  We propose a neural network-based meta-learning method to efficiently solve
partial differential equation (PDE) problems. The proposed method is designed
to meta-learn how to solve a wide variety of PDE problems, and uses the
knowledge for solving newly given PDE problems. We encode a PDE problem into a
problem representation using neural networks, where governing equations are
represented by coefficients of a polynomial function of partial derivatives,
and boundary conditions are represented by a set of point-condition pairs. We
use the problem representation as an input of a neural network for predicting
solutions, which enables us to efficiently predict problem-specific solutions
by the forwarding process of the neural network without updating model
parameters. To train our model, we minimize the expected error when adapted to
a PDE problem based on the physics-informed neural network framework, by which
we can evaluate the error even when solutions are unknown. We demonstrate that
our proposed method outperforms existing methods in predicting solutions of PDE
problems.
