Meta-learning PINN loss functions

We propose a meta-learning technique for offline discovery of physics-informed neural network (PINN) loss functions. extend earlier works on meta-learning, and develop gradient-based algorithm addressing diverse task distributions based parametrized partial differential equations (PDEs) that are solved with PINNs. Furthermore, new theory we identify two desirable properties meta-learned losses in PINN problems, which enforce by proposing regularization method or using specific parametrization the function. In computational examples, employed at test time regression PDE distributions. Our results indicate significant performance improvement can be achieved shared-among-tasks offline-learned function even out-of-distribution meta-testing. this case, solve tasks do not belong to distribution used meta-training, also employ architectures different from architecture meta-training. To better understand capabilities limitations proposed method, consider various parametrizations describe design options how they may affect performance.