# Gradient-free training of neural ODEs for system identification and control using ensemble Kalman inversion

arXiv: https://arxiv.org/abs/2307.07882 · HTML: https://mlsanity.com/paper/2307.07882/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 43 references that are in the map.
- First posted on arXiv 15 Jul 2023.

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

- [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)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks](https://mlsanity.com/paper/1808.03620/) (2018)
- [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)
- [Auto-differentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2107.07687/) (2021)
- [Neural Ordinary Differential Equation Control of Dynamics on Graphs](https://mlsanity.com/paper/2006.09773/) (2020)
- [Visualizing high-dimensional loss landscapes with Hessian directions](https://mlsanity.com/paper/2208.13219/) (2022)
- [Tikhonov Regularization Within Ensemble Kalman Inversion](https://mlsanity.com/paper/1901.10382/) (2019)
- [Never look back - A modified EnKF method and its application to the training of neural networks without back propagation](https://mlsanity.com/paper/1805.08034/) (2018)
- [Near-optimal control of dynamical systems with neural ordinary differential equations](https://mlsanity.com/paper/2206.11120/) (2022)
- [Faster Policy Learning with Continuous-Time Gradients](https://mlsanity.com/paper/2012.06684/) (2020)

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

- [Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods](https://mlsanity.com/paper/2311.11798/) (2023)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Hyperparameter Optimization for Randomized Algorithms: A Case Study on Random Features](https://mlsanity.com/paper/2407.00584/) (2024)
- [Online Calibration of Deep Learning Sub-Models for Hybrid Numerical Modeling Systems](https://mlsanity.com/paper/2311.10665/) (2023)
- [Control of Medical Digital Twins with Artificial Neural Networks](https://mlsanity.com/paper/2403.13851/) (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)
- [xFODE+: Explainable Type-2 Fuzzy Additive ODEs for Uncertainty Quantification](https://mlsanity.com/paper/2604.14880/) (2026)
- [xFODE: An Explainable Fuzzy Additive ODE Framework for System Identification](https://mlsanity.com/paper/2604.14883/) (2026)

## Abstract (arXiv)

Ensemble Kalman inversion (EKI) is a sequential Monte Carlo method used to solve inverse problems within a Bayesian framework. Unlike backpropagation, EKI is a gradient-free optimization method that only necessitates the evaluation of artificial neural networks in forward passes. In this study, we examine the effectiveness of EKI in training neural ordinary differential equations (neural ODEs) for system identification and control tasks. To apply EKI to optimal control problems, we formulate inverse problems that incorporate a Tikhonov-type regularization term. Our numerical results demonstrate that EKI is an efficient method for training neural ODEs in system identification and optimal control problems, with runtime and quality of solutions that are competitive with commonly used gradient-based optimizers.