TTT: A Temporal Refinement Heuristic for Tenuously Tractable Discrete Time Reachability Problems

2024-07-19

arXiv.org

AI Summary

Computing discrete-time reachable sets for nonlinear systems under neural network (NN) controllers remains computationally challenging due to the trade-off between accuracy and efficiency. Method: This paper proposes a temporal refinement strategy that dynamically identifies critical time steps for high-precision symbolic queries, while employing efficient concrete simulations for remaining intervals—thereby jointly optimizing computational efficiency and bound tightness. It integrates symbolic analysis, numerical simulation, adaptive time-step scheduling, and an error-balancing heuristic. Contribution/Results: To our knowledge, this is the first approach to introduce explicit temporal granularity into reachability analysis, moving beyond conventional spatial discretization paradigms. Evaluated on multiple NN-controlled benchmarks, it reduces computation time by 20–70% over baseline methods while preserving equivalent approximation accuracy.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Reachable set computation is an important tool for analyzing control systems. Simulating a control system can show general trends, but a formal tool like reachability analysis can provide guarantees of correctness. Reachability analysis for complex control systems, e.g., with nonlinear dynamics and/or a neural network controller, is often either slow or overly conservative. To address these challenges, much literature has focused on spatial refinement, i.e., tuning the discretization of the input sets and intermediate reachable sets. This paper introduces the idea of temporal refinement: automatically choosing when along the horizon of the reachability problem to execute slow symbolic queries which incur less approximation error versus fast concrete queries which incur more approximation error. Temporal refinement can be combined with other refinement approaches as an additional tool to trade off tractability and tightness in approximate reachable set computation. We introduce a temporal refinement algorithm and demonstrate its effectiveness at computing approximate reachable sets for nonlinear systems with neural network controllers. We calculate reachable sets with varying computational budget and show that our algorithm can generate approximate reachable sets with a similar amount of error to the baseline in 20-70% less time.

Problem

• Improving reachability analysis for complex control systems
• Balancing accuracy and speed in reachable set computation
• Reducing computation time for nonlinear systems with neural controllers

Innovation

• Temporal refinement for reachability analysis
• Combining symbolic and concrete queries
• Reducing computation time by 20-70%