# Certified Approximate Reachability (CARe): Formal Error Bounds on Deep Learning of Reachable Sets

arXiv: https://arxiv.org/abs/2503.23912 · HTML: https://mlsanity.com/paper/2503.23912/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 36 references that are in the map.
- First posted on arXiv 31 Mar 2025.

## Builds on (most cited first, 12 of 12)

- [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)
- [Robust Control Barrier-Value Functions for Safety-Critical Control](https://mlsanity.com/paper/2104.02808/) (2021)
- [Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction](https://mlsanity.com/paper/2312.08604/) (2023)
- [Decomposition of Reachable Sets and Tubes for a Class of Nonlinear Systems](https://mlsanity.com/paper/1611.00122/) (2016)
- [FaSTrack: a Modular Framework for Real-Time Motion Planning and Guaranteed Safe Tracking](https://mlsanity.com/paper/2102.07039/) (2021)
- [A Physics-Informed Machine Learning Framework for Safe and Optimal Control of Autonomous Systems](https://mlsanity.com/paper/2502.11057/) (2025)
- [Generating Formal Safety Assurances for High-Dimensional Reachability](https://mlsanity.com/paper/2209.12336/) (2022)
- [Multi-objective minimum time optimal control for low-thrust trajectory design](https://mlsanity.com/paper/2103.08813/) (2021)
- [Verification of safety critical control policies using kernel methods](https://mlsanity.com/paper/2203.12407/) (2022)
- [Safety-Aware Hybrid Control of Airborne Wind Energy Systems](https://mlsanity.com/paper/2212.07199/) (2022)
- [Multi-objective low-thrust spacecraft trajectory design using reachability analysis](https://mlsanity.com/paper/2212.07209/) (2022)

## Built on by (most cited first, 2 of 2)

- [Certified Neural Approximations of Nonlinear Dynamics](https://mlsanity.com/paper/2505.15497/) (2025)
- [Certifying Hamilton-Jacobi Reachability Learned via Reinforcement Learning](https://mlsanity.com/paper/2602.16475/) (2026)

## Abstract (arXiv)

Recent approaches to leveraging deep learning for computing reachable sets of continuous-time dynamical systems have gained popularity over traditional level-set methods, as they overcome the curse of dimensionality. However, as with level-set methods, considerable care needs to be taken in limiting approximation errors, particularly since no guarantees are provided during training on the accuracy of the learned reachable set. To address this limitation, we introduce an epsilon-approximate Hamilton-Jacobi Partial Differential Equation (HJ-PDE), which establishes a relationship between training loss and accuracy of the true reachable set. To formally certify this approximation, we leverage Satisfiability Modulo Theories (SMT) solvers to bound the residual error of the HJ-based loss function across the domain of interest. Leveraging Counter Example Guided Inductive Synthesis (CEGIS), we close the loop around learning and verification, by fine-tuning the neural network on counterexamples found by the SMT solver, thus improving the accuracy of the learned reachable set. To the best of our knowledge, Certified Approximate Reachability (CARe) is the first approach to provide soundness guarantees on learned reachable sets of continuous dynamical systems.