# Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay

arXiv: https://arxiv.org/abs/2307.11436 · HTML: https://mlsanity.com/paper/2307.11436/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 56 references that are in the map.
- First posted on arXiv 21 Jul 2023.

## Builds on (most cited first, 19 of 19)

- [Fast R-CNN](https://mlsanity.com/paper/1504.08083/) (2015)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Neural-Fly Enables Rapid Learning for Agile Flight in Strong Winds](https://mlsanity.com/paper/2205.06908/) (2022)
- [Policy Gradient Methods for the Noisy Linear Quadratic Regulator over a Finite Horizon](https://mlsanity.com/paper/2011.10300/) (2020)
- [Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem](https://mlsanity.com/paper/1912.11899/) (2019)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Neural Operators for Bypassing Gain and Control Computations in PDE Backstepping](https://mlsanity.com/paper/2302.14265/) (2023)
- [Density Constrained Reinforcement Learning](https://mlsanity.com/paper/2106.12764/) (2021)
- [Deep Reinforcement Learning with Shallow Controllers: An Experimental Application to PID Tuning](https://mlsanity.com/paper/2111.07171/) (2021)
- [Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs](https://mlsanity.com/paper/2303.10506/) (2023)
- [Meta-Reinforcement Learning for the Tuning of PI Controllers: An Offline Approach](https://mlsanity.com/paper/2203.09661/) (2022)
- [Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs](https://mlsanity.com/paper/1504.07491/) (2015)
- [Boundary Control of Coupled Reaction-Advection-Diffusion Systems with Spatially-Varying Coefficients](https://mlsanity.com/paper/1603.04914/) (2016)
- [Delay-robust control design for two heterodirectional linear coupled hyperbolic PDEs](https://mlsanity.com/paper/1709.04274/) (2017)
- [Delay-Compensated Control of Sandwiched ODE-PDE-ODE Hyperbolic Systems for Oil Drilling and Disaster Relief](https://mlsanity.com/paper/1910.05948/) (2019)
- [Distributed Reinforcement Learning for Flexible and Efficient UAV Swarm Control](https://mlsanity.com/paper/2103.04666/) (2021)
- [Joint Speed Control and Energy Replenishment Optimization for UAV-assisted IoT Data Collection with Deep Reinforcement Transfer Learning](https://mlsanity.com/paper/2106.10423/) (2021)
- [Delay-Compensated Distributed PDE Control of Traffic with Connected/Automated Vehicles](https://mlsanity.com/paper/2107.08651/) (2021)

## Built on by (most cited first, 13 of 13)

- [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)
- [Neural Operators for Boundary Stabilization of Stop-and-go Traffic](https://mlsanity.com/paper/2312.10374/) (2023)
- [Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernels](https://mlsanity.com/paper/2407.01745/) (2024)
- [Backstepping Neural Operators for 2× 2 Hyperbolic PDEs](https://mlsanity.com/paper/2312.16762/) (2023)
- [Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictors](https://mlsanity.com/paper/2509.26443/) (2025)
- [Delay-adaptive Control of Nonlinear Systems with Approximate Neural Operator Predictors](https://mlsanity.com/paper/2508.20367/) (2025)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [Neural Operator based Reinforcement Learning for Control of first-order PDEs with Spatially-Varying State Delay](https://mlsanity.com/paper/2501.18201/) (2025)
- [Employing Deep Neural Operators for PDE control by decoupling training and optimization](https://mlsanity.com/paper/2506.04742/) (2025)
- [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)

The recently introduced DeepONet operator-learning framework for PDE control is extended from the results for basic hyperbolic and parabolic PDEs to an advanced hyperbolic class that involves delays on both the state and the system output or input. The PDE backstepping design produces gain functions that are outputs of a nonlinear operator, mapping functions on a spatial domain into functions on a spatial domain, and where this gain-generating operator's inputs are the PDE's coefficients. The operator is approximated with a DeepONet neural network to a degree of accuracy that is provably arbitrarily tight. Once we produce this approximation-theoretic result in infinite dimension, with it we establish stability in closed loop under feedback that employs approximate gains. In addition to supplying such results under full-state feedback, we also develop DeepONet-approximated observers and output-feedback laws and prove their own stabilizing properties under neural operator approximations. With numerical simulations we illustrate the theoretical results and quantify the numerical effort savings, which are of two orders of magnitude, thanks to replacing the numerical PDE solving with the DeepONet.