# Robust Machine Learning Modeling for Predictive Control Using Lipschitz-Constrained Neural Networks

arXiv: https://arxiv.org/abs/2308.13721 · HTML: https://mlsanity.com/paper/2308.13721/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Aug 2023.

## Built on by (most cited first, 8 of 8)

- [ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks](https://mlsanity.com/paper/2404.04375/) (2024)
- [Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activations](https://mlsanity.com/paper/2401.14033/) (2024)
- [Stability and Performance Analysis of Discrete-Time ReLU Recurrent Neural Networks](https://mlsanity.com/paper/2405.05236/) (2024)
- [A Complete Set of Quadratic Constraints for Repeated ReLU and Generalizations](https://mlsanity.com/paper/2407.06888/) (2024)
- [Input Convex Lipschitz Recurrent Neural Networks for Robust and Efficient Process Modeling and Optimization](https://mlsanity.com/paper/2401.07494/) (2024)
- [Least Squares and Marginal Log-Likelihood Model Predictive Control using Normalizing Flows](https://mlsanity.com/paper/2409.17632/) (2024)
- [ECLipsE-Gen-Local: Efficient Compositional Local Lipschitz Estimates for Deep Neural Networks](https://mlsanity.com/paper/2510.05261/) (2025)
- [On the Conditions for Domain Stability for Machine Learning: a Mathematical Approach](https://mlsanity.com/paper/2412.00464/) (2024)

## Abstract (arXiv)

Neural networks (NNs) have emerged as a state-of-the-art method for modeling nonlinear systems in model predictive control (MPC). However, the robustness of NNs, in terms of sensitivity to small input perturbations, remains a critical challenge for practical applications. To address this, we develop Lipschitz-Constrained Neural Networks (LCNNs) for modeling nonlinear systems and derive rigorous theoretical results to demonstrate their effectiveness in approximating Lipschitz functions, reducing input sensitivity, and preventing over-fitting. Specifically, we first prove a universal approximation theorem to show that LCNNs using SpectralDense layers can approximate any 1-Lipschitz target function. Then, we prove a probabilistic generalization error bound for LCNNs using SpectralDense layers by using their empirical Rademacher complexity. Finally, the LCNNs are incorporated into the MPC scheme, and a chemical process example is utilized to show that LCNN-based MPC outperforms MPC using conventional feedforward NNs in the presence of training data noise.