Numerically Informed Convolutional Operator Network with Subproblem Decomposition for Poisson Equations

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.

💡 Research Summary

The paper introduces NICON (Numerically Informed Convolutional Operator Network), a framework that tightly integrates classical numerical discretizations—finite difference (FD) and finite element (FE)—with convolutional neural operator architectures for solving parametrized Poisson equations. Two concrete models are presented: FD‑CON, which uses a 5‑point (or 9‑point) finite‑difference stencil to construct a residual loss ‖−Δₕuₕ−f‖², and FE‑CON, which builds the weak‑form residual ‖A w−b‖² based on the FEM stiffness matrix and load vector. Both share a U‑Net‑style CNN backbone, but their loss functions directly encode the respective numerical schemes, forcing the network to satisfy consistency, stability, and convergence properties inherent to the underlying discretizations.

A rigorous error analysis is carried out in the H¹‑seminorm. For FD‑CON the authors prove an error bound of order O(h²)·‖L‖, where h is the mesh size and ‖L‖ is the L²‑norm of the training residual. For FE‑CON a bound of order O(h)·‖L‖ is derived, mirroring classic FEM convergence theory. Crucially, the analysis shows that if the training loss decays proportionally to a power of h, the total approximation error inherits the same convergence rate. This insight leads to a “loss‑mesh proportional training strategy”: learning‑rate, batch‑size, and regularisation parameters are scaled with h so that, for a fixed number of training samples, the residual is driven down to O(hⁿ). The strategy guarantees optimal convergence under grid refinement without increasing the dataset size.

To further improve data efficiency, the authors exploit the linearity of the Poisson equation and split the original problem into two sub‑problems: one with pure Dirichlet boundary conditions and another with pure Neumann conditions. Separate FD‑CON/FE‑CON networks are trained on each sub‑problem, and the final solution is obtained by a linear combination of the two predictions. This decomposition reduces the functional complexity each network must learn, leading to a substantial reduction in required training samples and improved generalisation, especially when boundary conditions vary arbitrarily.

Extensive experiments on a 2‑D square domain with mixed (Dirichlet–Neumann) boundaries and random source terms validate the theory. Grid sizes from 16×16 to 128×128 are tested. Both FD‑CON and FE‑CON achieve errors that follow the predicted convergence rates; FE‑CON reaches an L² error of ~1.2 × 10⁻³ on the finest grid. Memory consumption is roughly an order of magnitude lower than a traditional FEM implementation, and inference time is 2–5× faster because the network replaces the costly linear solve with a single forward pass. The sub‑problem decomposition further cuts error by about 30 % for the same training budget. A comparison table shows that, while FEM requires storing the full inverse stiffness matrix (e.g., >1 GB for 128×128), FE‑CON needs only a few megabytes.

The paper acknowledges limitations: the current study is confined to 2‑D polygonal domains and linear Poisson equations; extending the approach to nonlinear PDEs, complex geometries, or multiphysics problems will require new residual formulations and possibly adaptive mesh strategies. Moreover, the loss‑mesh proportional strategy assumes a sufficient number of training samples; in extremely low‑data regimes additional regularisation or physics‑informed priors may be needed.

In summary, NICON demonstrates that embedding numerical discretization knowledge into the loss function of convolutional operator networks yields provable convergence under grid refinement, dramatically reduces data and memory requirements, and offers a practical pathway to high‑fidelity, fast inference for parametrized PDEs. Future work will explore higher‑dimensional domains, nonlinear operators, and adaptive sub‑problem decompositions.