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