Stability and Discretization Error of State Space Model Neural Operators

2026-05-17

AI Summary

This work addresses the unclear relationship between continuous theory and discrete implementation in neural operators for solving partial differential equations, particularly concerning stability and discretization error. For the first time, rigorous discretization error bounds are established for State-Space Neural Operators (SS-NOs) and Fourier Neural Operators (FNOs). By leveraging functional analysis, the regularity of solutions is explicitly linked to input discretization, and Input-to-State Stability (ISS) theory is introduced to quantify how discretization affects stability in the continuous domain. Numerical experiments on one- and two-dimensional benchmark problems validate the tightness of the derived theoretical bounds, demonstrating that SS-NOs exhibit both robustness and numerical stability across varying resolutions.

Abstract

Neural operators have emerged as a powerful, discretization-invariant framework for solving partial differential equations (PDEs). Although established approaches like the Deep Operator Network (DeepONet) have successfully achieved universal approximation for operators, and architectures such as Fourier Neural Operators (FNOs) have shown algebraic convergence rates, a precise theoretical connection between the continuous theory and its discrete numerical implementation remains a challenge. Specifically, the relationship between the continuous formulation and the discrete numerical stability has yet to be fully explored. In this paper, we address this gap by establishing theoretical guarantees for the discretization error and stability of neural operator approximation schemes. We prove analytical bounds that link solution regularity to input discretization, providing a formal quantification of neural operator accuracy under real-world numerical constraints. We derive these bounds to the specific cases of State Space Model-based Neural Operators (SS-NOs) and FNOs, thus providing a new discretization error theorem for these models. Additionally, through an input-to-state stability (ISS) analysis, we formally assess the impact of discretization on the stability of SS-NOs results obtained in the continuous domain. Our empirical experiments on 1D and 2D benchmarks validate our theoretical bounds and show the robustness of SS-NOs under varying resolutions.

Problem

• neural operators
• discretization error
• numerical stability
• state space models
• partial differential equations

Innovation

• discretization error
• neural operators
• state space models
• numerical stability
• input-to-state stability