# Bridging Dimensions: Confident Reachability for High-Dimensional Controllers

arXiv: https://arxiv.org/abs/2311.04843 · HTML: https://mlsanity.com/paper/2311.04843/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 67 references that are in the map.
- First posted on arXiv 8 Nov 2023.

## Builds on (most cited first, 30 of 37)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Knowledge Distillation: A Survey](https://mlsanity.com/paper/2006.05525/) (2020)
- [End-to-end Driving via Conditional Imitation Learning](https://mlsanity.com/paper/1710.02410/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2004.05519/) (2020)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [Verisig: verifying safety properties of hybrid systems with neural network controllers](https://mlsanity.com/paper/1811.01828/) (2018)
- [Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators](https://mlsanity.com/paper/1903.01014/) (2019)
- [Conformal Prediction Under Feedback Covariate Shift for Biomolecular Design](https://mlsanity.com/paper/2202.03613/) (2022)
- [Towards Formal XAI: Formally Approximate Minimal Explanations of Neural Networks](https://mlsanity.com/paper/2210.13915/) (2022)
- [Conformal Prediction for Time Series with Modern Hopfield Networks](https://mlsanity.com/paper/2303.12783/) (2023)
- [Sample-Efficient Safety Assurances using Conformal Prediction](https://mlsanity.com/paper/2109.14082/) (2021)
- [POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems](https://mlsanity.com/paper/2106.13867/) (2021)
- [Conformal Prediction for STL Runtime Verification](https://mlsanity.com/paper/2211.01539/) (2022)
- [Conformal Prediction Regions for Time Series using Linear Complementarity Programming](https://mlsanity.com/paper/2304.01075/) (2023)
- [Verification of Image-based Neural Network Controllers Using Generative Models](https://mlsanity.com/paper/2105.07091/) (2021)
- [Verifying Controllers with Convolutional Neural Network-based Perception: A Case for Intelligible, Safe, and Precise Abstractions](https://mlsanity.com/paper/2111.05534/) (2021)
- [NNLander-VeriF: A Neural Network Formal Verification Framework for Vision-Based Autonomous Aircraft Landing](https://mlsanity.com/paper/2203.15841/) (2022)
- [Closed-loop Analysis of Vision-based Autonomous Systems: A Case Study](https://mlsanity.com/paper/2302.04634/) (2023)
- [DRYVR:Data-driven verification and compositional reasoning for automotive systems](https://mlsanity.com/paper/1702.06902/) (2017)
- [DiffRNN: Differential Verification of Recurrent Neural Networks](https://mlsanity.com/paper/2007.10135/) (2020)
- [A Simple and Efficient Sampling-based Algorithm for General Reachability Analysis](https://mlsanity.com/paper/2112.05745/) (2021)
- [POLAR-Express: Efficient and Precise Formal Reachability Analysis of Neural-Network Controlled Systems](https://mlsanity.com/paper/2304.01218/) (2023)
- [Distributionally Robust Statistical Verification with Imprecise Neural Networks](https://mlsanity.com/paper/2308.14815/) (2023)
- [PAC Model Checking of Black-Box Continuous-Time Dynamical Systems](https://mlsanity.com/paper/2007.10141/) (2020)
- [Assured Autonomy: Path Toward Living With Autonomous Systems We Can Trust](https://mlsanity.com/paper/2010.14443/) (2020)

## Built on by (most cited first, 5 of 5)

- [Four Principles for Physically Interpretable World Models](https://mlsanity.com/paper/2503.02143/) (2025)
- [Zero-shot Safety Prediction for Autonomous Robots with Foundation World Models](https://mlsanity.com/paper/2404.00462/) (2024)
- [NeuroStrata: Harnessing Neurosymbolic Paradigms for Improved Design, Testability, and Verifiability of Autonomous CPS](https://mlsanity.com/paper/2502.12267/) (2025)
- [Language-Enhanced Latent Representations for Out-of-Distribution Detection in Autonomous Driving](https://mlsanity.com/paper/2405.01691/) (2024)
- [Deterministic World Models for Closed-loop Reachability Analysis of End-to-End Vision-based Control](https://mlsanity.com/paper/2512.08991/) (2025)

## Abstract (arXiv)

Autonomous systems are increasingly implemented using end-to-end learning-based controllers. Such controllers make decisions that are executed on the real system, with images as one of the primary sensing modalities. Deep neural networks form a fundamental building block of such controllers. Unfortunately, the existing neural-network verification tools do not scale to inputs with thousands of dimensions -- especially when the individual inputs (such as pixels) are devoid of clear physical meaning. This paper takes a step towards connecting exhaustive closed-loop verification with high-dimensional controllers. Our key insight is that the behavior of a high-dimensional controller can be approximated with several low-dimensional controllers. To balance the approximation accuracy and verifiability of our low-dimensional controllers, we leverage the latest verification-aware knowledge distillation. Then, we inflate low-dimensional reachability results with statistical approximation errors, yielding a high-confidence reachability guarantee for the high-dimensional controller. We investigate two inflation techniques -- based on trajectories and control actions -- both of which show convincing performance in three OpenAI gym benchmarks.