# Near-optimal control of dynamical systems with neural ordinary differential equations

arXiv: https://arxiv.org/abs/2206.11120 · HTML: https://mlsanity.com/paper/2206.11120/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 54 references that are in the map.
- First posted on arXiv 22 Jun 2022.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Qualitatively characterizing neural network optimization problems](https://mlsanity.com/paper/1412.6544/) (2014)
- [On Empirical Comparisons of Optimizers for Deep Learning](https://mlsanity.com/paper/1910.05446/) (2019)
- [Visualizing and Understanding the Effectiveness of BERT](https://mlsanity.com/paper/1908.05620/) (2019)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [Deep neural networks, generic universal interpolation, and controlled ODEs](https://mlsanity.com/paper/1908.07838/) (2019)
- [Neural Ordinary Differential Equation Control of Dynamics on Graphs](https://mlsanity.com/paper/2006.09773/) (2020)
- [Exploring the Geometry and Topology of Neural Network Loss Landscapes](https://mlsanity.com/paper/2102.00485/) (2021)
- [Control of Stochastic Quantum Dynamics by Differentiable Programming](https://mlsanity.com/paper/2101.01190/) (2021)

## Built on by (most cited first, 9 of 9)

- [Gradient-free training of neural ODEs for system identification and control using ensemble Kalman inversion](https://mlsanity.com/paper/2307.07882/) (2023)
- [FI-ODE: Certifiably Robust Forward Invariance in Neural ODEs](https://mlsanity.com/paper/2210.16940/) (2022)
- [Control of Medical Digital Twins with Artificial Neural Networks](https://mlsanity.com/paper/2403.13851/) (2024)
- [Reevaluating Meta-Learning Optimization Algorithms Through Contextual Self-Modulation](https://mlsanity.com/paper/2410.01655/) (2024)
- [Enhancing Dynamical System Modeling through Interpretable Machine Learning Augmentations: A Case Study in Cathodic Electrophoretic Deposition](https://mlsanity.com/paper/2401.08414/) (2024)
- [Mean-Field Model for Two-Layer Neural Networks Trained with Consensus-Based Optimization](https://mlsanity.com/paper/2511.21466/) (2025)
- [Neural Control: Concurrent System Identification and Control Learning with Neural ODE](https://mlsanity.com/paper/2401.01836/) (2024)
- [Statistical Mechanics and Artificial Neural Networks: Principles, Models, and Applications](https://mlsanity.com/paper/2405.10957/) (2024)
- [Control of dynamical systems with neural networks](https://mlsanity.com/paper/2510.12810/) (2025)

## Abstract (arXiv)

Optimal control problems naturally arise in many scientific applications where one wishes to steer a dynamical system from a certain initial state x0 to a desired target state x^* in finite time T. Recent advances in deep learning and neural network-based optimization have contributed to the development of methods that can help solve control problems involving high-dimensional dynamical systems. In particular, the framework of neural ordinary differential equations (neural ODEs) provides an efficient means to iteratively approximate continuous time control functions associated with analytically intractable and computationally demanding control tasks. Although neural ODE controllers have shown great potential in solving complex control problems, the understanding of the effects of hyperparameters such as network structure and optimizers on learning performance is still very limited. Our work aims at addressing some of these knowledge gaps to conduct efficient hyperparameter optimization. To this end, we first analyze how truncated and non-truncated backpropagation through time affect runtime performance and the ability of neural networks to learn optimal control functions. Using analytical and numerical methods, we then study the role of parameter initializations, optimizers, and neural-network architecture. Finally, we connect our results to the ability of neural ODE controllers to implicitly regularize control energy.