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.

我们提出了一种基于神经网络的元学习方法，用于高效解决偏微分方程（PDE）问题。该方法通过元学习来解决各种各样的 PDE 问题，并将这些知识用于解决新的 PDE 问题。我们使用神经网络将 PDE 问题编码成问题表示，其中，控制方程由偏导数的多项式函数的系数表示，边界条件由一组点条件对表示。我们将问题表示作为神经网络的输入来预测解决方案，通过神经网络的前向过程，我们能够高效地预测特定问题的解决方案，而无需更新模型参数。为了训练我们的模型，我们最小化在基于物理知识的神经网络框架中适应 PDE 问题时的预期误差，通过这种方式，即使解决方案未知，我们也能评估误差。我们证明了我们提出的方法在预测 PDE 问题的解决方案方面优于现有方法。