MORL4PDEs: Data-driven discovery of PDEs based on multi-objective optimization and reinforcement learning

Extracting fundamental behavior patterns or governing equations from data can deepen our understanding and insights into physical systems, it will lead to the better control and application of these systems in science and engineering. Currently, most existing methods in extracting governing equations require a candidate function term library in advance, which results in the limitations of those learned equations. To overcome this problem in this paper we propose a new method for data-driven discovery of parsimonious partial differential equations (PDEs) by utilizing symbolic regression based on multi-objective optimization and reinforcement learning, we call the MORL4PDEs in short. Specifically, neural network agent aims to generate the pre-order traversal sequence of a binary tree, and through which we can obtain the expression for each PDE. Then the resulting individuals can be used as the initial population in the multi-objective genetic algorithm to ensure the accuracy and parsimony of the equations, whose plausibility is guaranteed according to the constraints generated from the rules of PDEs. Meanwhile, the neural network is optimized through reinforcement learning with the final expression of each PDE as a reward. Finally, several experiments are conduct to demonstrate the effectiveness of the proposed method, and the results show MORL4PDEs can identify governing equations in different dynamic systems, including those PDEs with complex forms and high-order derivatives.