# Neural Operator Feedback for a First-Order PIDE with Spatially-Varying State Delay

arXiv: https://arxiv.org/abs/2412.08219 · HTML: https://mlsanity.com/paper/2412.08219/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2024.

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

- [Soft Actor-Critic with Backstepping-Pretrained DeepONet for control of PDEs](https://mlsanity.com/paper/2507.04232/) (2025)
- [Backstepping-Guided Reinforcement Learning for Wide-Range Saint-Venant Canal Regulation](https://mlsanity.com/paper/2608.20089/) (2026)

## Abstract (arXiv)

A transport PDE with a spatial integral and recirculation with constant delay has been a benchmark for neural operator approximations of PDE backstepping controllers. Introducing a spatially-varying delay into the model gives rise to a gain operator defined through integral equations which the operator's input -- the varying delay function -- enters in previously unencountered manners, including in the limits of integration and as the inverse of the `delayED time' function. This, in turn, introduces novel mathematical challenges in estimating the operator's Lipschitz constant. The backstepping kernel function having two branches endows the feedback law with a two-branch structure, where only one of the two feedback branches depends on both of the kernel branches. For this rich feedback structure, we propose a neural operator approximation of such a two-branch feedback law and prove the approximator to be semiglobally practically stabilizing. With numerical results we illustrate the training of the neural operator and its stabilizing capability.