SHoP: A Deep Learning Framework for Solving High-order Partial Differential Equations
This work addresses a bottleneck in computational science for researchers and engineers who need accurate solutions to high-order PDEs, representing a novel method rather than an incremental improvement.
The authors tackled the problem of solving high-order partial differential equations (PDEs) by proposing SHoP, a deep learning framework that efficiently and accurately computes high-order derivatives and provides explicit solutions through Taylor series expansion, achieving effective results across four high-order PDEs with different dimensions.
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.