A high-precision PINN method for solving a generalized Burgers–Fisher equation

Abstract

This article presents a high-precision physics informed neural network (HpPINN) method to solve a class of Burgers–Fisher equation. The main difficulty is how to optimize the model to obtain highly accurate prediction solutions. For this purpose, we firstly introduce a new weighting function (WF), and use strategies such as local adaptive activation function (LAAF), training point resampler and combinatorial optimizers to train the model. The experimental results show that the accuracy of the training and test errors can reach $10^{-10}$, and the relative $L2$ error can achieve $10^{-6}$. Compared to previous results, the accuracy of the HpPINN method improves 10 times nearly. Second, we discuss the computational performance of the combinatorial optimizer in different cases. Assume that the maximum number of iterations is fixed at 10 000 times, this research shows that when the number of iterations $i_A$ with the Adam optimizer is 500 and the number of iterations $i_L$ with the L-BFGS optimizer is 9500, the training error and the relative $L2$ error are smaller. In addition, we also find that when $i_A$ gradually decreases from 1500 to 500, the corresponding relative $L2$ error also gradually decreases, then we may infer that the relative $L2$ error is locally monotonically increasing with respect to $i_A$. Finally, we discuss the HpPINN method with WF and without WF respectively. By control group, we verify that the validity of the HpPINN method mainly depends on the newly introduced weighting function.