# BERN-NN: Tight Bound Propagation For Neural Networks Using Bernstein Polynomial Interval Arithmetic

arXiv: https://arxiv.org/abs/2211.14438 · HTML: https://mlsanity.com/paper/2211.14438/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 45 references that are in the map.
- First posted on arXiv 22 Nov 2022.

## Builds on (most cited first, 25 of 25)

- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification](https://mlsanity.com/paper/2103.06624/) (2021)
- [Algorithms for Verifying Deep Neural Networks](https://mlsanity.com/paper/1903.06758/) (2019)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [Formal Security Analysis of Neural Networks using Symbolic Intervals](https://mlsanity.com/paper/1804.10829/) (2018)
- [Efficient Formal Safety Analysis of Neural Networks](https://mlsanity.com/paper/1809.08098/) (2018)
- [Branch and Bound for Piecewise Linear Neural Network Verification](https://mlsanity.com/paper/1909.06588/) (2019)
- [NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2004.05519/) (2020)
- [An approach to reachability analysis for feed-forward ReLU neural networks](https://mlsanity.com/paper/1706.07351/) (2017)
- [A Dual Approach to Scalable Verification of Deep Networks](https://mlsanity.com/paper/1803.06567/) (2018)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [Maximum Resilience of Artificial Neural Networks](https://mlsanity.com/paper/1705.01040/) (2017)
- [Measuring Neural Net Robustness with Constraints](https://mlsanity.com/paper/1605.07262/) (2016)
- [Verisig: verifying safety properties of hybrid systems with neural network controllers](https://mlsanity.com/paper/1811.01828/) (2018)
- [Output Reachable Set Estimation and Verification for Multi-Layer Neural Networks](https://mlsanity.com/paper/1708.03322/) (2017)
- [The Second International Verification of Neural Networks Competition (VNN-COMP 2021): Summary and Results](https://mlsanity.com/paper/2109.00498/) (2021)
- [Formal Verification of Neural Network Controlled Autonomous Systems](https://mlsanity.com/paper/1810.13072/) (2018)
- [POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems](https://mlsanity.com/paper/2106.13867/) (2021)
- [Reachable Set Computation and Safety Verification for Neural Networks with ReLU Activations](https://mlsanity.com/paper/1712.08163/) (2017)
- [Formal Analysis and Redesign of a Neural Network-Based Aircraft Taxiing System with VerifAI](https://mlsanity.com/paper/2005.07173/) (2020)
- [Reachable Set Estimation and Safety Verification for Piecewise Linear Systems with Neural Network Controllers](https://mlsanity.com/paper/1802.06981/) (2018)
- [Neurosymbolic Motion and Task Planning for Linear Temporal Logic Tasks](https://mlsanity.com/paper/2210.05180/) (2022)
- [Provably Correct Training of Neural Network Controllers Using Reachability Analysis](https://mlsanity.com/paper/2102.10806/) (2021)

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

- [DeepBern-Nets: Taming the Complexity of Certifying Neural Networks using Bernstein Polynomial Activations and Precise Bound Propagation](https://mlsanity.com/paper/2305.13508/) (2023)
- [Real-Time Safe Control of Neural Network Dynamic Models with Sound Approximation](https://mlsanity.com/paper/2404.13456/) (2024)
- [Automated Reachability Analysis of Neural Network-Controlled Systems via Adaptive Polytopes](https://mlsanity.com/paper/2212.07553/) (2022)
- [Language Models That Walk the Talk: A Framework for Formal Fairness Certificates](https://mlsanity.com/paper/2505.12767/) (2025)
- [GeoCert: Certified Geometric AI for Reliable Forecasting](https://mlsanity.com/paper/2604.23474/) (2026)

## Abstract (arXiv)

In this paper, we present BERN-NN as an efficient tool to perform bound propagation of Neural Networks (NNs). Bound propagation is a critical step in wide range of NN model checkers and reachability analysis tools. Given a bounded input set, bound propagation algorithms aim to compute tight bounds on the output of the NN. So far, linear and convex optimizations have been used to perform bound propagation. Since neural networks are highly non-convex, state-of-the-art bound propagation techniques suffer from introducing large errors. To circumvent such drawback, BERN-NN approximates the bounds of each neuron using a class of polynomials called Bernstein polynomials. Bernstein polynomials enjoy several interesting properties that allow BERN-NN to obtain tighter bounds compared to those relying on linear and convex approximations. BERN-NN is efficiently parallelized on graphic processing units (GPUs). Extensive numerical results show that bounds obtained by BERN-NN are orders of magnitude tighter than those obtained by state-of-the-art verifiers such as linear programming and linear interval arithmetic. Moreoveer, BERN-NN is both faster and produces tighter outputs compared to convex programming approaches like alpha-CROWN.