Enhancing the Performance of DeepReach on High-Dimensional Systems through Optimizing Activation Functions

TL;DR

This work tackles the challenge of high-dimensional Hamilton-Jacobi reachability by enhancing DeepReach through activation-function design. By intertwining sine and ReLU activations across a 3-hidden-layer network, the authors aim to better approximate the value function $V(x,t)$ that solves the Hamilton-Jacobi-Isaacs PDE, validating on a 3D Air3D case and a 9D multi-vehicle problem. They find that increasing sine-layer count improves BRT accuracy, and that placing sine activations on the first and last layers significantly impacts performance, achieving a best observed violation rate around $18.43\%$. These results suggest a practical path to scaling learning-based reachability to higher-dimensional systems and point to future work on architectural search and error-correction enhancements for real-time safety verification.

Abstract

With the continuous advancement in autonomous systems, it becomes crucial to provide robust safety guarantees for safety-critical systems. Hamilton-Jacobi Reachability Analysis is a formal verification method that guarantees performance and safety for dynamical systems and is widely applicable to various tasks and challenges. Traditionally, reachability problems are solved by using grid-based methods, whose computational and memory cost scales exponentially with the dimensionality of the system. To overcome this challenge, DeepReach, a deep learning-based approach that approximately solves high-dimensional reachability problems, is proposed and has shown lots of promise. In this paper, we aim to improve the performance of DeepReach on high-dimensional systems by exploring different choices of activation functions. We first run experiments on a 3D system as a proof of concept. Then we demonstrate the effectiveness of our approach on a 9D multi-vehicle collision problem.

Figures (5)

• Figure 1: Baseline experiment on air3D
• Figure 2: The slices of BRT for t=0.9 and theta=$\pi/2$ on air3D. The BRT in crimson is obtained from helperOC as the “ground truth” to compare with results from DeepReach (in light red). The four figures correspond to the four experiments we ran using different combinations of Sine and ReLU: upper left (exp1), upper right (exp2), bottom left (exp3), bottom right (exp4)
• Figure 3: Base case experiments on 9D multi-vehicle collision. left (exp10) right (exp9). The brown region is the union of pairwise BRTs whereas the light blue region is the learned from the whole 9D system. The overlapping region between the two BRTs is purple.
• Figure 4: Upper left (exp5), upper right (exp6), bottom left (exp7), bottom right (exp8). The coloring used is the same as in Figure 3.
• Figure 5: x-axis: activation structures of DeepReach; y-axis: violation rate in percentage