[Paper Review] Neural Network Based Model Predictive Control for an Autonomous Vehicle

Maria Luiza Costa Vianna, Éric Goubault+1 more|arXiv (Cornell University)|Jul 30, 2021

Advanced Control Systems Optimization20 references4 citations

TL;DR

This paper proposes training small, efficient neural networks via supervised and reinforcement learning to replace computationally expensive Model Predictive Control (MPC) in autonomous vehicles. Using a bicycle model, it demonstrates that a shallow 3-layer network with sigmoid activations achieves sub-1 cm tracking error, outperforming reinforcement learning and enabling real-time deployment and formal verification.

ABSTRACT

We study learning based controllers as a replacement for model predictive controllers (MPC) for the control of autonomous vehicles. We concentrate for the experiments on the simple yet representative bicycle model. We compare training by supervised learning and by reinforcement learning. We also discuss the neural net architectures so as to obtain small nets with the best performances. This work aims at producing controllers that can both be embedded on real-time platforms and amenable to verification by formal methods techniques.

Motivation & Objective

• Replace high-complexity MPC controllers in autonomous vehicles with lightweight neural network controllers.
• Investigate whether supervised and reinforcement learning can accurately approximate MPC behavior for real-time deployment.
• Optimize neural network architecture to minimize size and computational cost while maintaining high control precision.
• Ensure the resulting neural controllers are amenable to formal verification for safety-critical applications.
• Evaluate generalization performance of trained networks on complex, unseen trajectories using data from simple training trajectories.

Proposed method

• Train feedforward neural networks using supervised learning on state-action pairs generated by an MPC controller.
• Use relative vehicle-to-trajectory state as input, including future reference points over a finite horizon (20 steps).
• Train with varying network architectures: depth (1–3 layers), width (10–100 neurons), and activation functions (ReLU, tanh, sigmoid).
• Apply reinforcement learning using an actor-critic framework with a two-layer network (400, 300 neurons) to learn MPC policy directly from environment interaction.
• Compare performance using mean, maximum, and standard deviation of tracking error on validation trajectories.
• Use formal verification-friendly network design principles to prioritize small, simple architectures for safety validation.

Experimental results

Research questions

• RQ1 Can a small neural network trained via supervised learning accurately replicate the behavior of an MPC controller for autonomous vehicle path tracking?
• RQ2 How do architectural choices—such as depth, width, and activation functions—affect the performance and efficiency of the neural controller?
• RQ3 Does reinforcement learning produce a controller of comparable accuracy to supervised learning when the system dynamics are known?
• RQ4 Can a network trained on simple trajectories (e.g., straight lines) generalize effectively to complex, real-world trajectories?
• RQ5 To what extent does the input representation (e.g., full future trajectory vs. relative state) influence learning performance and generalization?

Key findings

• A 3-layer neural network with 10 neurons per layer and sigmoid activation in hidden layers achieved the lowest mean error (0.059 cm) and standard deviation (0.03 cm), outperforming ReLU and tanh.
• Supervised learning achieved a maximum tracking error of 0.39 cm, significantly lower than reinforcement learning’s 0.97 cm, indicating superior precision.
• The neural network trained on simple trajectories (e.g., straight lines) generalized effectively to complex trajectories, with a mean error of 0.077 cm on validation data.
• Adding more layers increased computational cost more than increasing neuron count, making deeper networks less efficient for this problem.
• Reinforcement learning required a much larger network (400, 300 neurons) to achieve acceptable performance, with evaluation time of 0.11ms per step, compared to 6 μs for the best supervised network.
• The final neural controller achieved less than 1 cm maximum deviation from the MPC trajectory, demonstrating strong approximation capability and suitability for real-time deployment.