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A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples

Jun Hu, Pengzhan Jin

TL;DR

The paper introduces a MIONet-based hybrid iterative framework that integrates neural operators with classical solvers to accelerate PDE solutions. It provides convergence conditions, spectral insights, and rate bounds that quantify how model inference errors interact with discretization, and demonstrates that periodically injecting MIONet corrections can dramatically reduce iteration counts. The approach is validated on 1D and 2D Poisson equations, including cases with inhomogeneous boundaries, achieving substantial speedups (up to ~30×) while remaining meshless and compatible with multigrid strategies. The work offers a principled path toward practical, fast PDE solvers that leverage learned operators for low-frequency error components.

Abstract

We propose a hybrid iterative method based on MIONet for PDEs, which combines the traditional numerical iterative solver and the recent powerful machine learning method of neural operator, and further systematically analyze its theoretical properties, including the convergence condition, the spectral behavior, as well as the convergence rate, in terms of the errors of the discretization and the model inference. We show the theoretical results for the frequently-used smoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We give an upper bound of the convergence rate of the hybrid method w.r.t. the model correction period, which indicates a minimum point to make the hybrid iteration converge fastest. Several numerical examples including the hybrid Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are presented to verify our theoretical results, and also reflect an excellent acceleration effect. As a meshless acceleration method, it is provided with enormous potentials for practice applications.

A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples

TL;DR

The paper introduces a MIONet-based hybrid iterative framework that integrates neural operators with classical solvers to accelerate PDE solutions. It provides convergence conditions, spectral insights, and rate bounds that quantify how model inference errors interact with discretization, and demonstrates that periodically injecting MIONet corrections can dramatically reduce iteration counts. The approach is validated on 1D and 2D Poisson equations, including cases with inhomogeneous boundaries, achieving substantial speedups (up to ~30×) while remaining meshless and compatible with multigrid strategies. The work offers a principled path toward practical, fast PDE solvers that leverage learned operators for low-frequency error components.

Abstract

We propose a hybrid iterative method based on MIONet for PDEs, which combines the traditional numerical iterative solver and the recent powerful machine learning method of neural operator, and further systematically analyze its theoretical properties, including the convergence condition, the spectral behavior, as well as the convergence rate, in terms of the errors of the discretization and the model inference. We show the theoretical results for the frequently-used smoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We give an upper bound of the convergence rate of the hybrid method w.r.t. the model correction period, which indicates a minimum point to make the hybrid iteration converge fastest. Several numerical examples including the hybrid Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are presented to verify our theoretical results, and also reflect an excellent acceleration effect. As a meshless acceleration method, it is provided with enormous potentials for practice applications.
Paper Structure (13 sections, 7 theorems, 129 equations, 10 figures, 5 tables, 1 algorithm)

This paper contains 13 sections, 7 theorems, 129 equations, 10 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Operator regression with MIONet
  3. A hybrid iterative method
  4. Theoretical analysis
  5. Convergence condition
  6. Error of model inference
  7. Spectral analysis
  8. Discussion for complicated cases
  9. Numerical experiments
  10. One-dimensional Poisson equation
  11. Two-dimensional Poisson equation
  12. Inhomogeneous boundary condition
  13. Conclusions

Key Result

Theorem 1

Suppose that $X_1,...,X_n,Y$ are Banach spaces, $K_i\subset X_i$ are compact sets, and $X_i$ have a Schauder basis with canonical projections $P_q^i=\psi_q^i\circ\varphi_q^i$ (truncate the previous q terms). Assume that $\mathcal{G}:K_1\times\cdots\times K_n\to Y$ is a continuous operator, then for In addition, if $\mathcal{G}$ is linear with respect to $v_i$, then linear $g_j^i$ is sufficient.

Figures (10)

  • Figure 1: An illustration of the architecture of MIONet for general operators. All the branch nets and the trunk net have the same output dimension, whose outputs are merged together via the Hadamard product and then a summation.
  • Figure 2: An illustration of the architecture of MIONet for the 2-d Poisson equation.
  • Figure 3: An example of ${\rm Rate}(M)$. In this case $\eta_1=0.999, \eta_2=0.5, \epsilon=0.1, R=10$.
  • Figure 4: An example of one prediction of MIONet for the 1-d Poisson equation.
  • Figure 5: Convergence rate of the Richardson-MIONet method versus the correction period $M$. The tendency is consistent with the theoretical results as Eq. (\ref{['eq:convergence_rate']}) and Figure \ref{['fig:rate_M']}.
  • ...and 5 more figures

Theorems & Definitions (12)

  • Definition 1: Schauder basis
  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2: Universal approximation theorem for MIONet
  • Theorem 3
  • Corollary 1
  • Corollary 2
  • proof
  • Theorem 4
  • ...and 2 more