NetGP: A Hybrid Framework Combining Genetic Programming and Deep Reinforcement Learning for PDE Solutions

Partial differential equations (PDEs) are fundamental in various scientific and engineering fields. Methods based on symbolic regression to solve PDEs have gained attention due to their inherent interpretability. However, existing symbolic regression methods rely solely on genetic programming (GP) during the search process, which presents opportunities for improvement in both precision and stability. We introduce a novel framework, itemd NetGP, which enhances symbolic regression for PDEs in three key aspects. First, NetGP employs prefix notation arrays to represent symbolic expressions, simplifying the evaluation process. Second, to improve the stability of the evolutionary process, deep reinforcement learning is integrated to generate new individuals. Additionally, a novel operator is proposed to avoid the generation of invalid expressions during crossover and mutation of array-based individuals. Empirical evaluations across five types of PDEs demonstrate that NetGP achieves outstanding accuracy and stability in solving these PDEs. The code can be found at https://github.com/grassdeerdeer/NetGP.