# Neural Operators for Bypassing Gain and Control Computations in PDE Backstepping

arXiv: https://arxiv.org/abs/2302.14265 · HTML: https://mlsanity.com/paper/2302.14265/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2023.

## Built on by (most cited first, 29 of 29)

- [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)
- [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)
- [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)
- [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)
- [On Bellman equations for continuous-time policy evaluation I: discretization and approximation](https://mlsanity.com/paper/2407.05966/) (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)
- [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)
- [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)
- [Neural Operators for Multi-Task Control and Adaptation](https://mlsanity.com/paper/2604.03449/) (2026)
- [FNOangle θ: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems](https://mlsanity.com/paper/2604.05187/) (2026)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [Moving-Horizon Estimators for Hyperbolic and Parabolic PDEs in 1-D](https://mlsanity.com/paper/2401.02516/) (2024)
- [Neural Operator based Reinforcement Learning for Control of first-order PDEs with Spatially-Varying State Delay](https://mlsanity.com/paper/2501.18201/) (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)
- [Sampling-Horizon Neural Operator Predictors for Nonlinear Control under Delayed Inputs](https://mlsanity.com/paper/2603.29119/) (2026)
- [Lotka-Sharpe Neural Operators for Control of Population PDEs](https://mlsanity.com/paper/2604.03892/) (2026)
- [Provable imitation learning for control of instability in partially-observed Vlasov--Poisson equations](https://mlsanity.com/paper/2605.05081/) (2026)
- [Families of Control-Cost-Parametrized Inverse-Optimal Universal Stabilizers](https://mlsanity.com/paper/2606.09047/) (2026)
- [Toward Goal-Agnostic Joint-Embedding Predictive Control of Partial Differential Equations](https://mlsanity.com/paper/2607.21644/) (2026)
- [Backstepping-Guided Reinforcement Learning for Wide-Range Saint-Venant Canal Regulation](https://mlsanity.com/paper/2608.20089/) (2026)
- [STCO: Conditional Neural Operators for Time-Dependent PDEs](https://mlsanity.com/paper/2608.20477/) (2026)

## Abstract (arXiv)

We introduce a framework for eliminating the computation of controller gain functions in PDE control. We learn the nonlinear operator from the plant parameters to the control gains with a (deep) neural network. We provide closed-loop stability guarantees (global exponential) under an NN-approximation of the feedback gains. While, in the existing PDE backstepping, finding the gain kernel requires (one offline) solution to an integral equation, the neural operator (NO) approach we propose learns the mapping from the functional coefficients of the plant PDE to the kernel function by employing a sufficiently high number of offline numerical solutions to the kernel integral equation, for a large enough number of the PDE model's different functional coefficients. We prove the existence of a DeepONet approximation, with arbitrarily high accuracy, of the exact nonlinear continuous operator mapping PDE coefficient functions into gain functions. Once proven to exist, learning of the NO is standard, completed "once and for all" (never online) and the kernel integral equation doesn't need to be solved ever again, for any new functional coefficient not exceeding the magnitude of the functional coefficients used for training. We also present an extension from approximating the gain kernel operator to approximating the full feedback law mapping, from plant parameter functions and state measurement functions to the control input, with semiglobal practical stability guarantees. Simulation illustrations are provided and code is available on github. This framework, eliminating real-time recomputation of gains, has the potential to be game changing for adaptive control of PDEs and gain scheduling control of nonlinear PDEs. The paper requires no prior background in machine learning or neural networks.