  Partial differential equations (PDEs) are often computationally challenging
to solve, and in many settings many related PDEs must be be solved either at
every timestep or for a variety of candidate boundary conditions, parameters,
or geometric domains. We present a meta-learning based method which learns to
rapidly solve problems from a distribution of related PDEs. We use
meta-learning (MAML and LEAP) to identify initializations for a neural network
representation of the PDE solution such that a residual of the PDE can be
quickly minimized on a novel task. We apply our meta-solving approach to a
nonlinear Poisson's equation, 1D Burgers' equation, and hyperelasticity
equations with varying parameters, geometries, and boundary conditions. The
resulting Meta-PDE method finds qualitatively accurate solutions to most
problems within a few gradient steps; for the nonlinear Poisson and
hyper-elasticity equation this results in an intermediate accuracy
approximation up to an order of magnitude faster than a baseline finite element
analysis (FEA) solver with equivalent accuracy. In comparison to other learned
solvers and surrogate models, this meta-learning approach can be trained
without supervision from expensive ground-truth data, does not require a mesh,
and can even be used when the geometry and topology varies between tasks.
