Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh
This addresses the computational bottleneck of solving many related PDEs in fields like physics and engineering, offering a faster, mesh-free approach that is incremental over existing learned solvers.
The authors tackled the challenge of solving related partial differential equations (PDEs) quickly by developing a meta-learning method that learns initializations for neural networks to minimize PDE residuals on novel tasks, achieving up to an order of magnitude faster solutions than baseline finite element analysis with equivalent accuracy for nonlinear Poisson and hyper-elasticity equations.
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.