Stability Analysis of a B-Spline Deep Neural Operator for Nonlinear Systems

TL;DR

This paper introduces a stability analysis method for neural operators using a B-spline-based neural architecture, enabling expressive modeling and post-training stability assessment through spectral analysis.

Contribution

It presents the Hybrid B-spline Deep Neural Operator (HBDNO), which maintains full expressive power while allowing stability analysis via control points and Koopman operator techniques.

Findings

HBDNO effectively preserves model expressiveness.

Control points serve as natural observables for stability analysis.

Spectral methods can assess stability of learned operators.

Abstract

This paper investigates the stability properties of neural operators through the structured representation offered by the Hybrid B-spline Deep Neural Operator (HBDNO). While existing stability-aware architectures typically enforce restrictive constraints that limit universality, HBDNO preserves full expressive power by representing outputs via B-spline control points. We show that these control points form a natural observable for post-training stability analysis. By applying Dynamic Mode Decomposition and connecting the resulting discrete dynamics to the Koopman operator framework, we provide a principled approach to spectral characterization of learned operators. Numerical results demonstrate the ability to assess stability and reveal future directions for safety-critical applications.