# Neural Flows: Efficient Alternative to Neural ODEs

arXiv: https://arxiv.org/abs/2110.13040 · HTML: https://mlsanity.com/paper/2110.13040/

## Key facts

- Cited by 101 papers in the map (data through 14 Oct 2026).
- Builds on 56 of its 87 references that are in the map.
- First posted on arXiv 25 Oct 2021.

## Builds on (most cited first, 30 of 56)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [DeepMind Control Suite](https://mlsanity.com/paper/1801.00690/) (2018)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Predicting Dynamic Embedding Trajectory in Temporal Interaction Networks](https://mlsanity.com/paper/1908.01207/) (2019)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Intensity-Free Learning of Temporal Point Processes](https://mlsanity.com/paper/1909.12127/) (2019)

## Built on by (most cited first, 30 of 101)

- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Understanding convolution on graphs via energies](https://mlsanity.com/paper/2206.10991/) (2022)
- [Rethinking Irregular Time Series Forecasting: A Simple yet Effective Baseline](https://mlsanity.com/paper/2505.11250/) (2025)
- [From Zero to Turbulence: Generative Modeling for 3D Flow Simulation](https://mlsanity.com/paper/2306.01776/) (2023)
- [Neural Laplace: Learning diverse classes of differential equations in the Laplace domain](https://mlsanity.com/paper/2206.04843/) (2022)
- [SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equations](https://mlsanity.com/paper/2502.02472/) (2025)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Generalization bounds for neural ordinary differential equations and deep residual networks](https://mlsanity.com/paper/2305.06648/) (2023)
- [Comprehensive Review of Neural Differential Equations for Time Series Analysis](https://mlsanity.com/paper/2502.09885/) (2025)
- [Time-IMM: A Dataset and Benchmark for Irregular Multimodal Multivariate Time Series](https://mlsanity.com/paper/2506.10412/) (2025)
- [Meta Temporal Point Processes](https://mlsanity.com/paper/2301.12023/) (2023)
- [Self-Consistent Velocity Matching of Probability Flows](https://mlsanity.com/paper/2301.13737/) (2023)
- [Gaussian Interpolation Flows](https://mlsanity.com/paper/2311.11475/) (2023)
- [Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patching](https://mlsanity.com/paper/2405.20799/) (2024)
- [Graph Neural Flows for Unveiling Systemic Interactions Among Irregularly Sampled Time Series](https://mlsanity.com/paper/2410.14030/) (2024)
- [HyperIMTS: Hypergraph Neural Network for Irregular Multivariate Time Series Forecasting](https://mlsanity.com/paper/2505.17431/) (2025)
- [Spatio-Temporal Graph Neural Point Process for Traffic Congestion Event Prediction](https://mlsanity.com/paper/2311.08635/) (2023)
- [Unleashing The Power of Pre-Trained Language Models for Irregularly Sampled Time Series](https://mlsanity.com/paper/2408.08328/) (2024)
- [TimeCHEAT: A Channel Harmony Strategy for Irregularly Sampled Multivariate Time Series Analysis](https://mlsanity.com/paper/2412.12886/) (2024)
- [Physiome-ODE: A Benchmark for Irregularly Sampled Multivariate Time Series Forecasting Based on Biological ODEs](https://mlsanity.com/paper/2502.07489/) (2025)
- [From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems](https://mlsanity.com/paper/2405.11542/) (2024)
- [Bridging Time and Frequency: A Joint Modeling Framework for Irregular Multivariate Time Series Forecasting](https://mlsanity.com/paper/2602.00582/) (2026)
- [LSCD: Lomb-Scargle Conditioned Diffusion for Time series Imputation](https://mlsanity.com/paper/2506.17039/) (2025)
- [ASTGI: Adaptive Spatio-Temporal Graph Interactions for Irregular Multivariate Time Series Forecasting](https://mlsanity.com/paper/2509.23313/) (2025)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [IVP-VAE: Modeling EHR Time Series with Initial Value Problem Solvers](https://mlsanity.com/paper/2305.06741/) (2023)
- [Lagrangian Flow Networks for Conservation Laws](https://mlsanity.com/paper/2305.16846/) (2023)
- [Neural Network Emulator for Atmospheric Chemical ODE](https://mlsanity.com/paper/2408.01829/) (2024)
- [Neural Spatiotemporal Point Processes: Trends and Challenges](https://mlsanity.com/paper/2502.09341/) (2025)
- [Neural Stochastic Flows: Solver-Free Modelling and Inference for SDE Solutions](https://mlsanity.com/paper/2510.25769/) (2025)

## Abstract (arXiv)

Neural ordinary differential equations describe how values change in time. This is the reason why they gained importance in modeling sequential data, especially when the observations are made at irregular intervals. In this paper we propose an alternative by directly modeling the solution curves - the flow of an ODE - with a neural network. This immediately eliminates the need for expensive numerical solvers while still maintaining the modeling capability of neural ODEs. We propose several flow architectures suitable for different applications by establishing precise conditions on when a function defines a valid flow. Apart from computational efficiency, we also provide empirical evidence of favorable generalization performance via applications in time series modeling, forecasting, and density estimation.