Reconstruction of dynamical systems from data without time labels

TL;DR

This work addresses reconstructing dynamical systems from data without time labels by treating observations as samples drawn from a time-distribution and matching them to simulated distributions via the Sliced Wasserstein distance. It develops a two-phase framework: (i) a forward-solver–based parameter identification using a dictionary of candidate terms with an $\ell_0$ sparsity penalty, and (ii) a distribution-matching phase that employs neural surrogates and physics-informed regularization, combined with alternating direction optimization. To handle long, complex trajectories, the method uses trajectory segmentation to break problems into simpler pieces, enabling robust recovery of both the underlying vector field and the missing time stamps. Experiments on illustrative and benchmark ODEs, including chaotic and high-dimensional systems and under noisy observations, demonstrate accurate recovery of states, parameters, and time labels, with strong robustness to variations in observation-time distributions and noise. The approach offers a practical route for extracting governing dynamics from unlabeled point clouds in domains such as microscopy and single-cell genomics, where time information is unavailable or unreliable.

Abstract

In this paper, we study the method to reconstruct dynamical systems from data without time labels. Data without time labels appear in many applications, such as molecular dynamics, single-cell RNA sequencing etc. Reconstruction of dynamical system from time sequence data has been studied extensively. However, these methods do not apply if time labels are unknown. Without time labels, sequence data becomes distribution data. Based on this observation, we propose to treat the data as samples from a probability distribution and try to reconstruct the underlying dynamical system by minimizing the distribution loss, sliced Wasserstein distance more specifically. Extensive experiment results demonstrate the effectiveness of the proposed method.

Figures (8)

• Figure 1: Loss landscape analysis of Cubilc2D problem: (a) The estimated $\sup_{t\in[0,10]}||\boldsymbol{x_{t}}||_2$ of parameter $(A_{12},A_{21})$ choice in $[1,3]\times[-3,-1]$; (b) The SWD loss landscape of $A_{12},A_{21}$ in $[1,3]\times[-3,-1]$ with $T=10$; (c) The SWD loss landscape of $A_{12},A_{21}$ in $[1,3]\times[-3,-1]$ with $T=0.4$; (d) ...
• Figure 2: Segmentation result of illustrative problem: (a) the 8 piece cluster result of Cubic2D ODE system initiate at $(2,0)$ with $T = 10$;(b) the 10 piece cluster result of Linear3D ODE system initiate at $(2,0,-1)$ with $T = 10$;(c) the 10 piece cluster result of Lorenz ODE system initiate at $(10,20,-10)$ with $T = 3$.
• Figure 3: Reconstruction result of illustrative system: (a)-(c)Comparison between learned curve at different stage and ground truth for all system;(d)-(f) SWD Loss, PINN Loss and Average Error for all system;(g)-(i) The error curve of parameters and the sum of unexisting terms for all system;(j)-(l)The absolute error of learned solution evaluated at uniform grid and the absolute error of reconstructed time label of the dataset.
• Figure 4: Reconstruction result of benchmark system:Reconstructing trajectories of four systems at different training stage:(a)Lorenz system; (b)LV4D system, plot the curve of first 3 dimension;(c)Duffing system. The point-wise error of learned solution function and time label of three training stage: (a)Lorenz system; (b)LV4D system; (c)Duffing system.
• Figure 5: Evaluation of reconstruction results from data with truncated normal distribution of observation time.(a):The evaluation metrics of reconstruction results in distribution matching phase;(b):The evaluation metrics of reconstruction results in parameter identification phase