The growing body of research shows how to replace classical partial differential equation (PDE) integrators with neural networks. The popular strategy is to generate the input-output pairs with a PDE solver, train the neural network in the regression setting, and use the trained model as a cheap surrogate for the solver. The bottleneck in this scheme is the number of expensive queries of a PDE solver needed to generate the dataset. To alleviate the problem, we propose a computationally cheap augmentation strategy based on general covariance and simple random coordinate transformations. Our approach relies on the fact that physical laws are independent of the coordinate choice, so the change in the coordinate system preserves the type of a parametric PDE and only changes PDE's data (e.g., initial conditions, diffusion coefficient). For tried neural networks and partial differential equations, proposed augmentation improves test error by 23% on average. The worst observed result is a 17% increase in test error for multilayer perceptron, and the best case is a 80% decrease for dilated residual network.

翻译：越来越多的研究表明如何用神经网络替代经典的偏微分方程（PDE）数值求解器。常见策略是：利用PDE求解器生成输入-输出对，在回归框架下训练神经网络，并将训练好的模型作为求解器的廉价替代品。该方案的瓶颈在于生成数据集所需的大量昂贵PDE求解器调用。为缓解此问题，我们提出一种基于广义协方差和简单随机坐标变换的计算廉价型数据增强策略。该方法基于物理定律与坐标选择无关的原理，因此坐标系变化保持参数化PDE的类型不变，仅改变PDE的数据（如初始条件、扩散系数）。对于所测试的神经网络与偏微分方程，所提增强方法平均降低测试误差23%。最差情形下多层感知机的测试误差增加17%，最佳情形下扩张残差网络的测试误差降低80%。