Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh

• URL: http://arxiv.org/abs/2211.01604v1
• Date: Thu, 3 Nov 2022 06:17:52 GMT
• Title: Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh
• Authors: Tian Qin, Alex Beatson, Deniz Oktay, Nick McGreivy, Ryan P. Adams
• Abstract summary: Partial differential equations (PDEs) are often computationally challenging to solve. We present a meta-learning based method which learns to rapidly solve problems from a distribution of related PDEs.
• Score: 24.572840023107574
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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