Multiscale Neural Networks for Approximating Green's Functions

TL;DR

This work addresses solving PDEs via Green's functions by introducing multiscale neural networks (MSNNs) that separate the learning of low-regularity near singularities from smooth global structure. Grounded in multiscale Barron-space theory, the authors derive approximation bounds showing that MSNNs can achieve accurate Green's-function representations with fewer and more moderate parameters than single-scale nets. Numerical experiments on Poisson-type problems demonstrate faster convergence and improved accuracy, and the method is demonstrated for solving PDEs by numerically integrating against learned Green's functions. The approach offers a scalable, efficient pathway to operator-learning in PDEs and opens avenues for fast solvers and high-dimensional Green's-function learning with potential extensions to more scales and preconditioning strategies.

Abstract

Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this paper, we address these challenges by leveraging multiscale NNs to learn Green's functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training.

Figures (11)

• Figure 1: The design of the multiscale NN, incorporating $\bm{x} - \bm{y}$ as an additional feature.
• Figure 1: FEM Green's function approximations at $y=0.95$ using $\varepsilon = 1.0, 0.1, 0.01, 0.001$ for the 1-D problem \ref{['eq:model problem']} with $c=0$ on $\Omega=[0,1]$.
• Figure 2: DD, dataset, and quadrature points in a rectangular domain. Left: DD using a coarse mesh $\mathcal{T}_C$ and $4$-way graph partitioning. Middle: dataset with $\bm{y}$ from one subdomain (the red star), $\bm{x}$ close to $\bm{y}$ (the blue squares), $\bm{x}$ uniformly sampled in $\Omega$ (the green pentagon), and $\bm{x}$ on boundary (the purple dots). Right: Quadrature points (the black dots) used in a fine mesh $\mathcal{T}_F$.
• Figure 2: FEM Green's function approximations using $\varepsilon = 1.0, 0.1, 0.01$ for the 1-D problem \ref{['eq:model problem']} with $c=0$ on $\Omega=[0,1]$.
• Figure 3: The overall procedure of solving PDEs by the multiscale NN learning approach for Green's function. A large-scale network in each subdomain is first trained, and both the large- and small-scale NNs in the subdomain are trained together. The PDE solution is computed by numerical integration with the learned Green's function and the right-hand-side function of the PDE. \newlabelfig:diagram0

Theorems & Definitions (22)

• Definition 1: Rademacher complexity anthony1999neural
• Lemma 1: shalev2014understanding
• Definition 2
• Definition 3
• Lemma 2: kim2019green
• Proposition 1
• Proof 1
• Remark 1
• Proposition 2
• Proof 2