# Neural ODEs Meet Concurrent Learning: Stable Online Learning with Lyapunov Guarantees

arXiv: https://arxiv.org/abs/2609.32289 · HTML: https://mlsanity.com/paper/2609.32289/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 41 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 24 of 24)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [DeepMind Control Suite](https://mlsanity.com/paper/1801.00690/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [Neural-Fly Enables Rapid Learning for Agile Flight in Strong Winds](https://mlsanity.com/paper/2205.06908/) (2022)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Adaptive-Control-Oriented Meta-Learning for Nonlinear Systems](https://mlsanity.com/paper/2103.04490/) (2021)
- [Regret Bounds for Adaptive Nonlinear Control](https://mlsanity.com/paper/2011.13101/) (2020)
- [Meta-Adaptive Nonlinear Control: Theory and Algorithms](https://mlsanity.com/paper/2106.06098/) (2021)
- [Differentiable Multiple Shooting Layers](https://mlsanity.com/paper/2106.03885/) (2021)
- [Multiple shooting for training neural differential equations on time series](https://mlsanity.com/paper/2109.06786/) (2021)
- [Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints](https://mlsanity.com/paper/2306.09739/) (2023)
- [Deep Model Reference Adaptive Control](https://mlsanity.com/paper/1909.08602/) (2019)
- [Learning Deep Dynamical Systems using Stable Neural ODEs](https://mlsanity.com/paper/2404.10622/) (2024)
- [Concurrent learning for parameter estimation using dynamic state-derivative estimators](https://mlsanity.com/paper/1507.08903/) (2015)
- [Implicit Regularization and Momentum Algorithms in Nonlinearly Parameterized Adaptive Control and Prediction](https://mlsanity.com/paper/1912.13154/) (2019)
- [FxTS-Net: Fixed-Time Stable Learning Framework for Neural ODEs](https://mlsanity.com/paper/2411.09118/) (2024)
- [System Identification and Control Using Lyapunov-Based Deep Neural Networks without Persistent Excitation: A Concurrent Learning Approach](https://mlsanity.com/paper/2505.10678/) (2025)

## Abstract (arXiv)

Neural ODEs learn dynamics from trajectory losses, but their adjoint gradients lack the regressor-times-parameter-error structure on which Lyapunov analyses of online adaptation rest, so training on streaming data comes without stability guarantees. We show that this structure is in fact present: the adjoint gradient decomposes exactly into a positive semi-definite trajectory operator acting on the parameter error plus a nonlinear perturbation with explicit, horizon-dependent bounds. A quadratic Lyapunov function then certifies online Neural ODE training over sliding windows under computable gain and horizon conditions, and the same certificate extends to stored data: its drift branch recovers concurrent learning, and its trajectory branch yields NODE-CL, a stored-segment Gauss-Newton method built on batched forward sensitivities that needs no state-derivative estimates. On four DeepMind Control Suite domains, NODE-CL attains the lowest median prediction error on three under velocity measurement noise, where observer-based concurrent learning degrades by up to 8x; with clean measurements it is best on the pendulum and within a factor of 1.6 of the best stored-data baseline on the cartpole and reacher.