Genetic Programming Symbolic Regression with Simplification-Pruning Operator for Solving Differential Equations

Differential equations (DEs) are important mathematical models for describing natural phenomena and engineering problems. Finding analytical solutions for DEs has theoretical and practical benefits. However, traditional methods for finding analytical solutions only work for some special forms of DEs, such as separable variables or transformable to ordinary differential equations. For general nonlinear DEs, analytical solutions are often hard to obtain. The current popular method based on neural networks requires a lot of data to train the network and only gives approximate solutions with errors and instability. It is also a black-box model that is not interpretable. To obtain analytical solutions for DEs, this paper proposes a symbolic regression algorithm based on genetic programming with the simplification-pruning operator (SP-GPSR). This method introduces a new operator that can simplify the individual expressions in the population and randomly remove some structures in the formulas. Moreover, this method uses multiple fitness functions that consider the accuracy of the analytic solution satisfying the sampled data and the differential equations. In addition, this algorithm also uses a hybrid optimization technique to improve search efficiency and convergence speed. This paper conducts experiments on two typical classes of DEs. The results show that the proposed method can effectively find analytical solutions for DEs with high accuracy and simplicity.