Stability and Discretization Error of State Space Model Neural Operators

2026-05-20 · Source: cs.NE updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, short

Summary

A new study establishes theoretical guarantees for the discretization error and stability of neural operator approximation schemes, specifically focusing on State Space Model-based Neural Operators (SS-NOs) and Fourier Neural Operators (FNOs). The research addresses a gap in understanding the connection between continuous theory and discrete numerical implementation for solving partial differential equations (PDEs) using neural operators. It provides analytical bounds that link solution regularity to input discretization, formally quantifying neural operator accuracy under numerical constraints. Furthermore, an input-to-state stability (ISS) analysis assesses the impact of discretization on the stability of SS-NOs' continuous domain results. Empirical experiments on 1D and 2D benchmarks validate these theoretical bounds and demonstrate the robustness of SS-NOs across varying resolutions.

Key takeaway

For AI scientists and research scientists developing or deploying neural operators for PDE solutions, understanding the theoretical bounds on discretization error and stability is crucial. Your model's accuracy and robustness under varying input resolutions can be formally quantified using the derived analytical bounds and ISS analysis. This enables more reliable deployment of SS-NOs and FNOs, ensuring predictable performance in real-world numerical constraints.

Key insights

Theoretical bounds quantify discretization error and stability in neural operators for solving PDEs.

Principles

• Solution regularity links to input discretization.
• Discretization impacts neural operator stability.

Method

The method involves deriving analytical bounds for discretization error and performing an input-to-state stability (ISS) analysis to assess discretization's impact on neural operator stability.

In practice

• Apply SS-NOs for robust PDE solutions.
• Consider input discretization for accuracy.

Topics

• Neural Operators
• State Space Model Neural Operators
• Fourier Neural Operators
• Discretization Error
• Stability Analysis

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.NE updates on arXiv.org.