# Towards Robust Neural Networks via Close-loop Control

arXiv: https://arxiv.org/abs/2102.01862 · HTML: https://mlsanity.com/paper/2102.01862/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 32 references that are in the map.
- First posted on arXiv 3 Feb 2021.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [DeepFool: a simple and accurate method to fool deep neural networks](https://mlsanity.com/paper/1511.04599/) (2015)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images](https://mlsanity.com/paper/1412.1897/) (2014)
- [A Survey of Deep Learning Techniques for Autonomous Driving](https://mlsanity.com/paper/1910.07738/) (2019)
- [Defense-GAN: Protecting Classifiers Against Adversarial Attacks Using Generative Models](https://mlsanity.com/paper/1805.06605/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [PixelDefend: Leveraging Generative Models to Understand and Defend against Adversarial Examples](https://mlsanity.com/paper/1710.10766/) (2017)
- [On Detecting Adversarial Perturbations](https://mlsanity.com/paper/1702.04267/) (2017)
- [You Only Propagate Once: Accelerating Adversarial Training via Maximal Principle](https://mlsanity.com/paper/1905.00877/) (2019)
- [Towards Robust Neural Networks via Random Self-ensemble](https://mlsanity.com/paper/1712.00673/) (2017)
- [An overview of deep learning in medical imaging focusing on MRI](https://mlsanity.com/paper/1811.10052/) (2018)
- [Exploring the Landscape of Spatial Robustness](https://mlsanity.com/paper/1712.02779/) (2017)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [On the Geometry of Adversarial Examples](https://mlsanity.com/paper/1811.00525/) (2018)
- [The Robust Manifold Defense: Adversarial Training using Generative Models](https://mlsanity.com/paper/1712.09196/) (2017)
- [Amata: An Annealing Mechanism for Adversarial Training Acceleration](https://mlsanity.com/paper/2012.08112/) (2020)
- [DDPNOpt: Differential Dynamic Programming Neural Optimizer](https://mlsanity.com/paper/2002.08809/) (2020)
- [A Differential Game Theoretic Neural Optimizer for Training Residual Networks](https://mlsanity.com/paper/2007.08880/) (2020)

## Built on by (most cited first, 23 of 23)

- [Delta Tuning: A Comprehensive Study of Parameter Efficient Methods for Pre-trained Language Models](https://mlsanity.com/paper/2203.06904/) (2022)
- [Evaluating the Adversarial Robustness of Adaptive Test-time Defenses](https://mlsanity.com/paper/2202.13711/) (2022)
- [Visual Prompting for Adversarial Robustness](https://mlsanity.com/paper/2210.06284/) (2022)
- [On Robust Prefix-Tuning for Text Classification](https://mlsanity.com/paper/2203.10378/) (2022)
- [A Closer Look at the Adversarial Robustness of Deep Equilibrium Models](https://mlsanity.com/paper/2306.01429/) (2023)
- [The Achilles' Heel of LLMs: How Altering a Handful of Neurons Can Cripple Language Abilities](https://mlsanity.com/paper/2510.10238/) (2025)
- [In Search of netUnicorn: A Data-Collection Platform to Develop Generalizable ML Models for Network Security Problems](https://mlsanity.com/paper/2306.08853/) (2023)
- [Self-Healing Robust Neural Networks via Closed-Loop Control](https://mlsanity.com/paper/2206.12963/) (2022)
- [PIFON-EPT: MR-Based Electrical Property Tomography Using Physics-Informed Fourier Networks](https://mlsanity.com/paper/2302.11883/) (2023)
- [Continual Adversarial Defense](https://mlsanity.com/paper/2312.09481/) (2023)
- [Towards Alternative Techniques for Improving Adversarial Robustness: Analysis of Adversarial Training at a Spectrum of Perturbations](https://mlsanity.com/paper/2206.06496/) (2022)
- [Robust Feature Inference: A Test-time Defense Strategy using Spectral Projections](https://mlsanity.com/paper/2307.11672/) (2023)
- [PID Control-Based Self-Healing to Improve the Robustness of Large Language Models](https://mlsanity.com/paper/2404.00828/) (2024)
- [Interpretability-Guided Test-Time Adversarial Defense](https://mlsanity.com/paper/2409.15190/) (2024)
- [Neural Observer with Lyapunov Stability Guarantee for Uncertain Nonlinear Systems](https://mlsanity.com/paper/2208.13006/) (2022)
- [Dynamic ensemble selection based on Deep Neural Network Uncertainty Estimation for Adversarial Robustness](https://mlsanity.com/paper/2308.00346/) (2023)
- [Mitigating distribution shift in machine learning-augmented hybrid simulation](https://mlsanity.com/paper/2401.09259/) (2024)
- [Test-time adversarial detection and robustness for localizing humans using ultra wide band channel impulse responses](https://mlsanity.com/paper/2211.05854/) (2022)
- [A Gradient-based Approach for Online Robust Deep Neural Network Training with Noisy Labels](https://mlsanity.com/paper/2306.05046/) (2023)
- [Asymptotically Fair Participation in Machine Learning Models: an Optimal Control Perspective](https://mlsanity.com/paper/2311.10223/) (2023)
- [Improving Adversarial Robustness via Feature Pattern Consistency Constraint](https://mlsanity.com/paper/2406.08829/) (2024)
- [Adversarial Examples in Environment Perception for Automated Driving (Review)](https://mlsanity.com/paper/2504.08414/) (2025)
- [Robust Convolution Neural ODEs via Contractivity-promoting regularization](https://mlsanity.com/paper/2508.11432/) (2025)

## Abstract (arXiv)

Despite their success in massive engineering applications, deep neural networks are vulnerable to various perturbations due to their black-box nature. Recent study has shown that a deep neural network can misclassify the data even if the input data is perturbed by an imperceptible amount. In this paper, we address the robustness issue of neural networks by a novel close-loop control method from the perspective of dynamic systems. Instead of modifying the parameters in a fixed neural network architecture, a close-loop control process is added to generate control signals adaptively for the perturbed or corrupted data. We connect the robustness of neural networks with optimal control using the geometrical information of underlying data to design the control objective. The detailed analysis shows how the embedding manifolds of state trajectory affect error estimation of the proposed method. Our approach can simultaneously maintain the performance on clean data and improve the robustness against many types of data perturbations. It can also further improve the performance of robustly trained neural networks against different perturbations. To the best of our knowledge, this is the first work that improves the robustness of neural networks with close-loop control.