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Efficient representation and approximation of model predictive control laws via deep learning

arXiv 1806.10644 ↗First posted 27 Jun 2018math.OCcited by 55

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Abstract (arXiv)

We show that artificial neural networks with rectifier units as activation functions can exactly represent the piecewise affine function that results from the formulation of model predictive control of linear time-invariant systems. The choice of deep neural networks is particularly interesting as they can represent exponentially many more affine regions compared to networks with only one hidden layer. We provide theoretical bounds on the minimum number of hidden layers and neurons per layer that a neural network should have to exactly represent a given model predictive control law. The proposed approach has a strong potential as an approximation method of predictive control laws, leading to better approximation quality and significantly smaller memory requirements than previous approaches, as we illustrate via simulation examples. We also suggest different alternatives to correct or quantify the approximation error. Since the online evaluation of neural networks is extremely simple, the approximated controllers can be deployed on low-power embedded devices with small storage capacity, enabling the implementation of advanced decision-making strategies for complex cyber-physical systems with limited computing capabilities.

Key facts. Cited by 55 papers in the map (data through 7 Oct 2026). First posted on arXiv 27 Jun 2018.

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Citing papers per year: 2020: 1; 2021: 10; 2022: 7; 2023: 8; 2024: 10; 2025: 12; 2026: 7.

55 papers.

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  2. 2020Imitation Learning with Stability and Safety Guaranteescited by 34
  3. 2023Learning to Warm-Start Fixed-Point Optimization Algorithmscited by 32
  4. 2021Integrated Decision and Control: Towards Interpretable and Computationally Efficient Driving Intelligencecited by 25
  5. 2025Synthesis of Model Predictive Control and Reinforcement Learning: Survey and Classificationcited by 19
  6. 2024Expressive Power of Graph Neural Networks for (Mixed-Integer) Quadratic Programscited by 16
  7. 2021Reliably-stabilizing piecewise-affine neural network controllerscited by 13
  8. 2023Automatic nonlinear MPC approximation with closed-loop guaranteescited by 12
  9. 2021Deep Learning Explicit Differentiable Predictive Control Laws for Buildingscited by 11
  10. 2023Improved uncertainty quantification for neural networks with Bayesian last layercited by 11
  11. 2022Stability Verification of Neural Network Controllers using Mixed-Integer Programmingcited by 10
  12. 2022Active Learning for Regression by Inverse Distance Weightingcited by 9
38 more
  1. 2023Approximate non-linear model predictive control with safety-augmented neural networkscited by 9
  2. 2021Ensuring DNN Solution Feasibility for Optimization Problems with Convex Constraints and Its Application to DC Optimal Power Flow Problemscited by 6
  3. 2022Neural Lyapunov Differentiable Predictive Controlcited by 6
  4. 2022Differentiable Predictive Control with Safety Guarantees: A Control Barrier Function Approachcited by 6
  5. 2022Model Predictive Control via On-Policy Imitation Learningcited by 6
  6. 2022Neural ODEs as Feedback Policies for Nonlinear Optimal Controlcited by 4
  7. 2023Data-driven decision-focused surrogate modelingcited by 4
  8. 2023An Improved Data Augmentation Scheme for Model Predictive Control Policy Approximationcited by 3
  9. 2023On the Sample Complexity of Imitation Learning for Smoothed Model Predictive Controlcited by 3
  10. 2024FedCoSR: Personalized Federated Learning with Contrastive Shareable Representations for Label Heterogeneity in Non-IID Datacited by 3
  11. 2024BO4IO: A Bayesian optimization approach to inverse optimization with uncertainty quantificationcited by 3
  12. 2024Self-Supervised Learning of Iterative Solvers for Constrained Optimizationcited by 3
  13. 2025Imitation Learning of MPC with Neural Networks: Error Guarantees and Sparsificationcited by 3
  14. 2025MPCritic: A plug-and-play MPC architecture for reinforcement learningcited by 3
  15. 2021Tailored neural networks for learning optimal value functions in MPCcited by 2
  16. 2025YANNs: Y-wise Affine Neural Networks for Exact and Efficient Representations of Piecewise Linear Functionscited by 2
  17. 2025Zero-Shot Transferable Solution Method for Parametric Optimal Control Problemscited by 2
  18. 2021Approximate Robust NMPC using Reinforcement Learningcited by 1
  19. 2021Robust Stability of Neural Network-controlled Nonlinear Systems with Parametric Variabilitycited by 1
  20. 2022Tailored max-out networks for learning convex PWQ functionscited by 1
  21. 2024On Building Myopic MPC Policies using Supervised Learningcited by 1
  22. 2024Mapping back and forth between model predictive control and neural networkscited by 1
  23. 2025System Identification and Control Using Lyapunov-Based Deep Neural Networks without Persistent Excitation: A Concurrent Learning Approachcited by 1
  24. 2025Partially-Supervised Neural Network Model For Quadratic Multiparametric Programmingcited by 1
  25. 2026Neural Operators for Multi-Task Control and Adaptationcited by 1
  26. 2021Online-Learning Deep Neuro-Adaptive Dynamic Inversion Controller for Model Free Controlcited by 0
  27. 2021Neural Network Based Model Predictive Control for an Autonomous Vehiclecited by 0
  28. 2021Error-free approximation of explicit linear MPC through lattice piecewise affine expressioncited by 0
  29. 2024Deep-MPC: A DAGGER-Driven Imitation Learning Strategy for Optimal Constrained Battery Chargingcited by 0
  30. 2024Improved Sample Complexity of Imitation Learning for Barrier Model Predictive Controlcited by 0
  31. 2024Safe and High-Performance Learning of Model Predicitve Control using Kernel-Based Interpolationcited by 0
  32. 2024CIKAN: Constraint Informed Kolmogorov-Arnold Networks for Autonomous Spacecraft Rendezvous using Time Shift Governorcited by 0
  33. 2025Bridging the Sim-to-real Gap: A Control Framework for Imitation Learning of Model Predictive Controlcited by 0
  34. 2025A Neural Network Framework for Discovering Closed-form Solutions to Quadratic Programs with Linear Constraintscited by 0
  35. 2025Data-driven Projection Generation for Efficiently Solving Heterogeneous Quadratic Programming Problemscited by 0
  36. 2025Zero-Shot Function Encoder-Based Differentiable Predictive Controlcited by 0
  37. 2025Optimizing Operation Recipes with Reinforcement Learning for Safe and Interpretable Control of Chemical Processescited by 0
  38. 2026Accounting for Optimal Control in the Sizing of Isolated Hybrid Renewable Energy Systems Using Imitation Learningcited by 0

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