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Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions

Ziang Chen, Liqiang Huang

TL;DR

This work tackles solving high-dimensional elliptic PDEs on the unit cube with homogeneous boundary conditions by encoding the PDE coefficients into Barron spaces and showing that the weak solutions can be approximated by shallow neural networks in the $H^1$ norm. The authors combine Sobolev gradient flows with Barron-norm estimates to prove exponential convergence of the flow and to bound the Barron norm growth, then use Barron-based approximation theory to convert these limits into explicit two-layer networks with cosine or ReLU activations. They provide explicit neuron-count bounds that scale as $k=O(d^{C|\log\varepsilon|})$ and hold for both Dirichlet and Neumann problems, thereby bypassing the curse of dimensionality under suitable coefficient regularity. The results extend prior Barron-function PDE theory to more general second-order elliptic equations and boundary conditions, offering a principled pathway for efficient high-dimensional PDE approximations in practice.

Abstract

We study the approximation complexity of high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube, within the framework of Barron spaces. Under the assumption that the coefficients belong to suitably defined Barron spaces, we prove that the solution can be efficiently approximated by two-layer neural networks, circumventing the curse of dimensionality. Our results demonstrate the expressive power of shallow networks in capturing high-dimensional PDE solutions under appropriate structural assumptions.

Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions

TL;DR

This work tackles solving high-dimensional elliptic PDEs on the unit cube with homogeneous boundary conditions by encoding the PDE coefficients into Barron spaces and showing that the weak solutions can be approximated by shallow neural networks in the norm. The authors combine Sobolev gradient flows with Barron-norm estimates to prove exponential convergence of the flow and to bound the Barron norm growth, then use Barron-based approximation theory to convert these limits into explicit two-layer networks with cosine or ReLU activations. They provide explicit neuron-count bounds that scale as and hold for both Dirichlet and Neumann problems, thereby bypassing the curse of dimensionality under suitable coefficient regularity. The results extend prior Barron-function PDE theory to more general second-order elliptic equations and boundary conditions, offering a principled pathway for efficient high-dimensional PDE approximations in practice.

Abstract

We study the approximation complexity of high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube, within the framework of Barron spaces. Under the assumption that the coefficients belong to suitably defined Barron spaces, we prove that the solution can be efficiently approximated by two-layer neural networks, circumventing the curse of dimensionality. Our results demonstrate the expressive power of shallow networks in capturing high-dimensional PDE solutions under appropriate structural assumptions.

Paper Structure

This paper contains 19 sections, 19 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Works
  4. Organization
  5. Main Results
  6. Barron Spaces
  7. Main Results
  8. Proof of the Main Results
  9. Step 1: Sobolev Gradient Flow
  10. The Construction
  11. Convergence Analysis
  12. Step 2: Barron Norm Estimates
  13. Sobolev Gradient Flow via the Inverse Elliptic Operator
  14. Analysis of the Barron Norm
  15. Step 3: Neural Network Approximation
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose the coefficients $A(x)$, $c(x)$, and $f(x)$ in equations Diri and Newu are suitable Barron functions. Then the weak solution $u^*$ to each equation can be approximated in the $H^1(\Omega)$-norm by a two-layer neural network of the form using at most $\mathcal{O}( d^{C|\log \varepsilon|})$ neurons, where $C>0$ depends only on the Barron norms of the PDE coefficients.

Theorems & Definitions (34)

  • Theorem 1.1: Main Results, Informal Version
  • Theorem 2.2
  • Definition 2.3: Barron Norms and Barron Functions
  • Proposition 2.4
  • Theorem 2.7: Main Result 1
  • Remark 2.8: Parameter Explanation
  • Theorem 2.9: Main Result 2
  • Proposition 3.1: Variational Characterization of Weak Solutions
  • proof
  • Theorem 3.2
  • ...and 24 more