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Efficient Manifold-Constrained Neural ODE for High-Dimensional Datasets

Muhao Guo, Haoran Li, Yang Weng

TL;DR

The paper addresses learning continuous-time dynamics in high-dimensional data by exploiting underlying manifold structure. It introduces a manifold-Constrained NODE that maps inputs to a latent manifold $\mathcal{M}$ via a structure-preserving encoder and evolves dynamics on $\mathcal{M}$ with a NODE, using a cross-entropy-based alignment loss together with a supervised loss. A theoretical existence result guarantees well-posed dynamics on the manifold, and joint optimization aligns geometry and task performance. Empirically, the method yields higher accuracy, fewer function evaluations, and faster convergence than baselines across image and time-series datasets, demonstrating a scalable, geometry-aware approach for efficient continuous-time modeling. Overall, this framework provides a principled way to incorporate manifold geometry into NODEs to tackle high-dimensional dynamical systems more efficiently.

Abstract

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.

Efficient Manifold-Constrained Neural ODE for High-Dimensional Datasets

TL;DR

The paper addresses learning continuous-time dynamics in high-dimensional data by exploiting underlying manifold structure. It introduces a manifold-Constrained NODE that maps inputs to a latent manifold via a structure-preserving encoder and evolves dynamics on with a NODE, using a cross-entropy-based alignment loss together with a supervised loss. A theoretical existence result guarantees well-posed dynamics on the manifold, and joint optimization aligns geometry and task performance. Empirically, the method yields higher accuracy, fewer function evaluations, and faster convergence than baselines across image and time-series datasets, demonstrating a scalable, geometry-aware approach for efficient continuous-time modeling. Overall, this framework provides a principled way to incorporate manifold geometry into NODEs to tackle high-dimensional dynamical systems more efficiently.

Abstract

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.

Paper Structure

This paper contains 17 sections, 1 theorem, 8 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Neural ODE (NODE)
  4. Efficiency of NODEs
  5. NODEs on Manifolds
  6. Background and Preliminaries
  7. Manifolds
  8. NODEs
  9. Manifold-Constrained NODE
  10. Numerical Experiments
  11. Experimental Setup
  12. Learning Dynamics on Spherical Manifold
  13. Image Classification with Manifold-Constrained NODE
  14. Series Classification with Manifold-Constrained NODE
  15. Dimensionality Sensitivity Analysis
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be a smooth, differentiable manifold embedded in $\mathbb{R}^m$, and let $f: \mathcal{M} \times \mathbb{R} \rightarrow T\mathcal{M}$ be a smooth, Lipschitz continuous vector field. For any initial condition $h_{\mathcal{G}}(0) = \mathcal{G}(x_0) \in \mathcal{M}$, there exists a uni with $h_{\mathcal{G}}(\tau) \in \mathcal{M}$ for all $\tau$.

Figures (7)

  • Figure 1: Illustration of Manifold NODE Methodology. Original high-dimensional data is projected into a lower-dimensional manifold space, preserving intrinsic data structures. This projection is achieved by minimizing the cross-entropy loss between edge probability distributions of the original and manifold spaces. In this compact manifold space, NODEs are utilized to efficiently and accurately learn the evolution of latent features. The transition of learned flow from the latent space to the output space enhances the interpretability and transparency of the learning process.
  • Figure 2: Top: Vector fields from a NODE in three-dimensional Euclidean space. Bottom: Vector fields from a Manifold-constrained NODE in spherical space.
  • Figure 3: Top: Trajectories in three-dimensional Euclidean space learned by NODE. Bottom: Trajectories learned by Manifold-constrained NODEs. Solid lines depict true trajectories, and dotted lines indicate the learned trajectories.
  • Figure 4: The left plot illustrates the graph structure constructed for the MNIST dataset, based on global probability. In this visualization, only edges with a probability exceeding $0.5$ are displayed. The right plot presents the weighted matrix between samples, with the samples organized according to their labels.
  • Figure 5: The figure shows the evolution of MNIST test samples in three-dimensional space: after one epoch of encoder training (left), one epoch of full model training (middle), and five epochs of full model training (right).
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1: Smooth mapping
  • Definition 2: Diffeomorphism
  • Definition 3: Diffeomorphic
  • Definition 4: Diffeomorphism in manifolds
  • Definition 5: Diffeomorphic of manifolds
  • Definition 6: Manifold-Constrained NODE
  • Theorem 1: Existence of Dynamics on Manifolds
  • proof