Backward Reachability Analysis of Neural Feedback Loops: Techniques for Linear and Nonlinear Systems

TL;DR

This paper develops backward reachability methods for neural feedback loops, enabling safety certification of systems with neural network controllers by leveraging forward analysis tools for both linear and nonlinear systems.

Contribution

It introduces a novel framework that uses forward NN analysis to efficiently approximate backward reachable sets in neural feedback loops, addressing nonlinearities and noninvertibility issues.

Findings

Effective algorithms for backward reachability in neural feedback loops.

Successful safety certification in a 6D system example.

Framework applicable to both linear and nonlinear systems.

Abstract

As neural networks (NNs) become more prevalent in safety-critical applications such as control of vehicles, there is a growing need to certify that systems with NN components are safe. This paper presents a set of backward reachability approaches for safety certification of neural feedback loops (NFLs), i.e., closed-loop systems with NN control policies. While backward reachability strategies have been developed for systems without NN components, the nonlinearities in NN activation functions and general noninvertibility of NN weight matrices make backward reachability for NFLs a challenging problem. To avoid the difficulties associated with propagating sets backward through NNs, we introduce a framework that leverages standard forward NN analysis tools to efficiently find over-approximations to backprojection (BP) sets, i.e., sets of states for which an NN policy will lead a system to a given target set. We present frameworks for calculating BP over approximations for both linear and nonlinear systems with control policies represented by feedforward NNs and propose computationally efficient strategies. We use numerical results from a variety of models to showcase the proposed algorithms, including a demonstration of safety certification for a 6D system.