# ReLU Networks for Model Predictive Control: Network Complexity and Performance Guarantees

arXiv: https://arxiv.org/abs/2601.16764 · HTML: https://mlsanity.com/paper/2601.16764/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 42 references that are in the map.
- First posted on arXiv 23 Jan 2026.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations](https://mlsanity.com/paper/1708.02691/) (2017)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [The Modern Mathematics of Deep Learning](https://mlsanity.com/paper/2105.04026/) (2021)
- [Learning an Approximate Model Predictive Controller with Guarantees](https://mlsanity.com/paper/1806.04167/) (2018)
- [Large Scale Model Predictive Control with Neural Networks and Primal Active Sets](https://mlsanity.com/paper/1910.10835/) (2019)
- [Safe and Fast Tracking on a Robot Manipulator: Robust MPC and Neural Network Control](https://mlsanity.com/paper/1912.10360/) (2019)
- [Reliably-stabilizing piecewise-affine neural network controllers](https://mlsanity.com/paper/2111.07183/) (2021)
- [Automatic nonlinear MPC approximation with closed-loop guarantees](https://mlsanity.com/paper/2312.10199/) (2023)
- [Stability Verification of Neural Network Controllers using Mixed-Integer Programming](https://mlsanity.com/paper/2206.13374/) (2022)
- [Neural Networks for Fast Optimisation in Model Predictive Control: A Review](https://mlsanity.com/paper/2309.02668/) (2023)
- [TinyMPC: Model-Predictive Control on Resource-Constrained Microcontrollers](https://mlsanity.com/paper/2310.16985/) (2023)
- [Approximation of Smoothness Classes by Deep Rectifier Networks](https://mlsanity.com/paper/2007.15645/) (2020)
- [A Sensitivity-based Data Augmentation Framework for Model Predictive Control Policy Approximation](https://mlsanity.com/paper/2009.07398/) (2020)
- [Deep Learning-Based Model Predictive Control for Resonant Power Converters](https://mlsanity.com/paper/1810.04872/) (2018)
- [Standoff Tracking Using DNN-Based MPC with Implementation on FPGA](https://mlsanity.com/paper/2212.10945/) (2022)

## Abstract (arXiv)

Recent years have witnessed a resurgence in using ReLU neural networks (NNs) to represent model predictive control (MPC) policies. However, determining the required network complexity to ensure closed-loop performance remains a fundamental open problem. This involves a critical precision-complexity trade-off: undersized networks may fail to capture the MPC policy, while oversized ones may outweigh the benefits of ReLU network approximation. In this work, we propose a projection-based method to enforce hard constraints and establish a state-dependent Lipschitz continuity property for the optimal MPC cost function, which enables sharp convergence analysis of the closed-loop system. For the first time, we derive explicit bounds on ReLU network width and depth for approximating MPC policies with guaranteed closed-loop performance. To further reduce network complexity and enhance closed-loop performance, we propose a non-uniform error framework with a state-aware scaling function to adaptively adjust both the input and output of the ReLU network. Our contributions provide a foundational step toward certifiable ReLU NN-based MPC.