# Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictors

arXiv: https://arxiv.org/abs/2509.26443 · HTML: https://mlsanity.com/paper/2509.26443/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 51 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 18 of 18)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [Neural Operators for Bypassing Gain and Control Computations in PDE Backstepping](https://mlsanity.com/paper/2302.14265/) (2023)
- [Data Complexity Estimates for Operator Learning](https://mlsanity.com/paper/2405.15992/) (2024)
- [Discretization Error of Fourier Neural Operators](https://mlsanity.com/paper/2405.02221/) (2024)
- [Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs](https://mlsanity.com/paper/2303.10506/) (2023)
- [Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay](https://mlsanity.com/paper/2307.11436/) (2023)
- [Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation](https://mlsanity.com/paper/2401.02511/) (2024)
- [Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems](https://mlsanity.com/paper/2411.18964/) (2024)
- [Deep Learning of Delay-Compensated Backstepping for Reaction-Diffusion PDEs](https://mlsanity.com/paper/2308.10501/) (2023)
- [Adaptive Neural-Operator Backstepping Control of a Benchmark Hyperbolic PDE](https://mlsanity.com/paper/2401.07862/) (2024)
- [Neural Operators for Boundary Stabilization of Stop-and-go Traffic](https://mlsanity.com/paper/2312.10374/) (2023)
- [Size Lowerbounds for Deep Operator Networks](https://mlsanity.com/paper/2308.06338/) (2023)
- [Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernels](https://mlsanity.com/paper/2407.01745/) (2024)
- [Predictor-Feedback Stabilization of Multi-Input Nonlinear Systems](https://mlsanity.com/paper/1508.05756/) (2015)
- [Safety-Critical Control with Input Delay in Dynamic Environment](https://mlsanity.com/paper/2112.08445/) (2021)
- [Global Stabilization of Chemostats with Nonzero Mortality and Substrate Dynamics](https://mlsanity.com/paper/2502.09310/) (2025)

## Built on by (most cited first, 2 of 2)

- [Sampling-Horizon Neural Operator Predictors for Nonlinear Control under Delayed Inputs](https://mlsanity.com/paper/2603.29119/) (2026)
- [Lotka-Sharpe Neural Operators for Control of Population PDEs](https://mlsanity.com/paper/2604.03892/) (2026)

## Abstract (arXiv)

This work establishes the first rigorous stability guarantees for approximate predictors in delay-adaptive control of nonlinear systems, addressing a key challenge in practical implementations where exact predictors are unavailable. We analyze two scenarios: (i) when the actuated input is directly measurable, and (ii) when it is estimated online. For the measurable input case, we prove semi-global practical asymptotic stability with an explicit bound proportional to the approximation error ε. For the unmeasured input case, we demonstrate local practical asymptotic stability, with the region of attraction explicitly dependent on both the initial delay estimate and the predictor approximation error. To bridge theory and practice, we show that neural operators-a flexible class of neural network-based approximators-can achieve arbitrarily small approximation errors, thus satisfying the conditions of our stability theorems. Numerical experiments on two nonlinear benchmark systems-a biological protein activator/repressor model and a micro-organism growth Chemostat model-validate our theoretical results. In particular, our numerical simulations confirm stability under approximate predictors, highlight the strong generalization capabilities of neural operators, and demonstrate a substantial computational speedup of up to 15x compared to a baseline fixed-point method.