# Deep Learning of Delay-Compensated Backstepping for Reaction-Diffusion PDEs

arXiv: https://arxiv.org/abs/2308.10501 · HTML: https://mlsanity.com/paper/2308.10501/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 46 references that are in the map.
- First posted on arXiv 21 Aug 2023.

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

- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Neural Operators for Bypassing Gain and Control Computations in PDE Backstepping](https://mlsanity.com/paper/2302.14265/) (2023)
- [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)
- [Delay-robust control design for two heterodirectional linear coupled hyperbolic PDEs](https://mlsanity.com/paper/1709.04274/) (2017)
- [Adaptive Control By Regulation-Triggered Batch Least-Squares Estimation of Non-Observable Parameters](https://mlsanity.com/paper/1811.10833/) (2018)
- [Delay-Compensated Control of Sandwiched ODE-PDE-ODE Hyperbolic Systems for Oil Drilling and Disaster Relief](https://mlsanity.com/paper/1910.05948/) (2019)
- [Delay-Compensated Distributed PDE Control of Traffic with Connected/Automated Vehicles](https://mlsanity.com/paper/2107.08651/) (2021)
- [Delay-Adaptive Boundary Control of Coupled Hyperbolic PDE-ODE Cascade Systems](https://mlsanity.com/paper/2301.12372/) (2023)
- [Time- and State-Dependent Input Delay-Compensated Bang-Bang Control of a Screw Extruder for 3D Printing](https://mlsanity.com/paper/1505.06375/) (2015)
- [Feedback stabilization of a 1D linear reaction-diffusion equation with delay boundary control](https://mlsanity.com/paper/1709.02735/) (2017)
- [Feedback Stabilization of a Class of Diagonal Infinite-Dimensional Systems with Delay Boundary Control](https://mlsanity.com/paper/1902.05086/) (2019)
- [Bilateral Boundary Control of Moving Shockwave in LWR Model of Congested Traffic](https://mlsanity.com/paper/1904.04303/) (2019)
- [Finite-dimensional control of the heat equation: Dirichlet actuation and point measurement](https://mlsanity.com/paper/2011.07256/) (2020)
- [Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs](https://mlsanity.com/paper/2104.13294/) (2021)
- [Predictor-Based Output Feedback Stabilization of an Input Delayed Parabolic PDE with Boundary Measurement](https://mlsanity.com/paper/2105.08431/) (2021)
- [Delay-Adaptive Control of First-order Hyperbolic PIDEs](https://mlsanity.com/paper/2307.04212/) (2023)

## Built on by (most cited first, 8 of 8)

- [Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation](https://mlsanity.com/paper/2401.02511/) (2024)
- [Adaptive Neural-Operator Backstepping Control of a Benchmark Hyperbolic PDE](https://mlsanity.com/paper/2401.07862/) (2024)
- [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)
- [Soft Actor-Critic with Backstepping-Pretrained DeepONet for control of PDEs](https://mlsanity.com/paper/2507.04232/) (2025)
- [Predictor-Based Output-Feedback Control of Linear Systems with Time-Varying Input and Measurement Delays via Neural-Approximated Prediction Horizons](https://mlsanity.com/paper/2603.29117/) (2026)

## Abstract (arXiv)

Deep neural networks that approximate nonlinear function-to-function mappings, i.e., operators, which are called DeepONet, have been demonstrated in recent articles to be capable of encoding entire PDE control methodologies, such as backstepping, so that, for each new functional coefficient of a PDE plant, the backstepping gains are obtained through a simple function evaluation. These initial results have been limited to single PDEs from a given class, approximating the solutions of only single-PDE operators for the gain kernels. In this paper we expand this framework to the approximation of multiple (cascaded) nonlinear operators. Multiple operators arise in the control of PDE systems from distinct PDE classes, such as the system in this paper: a reaction-diffusion plant, which is a parabolic PDE, with input delay, which is a hyperbolic PDE. The DeepONet-approximated nonlinear operator is a cascade/composition of the operators defined by one hyperbolic PDE of the Goursat form and one parabolic PDE on a rectangle, both of which are bilinear in their input functions and not explicitly solvable. For the delay-compensated PDE backstepping controller, which employs the learned control operator, namely, the approximated gain kernel, we guarantee exponential stability in the L2 norm of the plant state and the H1 norm of the input delay state. Simulations illustrate the contributed theory.