# Deep Operator Neural Network Model Predictive Control

arXiv: https://arxiv.org/abs/2505.18008 · HTML: https://mlsanity.com/paper/2505.18008/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 56 references that are in the map.
- First posted on arXiv 23 May 2025.

## Builds on (most cited first, 22 of 22)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [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)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpoint](https://mlsanity.com/paper/1912.04862/) (2019)
- [Real-time Neural-MPC: Deep Learning Model Predictive Control for Quadrotors and Agile Robotic Platforms](https://mlsanity.com/paper/2203.07747/) (2022)
- [Data-Enabled Predictive Control: In the Shallows of the DeePC](https://mlsanity.com/paper/1811.05890/) (2018)
- [Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators](https://mlsanity.com/paper/2109.10793/) (2021)
- [Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs](https://mlsanity.com/paper/2303.10506/) (2023)
- [On Recurrent Neural Networks for learning-based control: recent results and ideas for future developments](https://mlsanity.com/paper/2111.13557/) (2021)
- [RAMP-Net: A Robust Adaptive MPC for Quadrotors via Physics-informed Neural Network](https://mlsanity.com/paper/2209.09025/) (2022)
- [Nonlinear MPC for Offset-Free Tracking of systems learned by GRU Neural Networks](https://mlsanity.com/paper/2103.02383/) (2021)
- [Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems](https://mlsanity.com/paper/2411.18964/) (2024)
- [Nonlinear MPC design for incrementally ISS systems with application to GRU networks](https://mlsanity.com/paper/2309.16428/) (2023)
- [Basis functions nonlinear data-enabled predictive control: Consistent and computationally efficient formulations](https://mlsanity.com/paper/2311.05360/) (2023)
- [Modelling and Learning Dynamics for Robotic Food-Cutting](https://mlsanity.com/paper/2003.09179/) (2020)
- [Robust constrained nonlinear Model Predictive Control with Gated Recurrent Unit model -- Extended version](https://mlsanity.com/paper/2307.10069/) (2023)
- [Neural Data-Enabled Predictive Control](https://mlsanity.com/paper/2406.08003/) (2024)
- [Robust offset-free constrained Model Predictive Control with Long Short-Term Memory Networks -- Extended version](https://mlsanity.com/paper/2303.17304/) (2023)
- [Choose Wisely: Data-driven Predictive Control for Nonlinear Systems Using Online Data Selection](https://mlsanity.com/paper/2503.18845/) (2025)

## Built on by (most cited first, 3 of 3)

- [Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem](https://mlsanity.com/paper/2603.11972/) (2026)
- [The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositions](https://mlsanity.com/paper/2602.21910/) (2026)
- [Sampling-Horizon Neural Operator Predictors for Nonlinear Control under Delayed Inputs](https://mlsanity.com/paper/2603.29119/) (2026)

## Abstract (arXiv)

In this paper, we consider the design of model predictive control (MPC) algorithms based on deep operator neural networks (DeepONets). These neural networks are capable of accurately approximating real and complex valued solutions of continuous time nonlinear systems without relying on recurrent architectures. The DeepONet architecture is made up of two feedforward neural networks: the branch network, which encodes the input function space, and the trunk network, which represents dependencies on temporal variables or initial conditions. Utilizing the original DeepONet architecture as a predictor within MPC for Multi Input Multi Output (MIMO) systems requires multiple branch networks, to generate multi output predictions, one for each input. Moreover, to predict multiple time steps into the future, the network has to be evaluated multiple times. Motivated by this, we introduce a multi step DeepONet (MS-DeepONet) architecture that computes in one shot multi step predictions of system outputs from multi step input sequences, which is better suited for MPC. We prove that the MS DeepONet is a universal approximator in terms of multi step sequence prediction. Additionally, we develop automated hyper parameter selection strategies and implement MPC frameworks using both the standard DeepONet and the proposed MS DeepONet architectures in PyTorch. The implementation is publicly available on GitHub. Simulation results demonstrate that MS-DeepONet consistently outperforms the standard DeepONet in learning and predictive control tasks across several nonlinear benchmark systems: the van der Pol oscillator, the quadruple tank process, and a cart pendulum unstable system, where it successfully learns and executes multiple swing up and stabilization policies.