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

TL;DR

This work tackles the challenge of explicitly deriving Green's functions by marrying symbolic regression with physics-informed, symmetry-based hard constraints. The authors implement DISCOVER to search open-form expression trees for $G(x,y)$, enforcing self-adjointness through a symmetry constraint and optimizing constants via BFGS within a structure-aware reinforcement-learning framework. Validation on Laplace and Helmholtz operators yields exact Green's functions and exact-like reconstructions for unknown cases (periodic Helmholtz and jump conditions) with errors down to $\mathcal{O}(10^{-10})$, while outperforming PySR and rational neural networks in both interpretability and eigenstructure fidelity. The approach demonstrates robustness to noise and offers a pathway to efficient analytic solutions for PDEs relevant to Poisson-type problems and multiphase flows, with future work aimed at extending to two- and three-dimensional settings.

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.

Figures (12)

• Figure 1: Overview of the mining of Green's function based on DISCOVER framework du2024discover. (a) Establishing open-form candidate library. (b) Structure-aware LSTM agent for generating possible expressions with imposing constraints. (c) Acquire expression from pre-order traversal. (d) Rebuild the symbolic tree based on symmetrical hard constraint. (e) Reward function based on physics-informed constraints in Eq. (\ref{['eq6']}).
• Figure 2: Schematic of learning Green's function based on DISCOVER du2024discover.
• Figure 3: Schematic diagram of binary tree representation and transformation method based on preorder traversal algorithm.
• Figure 4: Schematic diagram of symmetry hard constraint.
• Figure 5: Laplace operator. (a) Exact Green's function. (b) Learned Green’s functions by DISCOVER and (c) PySR and (d) rational neural network (black box models). (e) Solution comparisons with exact $u$, $u$ calculated by DISCOVER, $u$ calculated by PySR, and $u$ calculated by rational neural network (black box models).