# Rational Neural Network Controllers

arXiv: https://arxiv.org/abs/2307.06287 · HTML: https://mlsanity.com/paper/2307.06287/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 52 references that are in the map.
- First posted on arXiv 12 Jul 2023.

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

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- [In-Datacenter Performance Analysis of a Tensor Processing Unit](https://mlsanity.com/paper/1704.04760/) (2017)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Adversarial Policies: Attacking Deep Reinforcement Learning](https://mlsanity.com/paper/1905.10615/) (2019)
- [Activation Functions in Deep Learning: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2109.14545/) (2021)
- [A Survey of Deep Learning Applications to Autonomous Vehicle Control](https://mlsanity.com/paper/1912.10773/) (2019)
- [Deep Learning for Medical Image Processing: Overview, Challenges and Future](https://mlsanity.com/paper/1704.06825/) (2017)
- [On the Power of Over-parametrization in Neural Networks with Quadratic Activation](https://mlsanity.com/paper/1803.01206/) (2018)
- [Vulnerability of Deep Reinforcement Learning to Policy Induction Attacks](https://mlsanity.com/paper/1701.04143/) (2017)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Pad'e Activation Units: End-to-end Learning of Flexible Activation Functions in Deep Networks](https://mlsanity.com/paper/1907.06732/) (2019)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [Neural networks and rational functions](https://mlsanity.com/paper/1706.03301/) (2017)
- [Enforcing robust control guarantees within neural network policies](https://mlsanity.com/paper/2011.08105/) (2020)
- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [Universal Approximation with Quadratic Deep Networks](https://mlsanity.com/paper/1808.00098/) (2018)
- [Certifiable Robustness to Adversarial State Uncertainty in Deep Reinforcement Learning](https://mlsanity.com/paper/2004.06496/) (2020)
- [QuadraLib: A Performant Quadratic Neural Network Library for Architecture Optimization and Design Exploration](https://mlsanity.com/paper/2204.01701/) (2022)
- [Recurrent Neural Network Controllers Synthesis with Stability Guarantees for Partially Observed Systems](https://mlsanity.com/paper/2109.03861/) (2021)
- [Neural Network Verification in Control](https://mlsanity.com/paper/2110.01388/) (2021)
- [PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units](https://mlsanity.com/paper/1909.05136/) (2019)
- [Synthesis of Stabilizing Recurrent Equilibrium Network Controllers](https://mlsanity.com/paper/2204.00122/) (2022)
- [Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multipliers](https://mlsanity.com/paper/2103.17106/) (2021)
- [Stability of Non-linear Neural Feedback Loops using Sum of Squares](https://mlsanity.com/paper/2204.03913/) (2022)
- [Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization](https://mlsanity.com/paper/2107.02379/) (2021)
- [Activation Adaptation in Neural Networks](https://mlsanity.com/paper/1901.09849/) (2019)
- [Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations](https://mlsanity.com/paper/1710.01358/) (2017)

## Abstract (arXiv)

Neural networks have shown great success in many machine learning related tasks, due to their ability to act as general function approximators. Recent work has demonstrated the effectiveness of neural networks in control systems (known as neural feedback loops), most notably by using a neural network as a controller. However, one of the big challenges of this approach is that neural networks have been shown to be sensitive to adversarial attacks. This means that, unless they are designed properly, they are not an ideal candidate for controllers due to issues with robustness and uncertainty, which are pivotal aspects of control systems. There has been initial work on robustness to both analyse and design dynamical systems with neural network controllers. However, one prominent issue with these methods is that they use existing neural network architectures tailored for traditional machine learning tasks. These structures may not be appropriate for neural network controllers and it is important to consider alternative architectures. This paper considers rational neural networks and presents novel rational activation functions, which can be used effectively in robustness problems for neural feedback loops. Rational activation functions are replaced by a general rational neural network structure, which is convex in the neural network's parameters. A method is proposed to recover a stabilising controller from a Sum of Squares feasibility test. This approach is then applied to a refined rational neural network which is more compatible with Sum of Squares programming. Numerical examples show that this method can successfully recover stabilising rational neural network controllers for neural feedback loops with non-linear plants with noise and parametric uncertainty.