Learning Singularity-Encoded Green's Functions with Application to Iterative Methods

2025-09-15

AI Summary

To address the numerical challenges in computing Green’s functions of high-dimensional elliptic PDEs—exacerbated by the curse of dimensionality and intrinsic singularities—this paper proposes an unsupervised singularity encoding learning framework. The method embeds a prior Green’s function estimate as an augmented variable into a higher-dimensional space and leverages the spectral bias of deep neural networks to accurately model singularities without labeled data. By integrating dimensional augmentation, neural parameterization, and projection-based reduction, it constructs an efficient surrogate model. This surrogate seamlessly integrates into iterative solvers, serving simultaneously as a preconditioner and a hybrid solver component. Experiments on 2D–4D problems demonstrate significant convergence acceleration—by several-fold—and precise resolution of singularity structure. The approach establishes a scalable, fully unsupervised paradigm for high-dimensional Green’s function modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Green's function provides an inherent connection between theoretical analysis and numerical methods for elliptic partial differential equations, and general absence of its closed-form expression necessitates surrogate modeling to guide the design of effective solvers. Unfortunately, numerical computation of Green's function remains challenging due to its doubled dimensionality and intrinsic singularity. In this paper, we present a novel singularity-encoded learning approach to resolve these problems in an unsupervised fashion. Our method embeds the Green's function within a one-order higher-dimensional space by encoding its prior estimate as an augmented variable, followed by a neural network parametrization to manage the increased dimensionality. By projecting the trained neural network solution back onto the original domain, our deep surrogate model exploits its spectral bias to accelerate conventional iterative schemes, serving either as a preconditioner or as part of a hybrid solver. The effectiveness of our proposed method is empirically verified through numerical experiments with two and four dimensional Green's functions, achieving satisfactory resolution of singularities and acceleration of iterative solvers.

Problem

• Learning Green's functions with encoded singularities for elliptic PDEs
• Resolving doubled dimensionality and singularity challenges in computation
• Accelerating iterative solvers through deep surrogate modeling

Innovation

• Singularity-encoded learning in augmented space
• Neural network parametrization for dimensionality management
• Deep surrogate model accelerates iterative solvers