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Numerically Informed Convolutional Operator Network with Subproblem Decomposition for Poisson Equations

Kyoungjin Jung, Jae Yong Lee, Dongwook Shin

TL;DR

This work addresses the challenge of learning PDE solution operators with reliable convergence properties under grid refinement by integrating classical numerical discretizations with neural operator learning. The authors introduce NICON, featuring two residual-based, physics-informed networks—FD-CON (finite difference) and FE-CON (finite element)—coupled with a domain-to-image mapping and a subproblem decomposition u = u1 + u2 that exploits linearity to improve training efficiency and generalization. They derive $H^1$-seminorm error estimates linking convergence to the decay rate of the training losses and propose strategies to achieve optimal rates; extensive experiments on Poisson and Helmholtz problems validate the theory, showing competitive accuracy on fine grids and improved generalization via decomposition. The framework demonstrates data-free training, scalability to complex geometries, and robustness to oscillatory solutions, offering a practical alternative to traditional solvers for parametric or repeated-PDE tasks. This work advances physics-informed operator learning by making it numerically principled and computationally tractable on high-resolution grids.

Abstract

Neural operators have shown remarkable performance in approximating solutions of partial differential equations. However, their convergence behavior under grid refinement is still not well understood from the viewpoint of numerical analysis. In this work, we propose a numerically informed convolutional operator network, called NICON, that explicitly couples classical finite difference and finite element methods with operator learning through residual-based training loss functions. We introduce two types of networks, FD-CON and FE-CON, which use residual-based loss functions derived from the corresponding numerical methods. We derive error estimates for FD-CON and FE-CON using finite difference and finite element analysis. These estimates show a direct relation between the convergence behavior and the decay rate of the training loss. From these analyses, we establish training strategies that guarantee optimal convergence rates under grid refinement. Several numerical experiments are presented to validate the theoretical results and show performance on fine grids.

Numerically Informed Convolutional Operator Network with Subproblem Decomposition for Poisson Equations

TL;DR

This work addresses the challenge of learning PDE solution operators with reliable convergence properties under grid refinement by integrating classical numerical discretizations with neural operator learning. The authors introduce NICON, featuring two residual-based, physics-informed networks—FD-CON (finite difference) and FE-CON (finite element)—coupled with a domain-to-image mapping and a subproblem decomposition u = u1 + u2 that exploits linearity to improve training efficiency and generalization. They derive -seminorm error estimates linking convergence to the decay rate of the training losses and propose strategies to achieve optimal rates; extensive experiments on Poisson and Helmholtz problems validate the theory, showing competitive accuracy on fine grids and improved generalization via decomposition. The framework demonstrates data-free training, scalability to complex geometries, and robustness to oscillatory solutions, offering a practical alternative to traditional solvers for parametric or repeated-PDE tasks. This work advances physics-informed operator learning by making it numerically principled and computationally tractable on high-resolution grids.

Abstract

Neural operators have shown remarkable performance in approximating solutions of partial differential equations. However, their convergence behavior under grid refinement is still not well understood from the viewpoint of numerical analysis. In this work, we propose a numerically informed convolutional operator network, called NICON, that explicitly couples classical finite difference and finite element methods with operator learning through residual-based training loss functions. We introduce two types of networks, FD-CON and FE-CON, which use residual-based loss functions derived from the corresponding numerical methods. We derive error estimates for FD-CON and FE-CON using finite difference and finite element analysis. These estimates show a direct relation between the convergence behavior and the decay rate of the training loss. From these analyses, we establish training strategies that guarantee optimal convergence rates under grid refinement. Several numerical experiments are presented to validate the theoretical results and show performance on fine grids.
Paper Structure (19 sections, 5 theorems, 58 equations, 15 figures, 1 table)

This paper contains 19 sections, 5 theorems, 58 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical Numerical Discretizations
  4. Machine-Learning Approaches: Related Works
  5. Remarks on Inference Efficiency
  6. Methodology
  7. Subproblem 1
  8. Subproblem 2
  9. FD-CON: Finite Difference Convolutional Operator Network
  10. FE-CON: Finite Element Convolutional Operator Network
  11. Error Analysis
  12. Numerical Experiments
  13. Network Architecture and Experimental Setup
  14. Performance Evaluation for Poisson Problem
  15. Numerical Verification of Error Estimates and Generalization
  16. ...and 4 more sections

Key Result

Theorem 3.1

Let $u \in H^2(\Omega) \cap H^1_D(\Omega)$ be the weak solution, and let $\hat{\boldsymbol{u}}$ be the FD-CON prediction. Then there exist positive constants $C_1$, $C_2$, and $C_3$, independent of $h$, such that where $\boldsymbol{u} = [\boldsymbol{u}_I^T, \boldsymbol{u}_B^T]^T$ is the reordered grid function grid_fn corresponding to the weak solution, i.e., $u_j = u(\eta_j), \ \eta_j \in \mathc

Figures (15)

  • Figure 1: Schematic overview of the Numerically Informed Convolutional Operator Networks (NICON).
  • Figure 2: Illustration of the mesh generation process based on image-domain mapping.
  • Figure 3: Input channels (a)--(c) and the predicted solution (d) for FD-CON. The source term $f$ is provided over the entire domain, while the Dirichlet and Neumann boundary conditions are encoded as separate channels. The predicted solution is post-processed to enforce the Dirichlet boundary condition.
  • Figure 4: Masks for source term $M_f$, Dirichlet boundary $M_D$, and Neumann boundary $M_N$
  • Figure 5: Resolution-dependent U-Net architecture for NICON.
  • ...and 10 more figures

Theorems & Definitions (13)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • ...and 3 more