# Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs

arXiv: https://arxiv.org/abs/2303.10506 · HTML: https://mlsanity.com/paper/2303.10506/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 111 references that are in the map.
- First posted on arXiv 18 Mar 2023.

## Builds on (most cited first, 30 of 54)

- [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)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [A Lyapunov-based Approach to Safe Reinforcement Learning](https://mlsanity.com/paper/1805.07708/) (2018)
- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Model-Based Multi-Agent RL in Zero-Sum Markov Games with Near-Optimal Sample Complexity](https://mlsanity.com/paper/2007.07461/) (2020)
- [Information Theoretic Regret Bounds for Online Nonlinear Control](https://mlsanity.com/paper/2006.12466/) (2020)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [On Gradient-Based Learning in Continuous Games](https://mlsanity.com/paper/1804.05464/) (2018)
- [Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem](https://mlsanity.com/paper/1912.11899/) (2019)
- [Learning Stability Certificates from Data](https://mlsanity.com/paper/2008.05952/) (2020)
- [Black-Box Control for Linear Dynamical Systems](https://mlsanity.com/paper/2007.06650/) (2020)
- [Policy Optimization Provably Converges to Nash Equilibria in Zero-Sum Linear Quadratic Games](https://mlsanity.com/paper/1906.00729/) (2019)
- [Provably Efficient Reinforcement Learning in Decentralized General-Sum Markov Games](https://mlsanity.com/paper/2110.05682/) (2021)
- [Learning Model Predictive Control for iterative tasks. A Data-Driven Control Framework](https://mlsanity.com/paper/1609.01387/) (2016)
- [Learning Stabilizable Nonlinear Dynamics with Contraction-Based Regularization](https://mlsanity.com/paper/1907.13122/) (2019)
- [Neural Operators for Bypassing Gain and Control Computations in PDE Backstepping](https://mlsanity.com/paper/2302.14265/) (2023)
- [Analysis of the Optimization Landscape of Linear Quadratic Gaussian (LQG) Control](https://mlsanity.com/paper/2102.04393/) (2021)
- [Learning Lyapunov Functions for Hybrid Systems](https://mlsanity.com/paper/2012.12015/) (2020)
- [TaSIL: Taylor Series Imitation Learning](https://mlsanity.com/paper/2205.14812/) (2022)
- [Reinforcement Learning with Fast Stabilization in Linear Dynamical Systems](https://mlsanity.com/paper/2007.12291/) (2020)
- [Global Convergence of Policy Gradient Primal-dual Methods for Risk-constrained LQRs](https://mlsanity.com/paper/2104.04901/) (2021)

## Built on by (most cited first, 18 of 18)

- [Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay](https://mlsanity.com/paper/2307.11436/) (2023)
- [PDE Control Gym: A Benchmark for Data-Driven Boundary Control of Partial Differential Equations](https://mlsanity.com/paper/2405.11401/) (2024)
- [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)
- [Adaptive Neural-Operator Backstepping Control of a Benchmark Hyperbolic PDE](https://mlsanity.com/paper/2401.07862/) (2024)
- [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)
- [Deep Operator Neural Network Model Predictive Control](https://mlsanity.com/paper/2505.18008/) (2025)
- [Delay compensation of multi-input distinct delay nonlinear systems via neural operators](https://mlsanity.com/paper/2509.17131/) (2025)
- [Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictors](https://mlsanity.com/paper/2509.26443/) (2025)
- [Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem](https://mlsanity.com/paper/2603.11972/) (2026)
- [Safe PDE Boundary Control with Neural Operators](https://mlsanity.com/paper/2411.15643/) (2024)
- [Delay-adaptive Control of Nonlinear Systems with Approximate Neural Operator Predictors](https://mlsanity.com/paper/2508.20367/) (2025)
- [Moving-Horizon Estimators for Hyperbolic and Parabolic PDEs in 1-D](https://mlsanity.com/paper/2401.02516/) (2024)
- [Soft Actor-Critic with Backstepping-Pretrained DeepONet for control of PDEs](https://mlsanity.com/paper/2507.04232/) (2025)
- [Lotka-Sharpe Neural Operators for Control of Population PDEs](https://mlsanity.com/paper/2604.03892/) (2026)
- [Backstepping-Guided Reinforcement Learning for Wide-Range Saint-Venant Canal Regulation](https://mlsanity.com/paper/2608.20089/) (2026)

## Abstract (arXiv)

Unlike ODEs, whose models involve system matrices and whose controllers involve vector or matrix gains, PDE models involve functions in those roles functional coefficients, dependent on the spatial variables, and gain functions dependent on space as well. The designs of gains for controllers and observers for PDEs, such as PDE backstepping, are mappings of system model functions into gain functions. These infinite dimensional nonlinear operators are given in an implicit form through PDEs, in spatial variables, which need to be solved to determine the gain function for each new functional coefficient of the PDE. The need for solving such PDEs can be eliminated by learning and approximating the said design mapping in the form of a neural operator. Learning the neural operator requires a sufficient number of prior solutions for the design PDEs, offline, as well as the training of the operator. In recent work, we developed the neural operators for PDE backstepping designs for first order hyperbolic PDEs. Here we extend this framework to the more complex class of parabolic PDEs. The key theoretical question is whether the controllers are still stabilizing, and whether the observers are still convergent, if they employ the approximate functional gains generated by the neural operator. We provide affirmative answers to these questions, namely, we prove stability in closed loop under gains produced by neural operators. We illustrate the theoretical results with numerical tests and publish our code on github. The neural operators are three orders of magnitude faster in generating gain functions than PDE solvers for such gain functions. This opens up the opportunity for the use of this neural operator methodology in adaptive control and in gain scheduling control for nonlinear PDEs.