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.

翻译：格林函数作为描述场中物理量相互作用关系的核函数，在多个学科领域具有重要的研究意义。它是流体力学中著名的毕奥-萨伐尔公式、压力泊松方程理论解等的基础。尽管格林函数至关重要，但其理论推导过程耗时费力。本研究采用DISCOVER方法——一种基于符号二叉树和强化学习的先进符号回归技术，成功识别了若干基本偏微分算子（包括拉普拉斯算子、亥姆霍兹算子以及具有跳跃条件的二阶微分算子）的未知格林函数。拉普拉斯算子和亥姆霍兹算子对于求解压力泊松方程尤为关键，而具有跳跃条件的二阶微分算子则是分析多相流和激波问题的基础。通过引入物理硬约束（特别是这些自伴算子固有的对称性），我们显著提升了DISCOVER框架的性能，其效能可能提高一倍。值得注意的是，针对拉普拉斯算子和亥姆霍兹算子发现的格林函数与真实格林函数完全吻合。此外，对于不存在已知精确格林函数的算子（如周期亥姆霍兹算子和具有跳跃条件的二阶微分算子），我们发现了解误差量级为10^(-10)的潜在格林函数。这项将符号回归应用于格林函数发现的研究，标志着利用人工智能加速科学发现（特别是在流体力学及相关领域）取得了关键性进展。