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

Abstract: The Green's function, serving as a kernel function that delineates the interaction relationships of physical quantities within a field, holds significant research implications across various disciplines. It forms the foundational basis for the renowned Biot-Savart formula in fluid dynamics, the theoretical solution of the pressure Poisson equation, and et al. Despite their importance, the theoretical derivation of the Green's function is both time-consuming and labor-intensive. In this study, we employed DISCOVER, an advanced symbolic regression method leveraging symbolic binary trees and reinforcement learning, to identify unknown Green's functions for several elementary partial differential operators, including Laplace operators, Helmholtz operators, and second-order differential operators with jump conditions. The Laplace and Helmholtz operators are particularly vital for resolving the pressure Poisson equation, while second-order differential operators with jump conditions are essential for analyzing multiphase flows and shock waves. By incorporating physical hard constraints, specifically symmetry properties inherent to these self-adjoint operators, we significantly enhanced the performance of the DISCOVER framework, potentially doubling its efficacy. Notably, the Green's functions discovered for the Laplace and Helmholtz operators precisely matched the true Green's functions. Furthermore, for operators without known exact Green's functions, such as the periodic Helmholtz operator and second-order differential operators with jump conditions, we identified potential Green's functions with solution error on the order of 10^-10. This application of symbolic regression to the discovery of Green's functions represents a pivotal advancement in leveraging artificial intelligence to accelerate scientific discoveries, particularly in fluid dynamics and related fields.