling, which in theory guarantees the effectiveness of our MAB. The
second one partitions the training into two stages, one is the stochastic phase
and the other deterministic phase; we update a GP reward for each phase to
enable different condition selections at the two stages to further bolster the
flexibility and performance. We have shown the advantage of METALIC on four
bench-mark PDE families.
