Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEs

TL;DR

This paper introduces a meta-learning approach using physics-informed neural networks to quickly and accurately solve new PDE problems by learning from a variety of existing PDE solutions.

Contribution

The paper presents a novel meta-learning framework that encodes PDE problems into representations and predicts solutions without parameter updates, improving efficiency and accuracy.

Findings

Outperforms existing methods in PDE solution prediction

Enables rapid problem-specific solutions without retraining

Uses physics-informed loss to evaluate solutions without known answers

Abstract

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.