FlowPath: Learning Data-Driven Manifolds with Invertible Flows for Robust Irregularly-sampled Time Series Classification

TL;DR

FlowPath tackles irregularly-sampled time series by learning a data-driven control path through an invertible neural flow, replacing fixed interpolants in Neural CDEs with a diffeomorphic, geometry-aware manifold. The approach guarantees smooth, information-preserving dynamics and provides theoretical guarantees on density preservation, existence/uniqueness, and generalization, while delivering statistically significant improvements in classification across 18 benchmarks and real-world HAR and medical datasets, especially under high missingness. By explicitly modeling the geometry of the control path in addition to the dynamics along it, FlowPath achieves robust performance and better generalization in ISTS tasks. This manifold-aware framework offers a practical, scalable solution for continuous-time modeling of irregular time series with broad applicability in real-world sensing and healthcare domains.

Abstract

Modeling continuous-time dynamics from sparse and irregularly-sampled time series remains a fundamental challenge. Neural controlled differential equations provide a principled framework for such tasks, yet their performance is highly sensitive to the choice of control path constructed from discrete observations. Existing methods commonly employ fixed interpolation schemes, which impose simplistic geometric assumptions that often misrepresent the underlying data manifold, particularly under high missingness. We propose FlowPath, a novel approach that learns the geometry of the control path via an invertible neural flow. Rather than merely connecting observations, FlowPath constructs a continuous and data-adaptive manifold, guided by invertibility constraints that enforce information-preserving and well-behaved transformations. This inductive bias distinguishes FlowPath from prior unconstrained learnable path models. Empirical evaluations on 18 benchmark datasets and a real-world case study demonstrate that FlowPath consistently achieves statistically significant improvements in classification accuracy over baselines using fixed interpolants or non-invertible architectures. These results highlight the importance of modeling not only the dynamics along the path but also the geometry of the path itself, offering a robust and generalizable solution for learning from irregular time series.

Figures (10)

• Figure 1: Test loss curves on 'BasicMotions' under regular and irregular scenarios using selected methods
• Figure 3: Qualitative comparison of path construction methods on a sample from the 'BasicMotions' dataset with 50% missingness. (a) Raw and sparse observations. (b) Path generated by a non-invertible MLP, exhibiting a biased curve toward the observed points. (c) Path produced by the proposed FlowPath, capturing a more structured and stable manifold. Each color denotes a separate data dimension.
• Figure 4: 2D projections of learned trajectories from sparse input. Off-diagonal plots show phase-space paths, and diagonal plots show marginal histograms. FlowPath (blue) better matches the ground truth (black) than the MLP (red).
• Figure 5: 1D and 2D Kernel Density Estimates (KDEs) for each method. FlowPath (blue) aligns more closely with the original distribution (black) than the MLP (red).
• Figure 6: 2D KDEs of the learned manifolds between Dim 1 and Dim 2, using the sparse observation shown in panel (b)

Theorems & Definitions (6)

• Theorem 1: Preservation of Probability Density
• Theorem 2: Existence and Uniqueness of Solutions
• Theorem 3: Generalization Bound
• proof
• proof
• proof