# Stable Neural Flows

arXiv: https://arxiv.org/abs/2003.08063 · HTML: https://mlsanity.com/paper/2003.08063/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Mar 2020.

## Built on by (most cited first, 25 of 25)

- [Closed-form Continuous-time Neural Models](https://mlsanity.com/paper/2106.13898/) (2021)
- [Constructing Neural Network-Based Models for Simulating Dynamical Systems](https://mlsanity.com/paper/2111.01495/) (2021)
- [Comprehensive Review of Neural Differential Equations for Time Series Analysis](https://mlsanity.com/paper/2502.09885/) (2025)
- [Robust Classification using Contractive Hamiltonian Neural ODEs](https://mlsanity.com/paper/2203.11805/) (2022)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Lyapunov-Stable Deep Equilibrium Models](https://mlsanity.com/paper/2304.12707/) (2023)
- [Accelerating Neural ODEs Using Model Order Reduction](https://mlsanity.com/paper/2105.14070/) (2021)
- [Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs](https://mlsanity.com/paper/2205.12550/) (2022)
- [Unconstrained Parametrization of Dissipative and Contracting Neural Ordinary Differential Equations](https://mlsanity.com/paper/2304.02976/) (2023)
- [DeNOTS: Stable Deep Neural ODEs for Time Series](https://mlsanity.com/paper/2408.08055/) (2024)
- [ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence](https://mlsanity.com/paper/2411.02292/) (2024)
- [Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methods](https://mlsanity.com/paper/2412.01181/) (2024)
- [Optimal Energy Shaping via Neural Approximators](https://mlsanity.com/paper/2101.05537/) (2021)
- [DualDynamics: Synergizing Implicit and Explicit Methods for Robust Irregular Time Series Analysis](https://mlsanity.com/paper/2401.04979/) (2024)
- [Learning Stochastic Optimal Policies via Gradient Descent](https://mlsanity.com/paper/2106.03780/) (2021)
- [The Vanishing Gradient Problem for Stiff Neural Differential Equations](https://mlsanity.com/paper/2508.01519/) (2025)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [ICODE: Modeling Dynamical Systems with Extrinsic Input Information](https://mlsanity.com/paper/2411.13914/) (2024)
- [Hybrid Energy-Based Models for Physical AI: Provably Stable Identification of Port-Hamiltonian Dynamics](https://mlsanity.com/paper/2604.00277/) (2026)
- [Modeling Nonlinear Dynamics in Continuous Time with Inductive Biases on Decay Rates and/or Frequencies](https://mlsanity.com/paper/2212.13033/) (2022)
- [Digital Twins for Patient Care via Knowledge Graphs and Closed-Form Continuous-Time Liquid Neural Networks](https://mlsanity.com/paper/2307.04772/) (2023)
- [Wavelet-Enhanced Neural ODE and Graph Attention for Interpretable Energy Forecasting](https://mlsanity.com/paper/2507.10132/) (2025)
- [Robust Convolution Neural ODEs via Contractivity-promoting regularization](https://mlsanity.com/paper/2508.11432/) (2025)
- [Fine-Tuning Flow Matching via Maximum Likelihood Estimation of Reconstructions](https://mlsanity.com/paper/2510.02081/) (2025)
- [Markovian ODE-guided scoring can assess the quality of offline reasoning traces in language models](https://mlsanity.com/paper/2603.01580/) (2026)

## Abstract (arXiv)

We introduce a provably stable variant of neural ordinary differential equations (neural ODEs) whose trajectories evolve on an energy functional parametrised by a neural network. Stable neural flows provide an implicit guarantee on asymptotic stability of the depth-flows, leading to robustness against input perturbations and low computational burden for the numerical solver. The learning procedure is cast as an optimal control problem, and an approximate solution is proposed based on adjoint sensivity analysis. We further introduce novel regularizers designed to ease the optimization process and speed up convergence. The proposed model class is evaluated on non-linear classification and function approximation tasks.