# Adaptive Neural-Operator Backstepping Control of a Benchmark Hyperbolic PDE

arXiv: https://arxiv.org/abs/2401.07862 · HTML: https://mlsanity.com/paper/2401.07862/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 34 references that are in the map.
- First posted on arXiv 15 Jan 2024.

## Builds on (most cited first, 10 of 10)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [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)
- [Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation](https://mlsanity.com/paper/2401.02511/) (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)
- [Size Lowerbounds for Deep Operator Networks](https://mlsanity.com/paper/2308.06338/) (2023)
- [Adaptive Boundary Control of Constant-Parameter Reaction-Diffusion PDEs Using Regulation-Triggered Finite-Time Identification](https://mlsanity.com/paper/1805.08404/) (2018)
- [Stability Results for the Continuity Equation](https://mlsanity.com/paper/1908.05995/) (2019)

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

- [Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems](https://mlsanity.com/paper/2411.18964/) (2024)
- [Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernels](https://mlsanity.com/paper/2407.01745/) (2024)
- [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)
- [Lotka-Sharpe Neural Operators for Control of Population PDEs](https://mlsanity.com/paper/2604.03892/) (2026)
- [Families of Control-Cost-Parametrized Inverse-Optimal Universal Stabilizers](https://mlsanity.com/paper/2606.09047/) (2026)

## Abstract (arXiv)

To stabilize PDEs, feedback controllers require gain kernel functions, which are themselves governed by PDEs. Furthermore, these gain-kernel PDEs depend on the PDE plants' functional coefficients. The functional coefficients in PDE plants are often unknown. This requires an adaptive approach to PDE control, i.e., an estimation of the plant coefficients conducted concurrently with control, where a separate PDE for the gain kernel must be solved at each timestep upon the update in the plant coefficient function estimate. Solving a PDE at each timestep is computationally expensive and a barrier to the implementation of real-time adaptive control of PDEs. Recently, results in neural operator (NO) approximations of functional mappings have been introduced into PDE control, for replacing the computation of the gain kernel with a neural network that is trained, once offline, and reused in real-time for rapid solution of the PDEs. In this paper, we present the first result on applying NOs in adaptive PDE control, presented for a benchmark 1-D hyperbolic PDE with recirculation. We establish global stabilization via Lyapunov analysis, in the plant and parameter error states, and also present an alternative approach, via passive identifiers, which avoids the strong assumptions on kernel differentiability. We then present numerical simulations demonstrating stability and observe speedups up to three orders of magnitude, highlighting the real-time efficacy of neural operators in adaptive control. Our code (Github) is made publicly available for future researchers.