Discover Green's function based on symbolic regression with physical hard constraints

The Green's function, a kernel function that describes interactions within a field, is crucial across various disciplines. It underpins key formulas such as the Biot–Savart law in fluid dynamics, solutions to the pressure Poisson equation, and closure models for turbulent Reynolds stresses. Despite its importance, deriving Green's functions theoretically is often labor-intensive and time-consuming. In this study, we present an enhanced symbolic regression method that integrates physical constraints and reinforcement learning within the “Deep Identification of Symbolically Concise Open-form Partial Differential Equations Via Enhanced Reinforcement-learning” (DISCOVER) framework. This method is designed to identify unknown Green's functions for differential operators, including the Laplace, Helmholtz, and second-order operators with jump conditions. The Laplace and Helmholtz operators are vital for solving the pressure Poisson equation, while second-order operators with jump conditions are crucial for analyzing multiphase flows and shock waves. By integrating physical constraints, particularly symmetry properties of self-adjoint operators, we significantly improved the model's performance. Our method accurately identified Green's functions for the Laplace and high-frequency Helmholtz operators and achieved a solution error on the order of 10−10 for periodic Helmholtz operators and second-order operators with jump conditions, all within 30 min of computation. The accuracy of our method outperforms several state-of-the-art baseline methods, including Symbolic Physics Learner and Reinforcement Symbolic Regression Machine. This demonstrates the method's capability in learning high frequency, complex periodic, and jump functions. This advancement in symbolic regression highlights the potential of artificial intelligence to accelerate scientific discovery and enhance the modeling of complex physical systems, particularly in fluid dynamics.