Stability and Discretization Error of State Space Model Neural Operators

TL;DR

This paper provides theoretical guarantees for the discretization error and stability of neural operator schemes, specifically for State Space Model-based Neural Operators and Fourier Neural Operators, validated through empirical experiments.

Contribution

It establishes the first formal bounds linking solution regularity to input discretization and analyzes the stability of neural operators under discretization effects.

Findings

Theoretical bounds accurately predict discretization error in neural operators.

Empirical results validate the robustness of SS-NOs across different resolutions.

Stability analysis confirms neural operators' robustness under discretization.

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.