# Efficient Manifold-Constrained Neural ODE for High-Dimensional Datasets

arXiv: https://arxiv.org/abs/2510.04138 · HTML: https://mlsanity.com/paper/2510.04138/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 42 references that are in the map.
- First posted on arXiv 5 Oct 2025.

## Builds on (most cited first, 19 of 19)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [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)
- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [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)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Parametric UMAP embeddings for representation and semi-supervised learning](https://mlsanity.com/paper/2009.12981/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [Bias in Large Language Models: Origin, Evaluation, and Mitigation](https://mlsanity.com/paper/2411.10915/) (2024)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Heavy Ball Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.04840/) (2021)
- [Opening the Blackbox: Accelerating Neural Differential Equations by Regularizing Internal Solver Heuristics](https://mlsanity.com/paper/2105.03918/) (2021)
- [Second-Order Neural ODE Optimizer](https://mlsanity.com/paper/2109.14158/) (2021)
- [Accelerating Neural ODEs Using Model Order Reduction](https://mlsanity.com/paper/2105.14070/) (2021)
- [TO-FLOW: Efficient Continuous Normalizing Flows with Temporal Optimization adjoint with Moving Speed](https://mlsanity.com/paper/2203.10335/) (2022)
- [Advantage Actor-Critic with Reasoner: Explaining the Agent's Behavior from an Exploratory Perspective](https://mlsanity.com/paper/2309.04707/) (2023)

## Built on by (most cited first, 2 of 2)

- [Capturing reduced-order quantum many-body dynamics out of equilibrium via neural ordinary differential equations](https://mlsanity.com/paper/2512.13913/) (2025)
- [Enabling Scalable Topology Inference in Distribution Systems via Constrained Multi-Source Inference](https://mlsanity.com/paper/2607.20480/) (2026)

## Abstract (arXiv)

Neural ordinary differential equations (NODE) have garnered significant attention for their design of continuous-depth neural networks and the ability to learn data/feature dynamics. However, for high-dimensional systems, estimating dynamics requires extensive calculations and suffers from high truncation errors for the ODE solvers. To address the issue, one intuitive approach is to consider the non-trivial topological space of the data distribution, i.e., a low-dimensional manifold. Existing methods often rely on knowledge of the manifold for projection or implicit transformation, restricting the ODE solutions on the manifold. Nevertheless, such knowledge is usually unknown in realistic scenarios. Therefore, we propose a novel approach to explore the underlying manifold to restrict the ODE process. Specifically, we employ a structure-preserved encoder to process data and find the underlying graph to approximate the manifold. Moreover, we propose novel methods to combine the NODE learning with the manifold, resulting in significant gains in computational speed and accuracy. Our experimental evaluations encompass multiple datasets, where we compare the accuracy, number of function evaluations (NFEs), and convergence speed of our model against existing baselines. Our results demonstrate superior performance, underscoring the effectiveness of our approach in addressing the challenges of high-dimensional datasets.