# Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernels

arXiv: https://arxiv.org/abs/2407.01745 · HTML: https://mlsanity.com/paper/2407.01745/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 1 Jul 2024.

## 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)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [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)
- [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)
- [Backstepping Neural Operators for 2× 2 Hyperbolic PDEs](https://mlsanity.com/paper/2312.16762/) (2023)
- [Adaptive Boundary Control of Constant-Parameter Reaction-Diffusion PDEs Using Regulation-Triggered Finite-Time Identification](https://mlsanity.com/paper/1805.08404/) (2018)
- [Finite-dimensional output stabilization for a class of linear distributed parameter systems -- a small-gain approach](https://mlsanity.com/paper/2104.06102/) (2021)
- [Towards a MATLAB Toolbox to compute backstepping kernels using the power series method](https://mlsanity.com/paper/2403.16070/) (2024)
- [Gain-Only Neural Operators for PDE Backstepping](https://mlsanity.com/paper/2403.19344/) (2024)

## Built on by (most cited first, 5 of 5)

- [Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems](https://mlsanity.com/paper/2411.18964/) (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)

## Abstract (arXiv)

Neural operator approximations of the gain kernels in PDE backstepping has emerged as a viable method for implementing controllers in real time. With such an approach, one approximates the gain kernel, which maps the plant coefficient into the solution of a PDE, with a neural operator. It is in adaptive control that the benefit of the neural operator is realized, as the kernel PDE solution needs to be computed online, for every updated estimate of the plant coefficient. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE (a reaction-diffusion equation with a spatially-varying and unknown reaction coefficient). We prove global stability and asymptotic regulation of the plant state for a Lyapunov design of parameter adaptation. The key technical challenge of the result is handling the 2D nature of the gain kernels and proving that the target system with two distinct sources of perturbation terms, due to the parameter estimation error and due to the neural approximation error, is Lyapunov stable. To verify our theoretical result, we present simulations achieving calculation speedups up to 45x relative to the traditional finite difference solvers for every timestep in the simulation trajectory.