Solving partial differential equations (PDEs) has been a fundamental problem in computational science and of wide applications for both scientific and engineering research. Due to its universal approximation property, neural network is widely used to approximate the solutions of PDEs. However, existing works are incapable of solving high-order PDEs due to insufficient calculation accuracy of higher-order derivatives, and the final network is a black box without explicit explanation. To address these issues, we propose a deep learning framework to solve high-order PDEs, named SHoP. Specifically, we derive the high-order derivative rule for neural network, to get the derivatives quickly and accurately; moreover, we expand the network into a Taylor series, providing an explicit solution for the PDEs. We conduct experimental validations four high-order PDEs with different dimensions, showing that we can solve high-order PDEs efficiently and accurately.

翻译：摘要：求解偏微分方程一直是计算科学中的基础问题，并在科学和工程研究中具有广泛应用。由于神经网络的通用近似特性，它被广泛用于逼近偏微分方程的解。然而，现有方法因高阶导数计算精度不足而无法求解高阶偏微分方程，且最终网络是缺乏明确解释的“黑箱”。针对这些问题，我们提出一种名为SHoP的深度学习框架以求解高阶偏微分方程。具体而言，我们推导了神经网络的高阶导数规则，从而快速准确地获取导数；此外，我们将网络展开为泰勒级数，为偏微分方程提供显式解。我们在四个不同维度的高阶偏微分方程上进行了实验验证，结果表明该方法能高效、准确地求解高阶偏微分方程。