# Backstepping Neural Operators for 2× 2 Hyperbolic PDEs

arXiv: https://arxiv.org/abs/2312.16762 · HTML: https://mlsanity.com/paper/2312.16762/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 57 references that are in the map.
- First posted on arXiv 28 Dec 2023.

## Builds on (most cited first, 17 of 17)

- [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 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)
- [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)
- [Control of Homodirectional and General Heterodirectional Linear Coupled Hyperbolic PDEs](https://mlsanity.com/paper/1504.07491/) (2015)
- [Adaptive Control By Regulation-Triggered Batch Least-Squares Estimation of Non-Observable Parameters](https://mlsanity.com/paper/1811.10833/) (2018)
- [Spill-Free Transfer and Stabilization of Viscous Liquid](https://mlsanity.com/paper/2108.11052/) (2021)
- [Delay-Adaptive Boundary Control of Coupled Hyperbolic PDE-ODE Cascade Systems](https://mlsanity.com/paper/2301.12372/) (2023)
- [Backstepping Stabilization of the Linearized Saint-Venant-Exner Model](https://mlsanity.com/paper/1505.06367/) (2015)
- [Bilateral Boundary Control of One-Dimensional First- and Second-Order PDEs using Infinite-Dimensional Backstepping](https://mlsanity.com/paper/1603.04913/) (2016)
- [Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation star](https://mlsanity.com/paper/1701.05067/) (2017)
- [Stop-and-Go Suppression in Two-Class Congested Traffic](https://mlsanity.com/paper/1905.06476/) (2019)
- [Boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space](https://mlsanity.com/paper/2004.12594/) (2020)
- [Output-Feedback Control of Viscous Liquid-Tank System and its Numerical Approximation](https://mlsanity.com/paper/2201.13272/) (2022)

## Built on by (most cited first, 4 of 4)

- [Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernels](https://mlsanity.com/paper/2407.01745/) (2024)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (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)

Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has proven capable of approximating PDE backstepping designs in which a single Goursat-form PDE governs a single feedback gain function. In boundary control of coupled PDEs, coupled Goursat-form PDEs govern two or more gain kernels-a PDE structure unaddressed thus far with DeepONet. In this paper, we explore the subject of approximating systems of gain kernel PDEs for hyperbolic PDE plants by considering a simple counter-convecting 2× 2 coupled system in whose control a 2× 2 kernel PDE system in Goursat form arises. Engineering applications include oil drilling, the Saint-Venant model of shallow water waves, and the Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow. We establish the continuity of the mapping from a total of five plant PDE functional coefficients to the kernel PDE solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and ensure that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. Taking into account anti-collocated boundary actuation and sensing, our L2-Globally-exponentially stabilizing (GES) approximate gain kernel-based output feedback design implies the deep learning of both the controller's and the observer's gains. Moreover, the encoding of the output-feedback law into DeepONet ensures semi-global practical exponential stability (SG-PES). The DeepONet operator speeds up the computation of the controller gains by multiple orders of magnitude. Its theoretically proven stabilizing capability is demonstrated through simulations.