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Randomized Radial Basis Function Neural Network for Solving Multiscale Elliptic Equations

Yuhang Wu, Ziyuan Liu, Wenjun Sun, Xu Qian

TL;DR

The paper introduces RRNN, a mesh-free approach for linear multiscale elliptic PDEs that combines domain decomposition with randomized Gaussian RBFs and a least-squares training of the output layer. By formulating subdomain weak problems and enforcing inter-domain continuity via stochastic boundary collocation, RRNN yields a global linear system that can be solved efficiently. Across one- and two-dimensional tests, RRNN outperforms gradient-descent-based neural PDE solvers in both accuracy and training time, and often surpasses LocELM while maintaining comparable costs to ELM-based methods. The method is particularly advantageous at small scale ratios $\varepsilon$, where capturing high-frequency features is most challenging, and offers a promising direction for scalable, accurate multiscale PDE solvers using neural-inspired architectures.

Abstract

To overcome these obstacles and improve computational accuracy and efficiency, this paper presents the Randomized Radial Basis Function Neural Network (RRNN), an innovative approach explicitly crafted for solving multiscale elliptic equations. The RRNN method commences by decomposing the computational domain into non-overlapping subdomains. Within each subdomain, the solution to the localized subproblem is approximated by a randomized radial basis function neural network with a Gaussian kernel. This network is distinguished by the random assignment of width and center coefficients for its activation functions, thereby rendering the training process focused solely on determining the weight coefficients of the output layer. For each subproblem, similar to the Petrov-Galerkin finite element method, a linear system will be formulated on the foundation of a weak formulation. Subsequently, a selection of collocation points is stochastically sampled at the boundaries of the subdomain, ensuring satisfying $C^0$ and $C^1$ continuity and boundary conditions to couple these localized solutions. The network is ultimately trained using the least squares method to ascertain the output layer weights. To validate the RRNN method's effectiveness, an extensive array of numerical experiments has been executed and the results demonstrate that the proposed method can improve the accuracy and efficiency well.

Randomized Radial Basis Function Neural Network for Solving Multiscale Elliptic Equations

TL;DR

The paper introduces RRNN, a mesh-free approach for linear multiscale elliptic PDEs that combines domain decomposition with randomized Gaussian RBFs and a least-squares training of the output layer. By formulating subdomain weak problems and enforcing inter-domain continuity via stochastic boundary collocation, RRNN yields a global linear system that can be solved efficiently. Across one- and two-dimensional tests, RRNN outperforms gradient-descent-based neural PDE solvers in both accuracy and training time, and often surpasses LocELM while maintaining comparable costs to ELM-based methods. The method is particularly advantageous at small scale ratios , where capturing high-frequency features is most challenging, and offers a promising direction for scalable, accurate multiscale PDE solvers using neural-inspired architectures.

Abstract

To overcome these obstacles and improve computational accuracy and efficiency, this paper presents the Randomized Radial Basis Function Neural Network (RRNN), an innovative approach explicitly crafted for solving multiscale elliptic equations. The RRNN method commences by decomposing the computational domain into non-overlapping subdomains. Within each subdomain, the solution to the localized subproblem is approximated by a randomized radial basis function neural network with a Gaussian kernel. This network is distinguished by the random assignment of width and center coefficients for its activation functions, thereby rendering the training process focused solely on determining the weight coefficients of the output layer. For each subproblem, similar to the Petrov-Galerkin finite element method, a linear system will be formulated on the foundation of a weak formulation. Subsequently, a selection of collocation points is stochastically sampled at the boundaries of the subdomain, ensuring satisfying and continuity and boundary conditions to couple these localized solutions. The network is ultimately trained using the least squares method to ascertain the output layer weights. To validate the RRNN method's effectiveness, an extensive array of numerical experiments has been executed and the results demonstrate that the proposed method can improve the accuracy and efficiency well.
Paper Structure (9 sections, 1 theorem, 26 equations, 11 figures, 9 tables, 1 algorithm)

This paper contains 9 sections, 1 theorem, 26 equations, 11 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Randomized radial basis function neural network
  3. Radial basis function neural network
  4. Randomized radial basis function neural network
  5. Method
  6. Numerical examples
  7. One dimensional
  8. Two dimensional
  9. Conclusion

Key Result

Theorem 2.1

Given any small positive value $\epsilon>0$, the activation function $\rho : \mathbb{R}^n \rightarrow \mathbb{R}$ which is infinitely differentiable in any interval, and $Q$ arbitrary distinct samples $(\boldsymbol{x}_i,\boldsymbol{u}_i) \in \mathbb{R}^n \times \mathbb{R}^m$, there exists $J \le Q$

Figures (11)

  • Figure 1: The structure of randomized radial basis function neural network for solving problems \ref{['1.1']}.
  • Figure 2: The coefficient $A^\varepsilon(x)$ and solution and their derivative of Example 4.2 in Section 4 when $\varepsilon=0.005.$
  • Figure 3: Variation of the coefficient $A^\varepsilon(x)$ after domain decomposition and normalization.
  • Figure 4: The coefficient $A^\varepsilon(x)$, numerical solution obtained by RRNN and reference solution, and absolute point-wise error for Example 4.1: (a)(b)(c) for $\varepsilon= 0.5$, (d)(e)(f) for $\varepsilon = 0.005$.
  • Figure 5: Comparison between RRNN and LocELM with different $Q$ for Example 4.2.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Remark 2.1
  • Theorem 2.1