Deep Reinforcement Learning-Based Symbolic Regression for PDE Discovery Using Spatio-Temporal Rewards
Xizhe Wang, Hongbo Zhao, Qianchuan Zhao, Benben Jiang · 2024
The discovery of partial differential equations (PDEs) directly from data has significant implications for automation and manufacturing fields. This approach leverages advanced computational techniques and data-driven methodologies to identify the underlying mathematical models that govern the physical processes in various systems. In this article, a deep symbolic regression framework is developed for PDE discovery which is enhanced by reinforcement learning with spatio-temporal rewards. Within this framework, a dense encoder-decoder network is proposed to adeptly extract the information of temporal dynamics related to PDEs from the data, and a refined reinforcement learning reward shaping technique is further put forward that incorporates the extracted dynamics, specifically designed for PDE symbolic regression tasks. An example of Cahn-Hilliard equation, which is widely used to describe the phase separation process in multi-component mixture systems, is used to validate the effectiveness of the proposed approach. The experimental results show that the proposed deep symbolic regression approach outperforms the baselines in identifying PDEs, achieving higher accuracy with less data. Additionally, we provide a thorough analysis on how adjustable parameters affect the PDE discovery performance, further verifying the effectiveness of the proposed approach.