Efficient Discovery Method for Partial Differential Equations Based on PDE-RLGA

Partial differential equations serve as essential tools for describing the evolution of complex systems in physics, engineering, and other fields. Traditional modeling methods often encounter difficulties such as local optima in structure search and weak noise resistance in complex scenarios. To address these issues, this paper proposes an efficient discovery method for partial differential equations by integrating reinforcement learning, genetic algorithm, sparse identification of nonlinear dynamics, and Bayesian optimization. The method first performs a global evolutionary search of candidate structures for partial differential equations through an improved genetic algorithm. Then, it applies a sparse identification algorithm to remove redundant terms by sparse regression. Finally, it uses a Bayesian optimization to optimize key hyperparameters such as regularization strength, crossover probability, and mutation probability, while quantifying uncertainty. Experimental results show that the proposed method performs well on the Tsinghua partial differential equation operator large model dataset and the fluid mechanics dataset. The structure accuracy of the discovered equations reaches 98.32%, and the training time for discovering equations is only 172.7 ms (excluding the time needed to train RL model), which is much faster than other compared algorithms. These results demonstrate that the collaboration among multiple algorithms enables accurate and efficient discovery of partial differential equations under complex environments, providing technical support for system modeling and rule mining in multiple fields.