Solving Differential Equations using Data-Driven Adaptive Numerical Method

Abstract: Accuracy and efficiency have always been of great concern when solving differential equations. One approach to improve accuracy is by introducing a neural network, whose role is to learn the local truncation error (LTE) of a numerical method. This LTE can then be used as a correction term to reduce the error that the numerical method causes when solving the differential equations. However, using this method to improve accuracy comes at a high computational cost. To reduce the cost and improve efficiency, one can consider using adaptive step sizes to compute the solution of the differential equation. In this thesis, the LTE computed by the neural network is used both as a correction term, and to construct an adaptive step size method. This thesis focuses in particular on implementing an adaptive scheme for a data-driven Euler method, namely the deep Euler method (DEM), and introduces the deep implicit Euler method using feed-forward neural networks. It is shown that the correction term improves the accuracy of the solutions. Moreover, the analysis of the computation time and the number of forward passes in the network shows that the adaptive step size method effectively reduces the computational cost caused by DEM.