# Imposing Exact Safety Specifications in Neural Reachable Tubes

**Authors:** Aditya Singh, Zeyuan Feng, Somil Bansal  
**arXiv:** [2404.00814](https://arxiv.org/abs/2404.00814)  
**Date:** 2024-03-31  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2404.00814v1/

## Key points

- The Hamilton-Jacobi (HJ) reachability analysis is a crucial verification tool for ensuring safety and performance guarantees in autonomous systems.
- It can handle nonlinear dynamical systems with bounded adversarial disturbances and constraints on states and inputs.
- Computational complexity of solving the partial differential equation (PDE) scales exponentially with state dimension, making it challenging for large-scale systems.
- DeepReach, a learning-based approach using neural networks, approximates high-dimensional reachable tubes but faces accuracy challenges as system complexity increases due to imprecise imposition of safety constraints during learning.
- Proposed variant of DeepReach exacts imposes safety constraints during learning by restructuring the value function as a weighted sum of boundary conditions and neural network output, leading to significant improvements in accuracy for tasks like rocket landing and multivehicle collision avoidance.
- Terminal time gradients play a crucial role in curriculum training stages, impacting model performance and suggesting opportunities for refining training strategies to enhance accuracy and efficiency.

## Summary

The Hamilton-Jacobi (HJ) reachability analysis is a crucial verification tool for ensuring safety and performance guarantees in autonomous systems. It is highly valued for its ability to handle nonlinear dynamical systems with bounded adversarial disturbances and constraints on states and inputs. However, the computational complexity of solving the partial differential equation (PDE) to compute a safety value function scales exponentially with the state dimension, making it challenging to apply directly in large-scale systems. Recently, a learning-based approach called DeepReach has been introduced to approximate high-dimensional reachable tubes using neural networks. While DeepReach has shown effectiveness, the accuracy of the learned solution tends to decrease as system complexity increases. One of the reasons for this decline is the imprecise imposition of safety constraints during the learning process, particularly related to the PDE's boundary conditions. DeepReach incorporates boundary conditions as soft constraints in the loss function, leaving room for errors during value function learning. Additionally, balancing the contributions from imposing boundary conditions and solving the PDE in the loss function can introduce inaccuracies in the overall learned solution. In response to these challenges, this work proposes a variant of DeepReach that exacts imposes safety constraints during learning by restructuring the overall value function as a weighted sum of boundary conditions and neural network output. This eliminates the need for a separate boundary loss during training and avoids adjustments in loss functions. The efficacy of this approach is demonstrated through significant improvements in accuracy for challenging high-dimensional reachability tasks such as rocket landing and multivehicle collision avoidance problems. Moreover, additional insights from related research suggest that terminal time gradients play a crucial role at different stages of curriculum training, emphasizing their impact on model performance. These findings highlight opportunities for further refinement in training strategies to enhance model accuracy and efficiency. Overall, by addressing issues related to safety constraint imposition and refining training methodologies based on key takeaways from previous studies, this refined approach offers promising advancements in improving accuracy and scalability for high-dimensional reachability tasks in autonomous systems.