Bridging Model Predictive Control and Deep Learning for Scalable Reachability Analysis

TL;DR

This work addresses the scalability and reliability of HJ reachability for safety-critical robotics by integrating model predictive control (MPC) with DeepReach-style learning. It uses a sampling-based MPC to generate approximate safety value function labels, which are then used to supervise and stabilize neural learning of the HJB-VI solution, with a curriculum-based training and iterative MPC dataset refinement guided by the learned policy. The approach is complemented by conformal prediction to provide probabilistic safety guarantees for the learned value function and its induced safe policy. Across four challenging high-dimensional systems (2D vertical drone, 13D quadrotor, 7D F1Tenth, 40D publisher-subscriber), the method yields larger verified safe sets and lower mean-squared errors than baselines, demonstrating improved robustness, data efficiency, and scalability for real-world safety assurances.

Abstract

Hamilton-Jacobi (HJ) reachability analysis is a widely used method for ensuring the safety of robotic systems. Traditional approaches compute reachable sets by numerically solving an HJ Partial Differential Equation (PDE) over a grid, which is computationally prohibitive due to the curse of dimensionality. Recent learning-based methods have sought to address this challenge by approximating reachability solutions using neural networks trained with PDE residual error. However, these approaches often suffer from unstable training dynamics and suboptimal solutions due to the weak learning signal provided by the residual loss. In this work, we propose a novel approach that leverages model predictive control (MPC) techniques to guide and accelerate the reachability learning process. Observing that HJ reachability is inherently rooted in optimal control, we utilize MPC to generate approximate reachability solutions at key collocation points, which are then used to tactically guide the neural network training by ensuring compliance with these approximations. Moreover, we iteratively refine the MPC generated solutions using the learned reachability solution, mitigating convergence to local optima. Case studies on a 2D vertical drone, a 13D quadrotor, a 7D F1Tenth car, and a 40D publisher-subscriber system demonstrate that bridging MPC with deep learning yields significant improvements in the robustness and accuracy of reachable sets, as well as corresponding safety assurances, compared to existing methods.

Figures (7)

• Figure 1: We propose a framework that efficiently generates approximate safety value function datasets using a sampling-based MPC approach and integrates these data labels to guide the learning of reachability solutions for high-dimensional autonomous systems. The learned value function is then verified using conformal prediction, providing probabilistic safety assurances for the system under the induced safe policy.
• Figure 2: Parameterized Vertical Drone: Value function slices at $K=12$. The brown lines represent the ground and the ceiling -- the failure set in this case. In (a), the solid black lines are the contours of ground-truth safe sets. In (b), all MPC data samples with $K\in (11,12)$ used for the proposed approach are shown. (c) illustrates the learned safe set using the neural CBF approach. For (d), (e), and (f), the dashed contours illustrate the learned safe sets and the solid contours represent the verified safe sets. Note that the black solid contours are missing in (d) and (e) since Vanilla DeepReach and MPC Distillation have an empty verified safe set.
• Figure 3: 13D Quadrotor: slices of learned value functions on the X-Y plane at $(p_z,q_\omega,q_x,q_y,q_z, v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)=(0.54, 0.44, -0.45, 0.27, -0.73, 5.00, -1.07, -3.34, 3.19, -2.80, 3.43)$. The brown circles represent the cylinder obstacle (not the failure set contour due to the quadrotor’s nonzero collision radius). Again, the solid contours are the verified safe sets while the dashed contours are the learned BRTs. The dashed lines are selected rollout trajectories, where red color indicates collisions.
• Figure 4: The x-axis shows the size of non-bootstrapped dataset. The blue line represents the mean recovered volumes along with their standard deviation across five random seeds, whereas the red line shows the mean false positive rates.
• Figure 5: F1Tenth: X-Y Plane Slices of the Learned Value Functions. The remaining state variables are $(\phi , v , \theta_{yaw} , \omega_{yaw} , \beta_{slip})=(0,8,0,0,0)$. The track boundaries are shown as the brown contours. The black solid contours are the verified safe sets while the dashed contours are the learned BRTs. Note that the black solid contours are missing in (a), (b), and (d) since these baselines have an empty verified safe set. On the other hand, the black solid contour and dashed contour overlap significantly for the proposed method, with $\delta=0.12$, highlighting the high accuracy of the learned safety value function.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3