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.

翻译：我们提出一种基于神经网络的元学习方法，用于高效求解偏微分方程（PDE）问题。该方法旨在元学习如何解决多种PDE问题，并利用所学知识求解新给定的PDE问题。我们利用神经网络将PDE问题编码为问题表示，其中控制方程由偏导数的多项式函数系数表示，边界条件由一组点-条件对表示。我们将该问题表示作为预测解的神经网络的输入，从而能够通过神经网络的前向传播过程高效预测问题特定的解，而无需更新模型参数。为训练模型，我们基于物理信息神经网络框架最小化当适应某一PDE问题时的期望误差，从而能在解未知的情况下评估误差。实验证明，我们提出的方法在预测PDE问题的解方面优于现有方法。