Interpretable Solutions for Multi-Physics PDEs Using T-NNGP

Abstract

Multiphysics simulation aims to predict and understand interactions between multiple physical phenomena, aiding in comprehending natural processes and guiding engineering design. The system of Partial Differential Equations (PDEs) is crucial for representing these physical fields, and solving these PDEs is fundamental to such simulations. However, current methods primarily yield numerical outputs, limiting interpretability and generalizability. We introduce T-NNGP, a hybrid genetic programming algorithm that integrates traditional numerical methods with deep learning to derive approximate symbolic expressions for multiple unknown functions within a system of PDEs. T-NNGP initially obtains numerical solutions using traditional methods, then generates candidate symbolic expressions via deep reinforcement learning, and finally optimizes these expressions using genetic programming. Furthermore, a universal decoupling strategy guides the search direction and addresses coupling problems, thereby accelerating the search process. Experimental results on three types of PDEs demonstrate that our method can reliably obtain human-understandable symbolic expressions that fit both the PDEs and the numerical solutions from traditional methods. This work advances multiphysics simulation by enhancing our ability to derive approximate symbolic solutions for PDEs, thereby improving our understanding of complex physical phenomena.