# Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximations

arXiv: https://arxiv.org/abs/2410.10174 · HTML: https://mlsanity.com/paper/2410.10174/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 53 references that are in the map.
- First posted on arXiv 14 Oct 2024.

## Builds on (most cited first, 29 of 29)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Understanding disentangling in β-VAE](https://mlsanity.com/paper/1804.03599/) (2018)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Deep Kalman Filters](https://mlsanity.com/paper/1511.05121/) (2015)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [A Disentangled Recognition and Nonlinear Dynamics Model for Unsupervised Learning](https://mlsanity.com/paper/1710.05741/) (2017)
- [Variational Autoencoders Pursue PCA Directions (by Accident)](https://mlsanity.com/paper/1812.06775/) (2018)
- [Continuous-Time Modeling of Counterfactual Outcomes Using Neural Controlled Differential Equations](https://mlsanity.com/paper/2206.08311/) (2022)
- [STEER: Simple Temporal Regularization For Neural ODEs](https://mlsanity.com/paper/2006.10711/) (2020)
- [Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs](https://mlsanity.com/paper/2310.02619/) (2023)
- [Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems](https://mlsanity.com/paper/2010.14685/) (2020)
- [Constructing Neural Network-Based Models for Simulating Dynamical Systems](https://mlsanity.com/paper/2111.01495/) (2021)
- [Tutorial: Deriving the Standard Variational Autoencoder (VAE) Loss Function](https://mlsanity.com/paper/1907.08956/) (2019)
- [Bayesian Neural Controlled Differential Equations for Treatment Effect Estimation](https://mlsanity.com/paper/2310.17463/) (2023)
- [Neural Ordinary Differential Equations for Nonlinear System Identification](https://mlsanity.com/paper/2203.00120/) (2022)
- [Stochasticity in Neural ODEs: An Empirical Study](https://mlsanity.com/paper/2002.09779/) (2020)
- [Interpretable learning of effective dynamics for multiscale systems](https://mlsanity.com/paper/2309.05812/) (2023)
- [Modulated Neural ODEs](https://mlsanity.com/paper/2302.13262/) (2023)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [A multimodal dynamical variational autoencoder for audiovisual speech representation learning](https://mlsanity.com/paper/2305.03582/) (2023)
- [Continuous-time Autoencoders for Regular and Irregular Time Series Imputation](https://mlsanity.com/paper/2312.16581/) (2023)
- [Latent Neural ODE for Integrating Multi-timescale measurements in Smart Distribution Grids](https://mlsanity.com/paper/2211.03882/) (2022)
- [Learnable & Interpretable Model Combination in Dynamical Systems Modeling](https://mlsanity.com/paper/2406.08093/) (2024)

## Built on by (most cited first, 4 of 4)

- [Hierarchical Koopman Diffusion: Fast Generation with Interpretable Diffusion Trajectory](https://mlsanity.com/paper/2510.12220/) (2025)
- [Data-driven stochastic reduced-order modeling of parametrized dynamical systems](https://mlsanity.com/paper/2601.10690/) (2026)
- [Parallel-in-Time Training of Recurrent Neural Networks for Dynamical Systems Reconstruction](https://mlsanity.com/paper/2605.12683/) (2026)
- [Differentiable Koopman Operator for Contrastive Learning on Dynamic Graphs](https://mlsanity.com/paper/2610.02990/) (2026)

## Abstract (arXiv)

Variational Autoencoders (VAEs) are a powerful framework for learning latent representations of reduced dimensionality, while Neural ODEs excel in learning transient system dynamics. This work combines the strengths of both to generate fast surrogate models with adjustable complexity reacting on time-varying inputs signals. By leveraging the VAE's dimensionality reduction using a nonhierarchical prior, our method adaptively assigns stochastic noise, naturally complementing known NeuralODE training enhancements and enabling probabilistic time series modeling. We show that standard Latent ODEs struggle with dimensionality reduction in systems with time-varying inputs. Our approach mitigates this by continuously propagating variational parameters through time, establishing fixed information channels in latent space. This results in a flexible and robust method that can learn different system complexities, e.g. deep neural networks or linear matrices. Hereby, it enables efficient approximation of the Koopman operator without the need for predefining its dimensionality. As our method balances dimensionality reduction and reconstruction accuracy, we call it Balanced Neural ODE (B-NODE). We demonstrate the effectiveness of this methods on several academic and real-world test cases, e.g. a power plant or MuJoCo data.