Meta-Auto-Decoder: A Meta-Learning Based Reduced Order Model for Solving Parametric Partial Differential Equations

TL;DR

Meta-Auto-Decoder (MAD) introduces a neural network-based nonlinear trial manifold for parametric PDEs, enabling mesh-free, unsupervised pre-training and rapid adaptation, outperforming existing deep learning methods in speed and accuracy.

Contribution

MAD leverages meta-learning to construct a nonlinear trial manifold for parametric PDEs, allowing mesh-free, unsupervised pre-training and fast adaptation.

Findings

Faster convergence than existing deep learning methods

Maintains high accuracy in solution approximation

Effective in handling heterogeneous PDE parameters

Abstract

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc. Typical reduced order modeling techniques accelarate solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the offline stage. These methods often need a predefined mesh as well as a series of precomputed solution snapshots, andmay struggle to balance between efficiency and accuracy due to the limitation of the linear ansatz. Utilizing the nonlinear representation of neural networks, we propose Meta-Auto-Decoder (MAD) to construct a nonlinear trial manifold, whose best possible performance is measured theoretically by the decoder width. Based on the meta-learning concept, the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage. Fast adaptation to new (possibly heterogeneous) PDE parameters is enabled by searching on this trial manifold, and optionally fine-tuning the trial manifold at the same time. Extensive numerical experiments show that the MAD method exhibits faster convergence speed without losing accuracy than other deep learning-based methods.