# Generating Formal Safety Assurances for High-Dimensional Reachability

arXiv: https://arxiv.org/abs/2209.12336 · HTML: https://mlsanity.com/paper/2209.12336/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 30 references that are in the map.
- First posted on arXiv 25 Sep 2022.

## Builds on (most cited first, 10 of 10)

- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [Funnel Libraries for Real-Time Robust Feedback Motion Planning](https://mlsanity.com/paper/1601.04037/) (2016)
- [Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures](https://mlsanity.com/paper/1910.09045/) (2019)
- [A Neural Network Approach for High-Dimensional Optimal Control Applied to Multi-Agent Path Finding](https://mlsanity.com/paper/2104.03270/) (2021)
- [A Classification-based Approach for Approximate Reachability](https://mlsanity.com/paper/1803.03237/) (2018)
- [Technical Report: Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures](https://mlsanity.com/paper/1305.7484/) (2013)
- [Numerical Verification of Affine Systems with up to a Billion Dimensions](https://mlsanity.com/paper/1804.01583/) (2018)
- [Synthesis of separable controlled invariant sets for modular local control design](https://mlsanity.com/paper/1510.02990/) (2015)
- [A New Simulation Metric to Determine Safe Environments and Controllers for Systems with Unknown Dynamics](https://mlsanity.com/paper/1902.10320/) (2019)

## Built on by (most cited first, 11 of 11)

- [The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systems](https://mlsanity.com/paper/2309.05837/) (2023)
- [Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction](https://mlsanity.com/paper/2312.08604/) (2023)
- [Data-Driven Reachability Analysis of Stochastic Dynamical Systems with Conformal Inference](https://mlsanity.com/paper/2309.09187/) (2023)
- [Human-AI Safety: A Descendant of Generative AI and Control Systems Safety](https://mlsanity.com/paper/2405.09794/) (2024)
- [AnySafe: Adapting Latent Safety Filters at Runtime via Safety Constraint Parameterization in the Latent Space](https://mlsanity.com/paper/2509.19555/) (2025)
- [Statistical Reachability Analysis of Stochastic Cyber-Physical Systems under Distribution Shift](https://mlsanity.com/paper/2407.11609/) (2024)
- [Value Approximation for Two-Player General-Sum Differential Games with State Constraints](https://mlsanity.com/paper/2311.16520/) (2023)
- [Learning Maximal Safe Sets Using Hypernetworks for MPC-based Local Trajectory Planning in Unknown Environments](https://mlsanity.com/paper/2410.20267/) (2024)
- [Reachability Barrier Networks: Learning Hamilton-Jacobi Solutions for Smooth and Flexible Control Barrier Functions](https://mlsanity.com/paper/2505.11755/) (2025)
- [Certified Approximate Reachability (CARe): Formal Error Bounds on Deep Learning of Reachable Sets](https://mlsanity.com/paper/2503.23912/) (2025)
- [Gait Switching and Enhanced Stabilization of Walking Robots with Deep Learning-based Reachability: A Case Study on Two-link Walker](https://mlsanity.com/paper/2409.16301/) (2024)

## Abstract (arXiv)

Providing formal safety and performance guarantees for autonomous systems is becoming increasingly important. Hamilton-Jacobi (HJ) reachability analysis is a popular formal verification tool for providing these guarantees, since it can handle general nonlinear system dynamics, bounded adversarial system disturbances, and state and input constraints. However, it involves solving a PDE, whose computational and memory complexity scales exponentially with respect to the state dimensionality, making its direct use on large-scale systems intractable. A recently proposed method called DeepReach overcomes this challenge by leveraging a sinusoidal neural PDE solver for high-dimensional reachability problems, whose computational requirements scale with the complexity of the underlying reachable tube rather than the state space dimension. Unfortunately, neural networks can make errors and thus the computed solution may not be safe, which falls short of achieving our overarching goal to provide formal safety assurances. In this work, we propose a method to compute an error bound for the DeepReach solution. This error bound can then be used for reachable tube correction, resulting in a safe approximation of the true reachable tube. We also propose a scenario-based optimization approach to compute a probabilistic bound on this error correction for general nonlinear dynamical systems. We demonstrate the efficacy of the proposed approach in obtaining probabilistically safe reachable tubes for high-dimensional rocket-landing and multi-vehicle collision-avoidance problems.