Forward and Backward Reachability Analysis of Closed-loop Recurrent Neural Networks via Hybrid Zonotopes

TL;DR

This paper introduces a hybrid zonotope-based method for exact forward and backward reachability analysis of closed-loop RNNs with ReLU activations, improving scalability and providing safety certification.

Contribution

It presents a novel hybrid zonotope approach that computes exact reachable sets without unrolling and introduces a tunable relaxation scheme for scalability.

Findings

Effective computation of reachable sets for RNNs.

Scalable relaxation scheme balances accuracy and complexity.

Certifies safety of closed-loop RNN systems.

Abstract

Recurrent neural networks (RNNs) are widely employed to model complex dynamical systems due to their hidden-state structure, which inherently captures temporal dependencies. This work presents a hybrid zonotope-based approach for computing exact forward and backward reachable sets of closed-loop RNN systems with ReLU activation functions. The method formulates state-pair sets to compute reachable sets as hybrid zonotopes without requiring unrolling. To improve scalability, a tunable relaxation scheme is proposed that ranks unstable ReLU units across all layers using a triangle-area score and selectively applies convex relaxations within a fixed binary limit in the hybrid zonotopes. This scheme enables an explicit tradeoff between computational complexity and approximation accuracy, with exact reachability as a special case. In addition, a sufficient condition is derived to certify the safety of closed-loop RNN systems. Numerical examples demonstrate the effectiveness of the proposed approach.