Meta-iLaD: Identifiable Latent Dynamics via Meta-Learning of Dynamics Environments

Yubo Ye; Sweekar Piya; Xiajun Jiang; Linwei Wang

cs.LGOct 19, 2026

Meta-iLaD: Identifiable Latent Dynamics via Meta-Learning of Dynamics Environments

Authors: Yubo Ye, Sweekar Piya, Xiajun Jiang, Linwei Wang

Organizations: Department of Computing and Information Sciences, Rochester Institute of Technology, Rochester, New York, USA · Department of Computer Science, Rowan University, Glassboro, New Jersey, USA

Abstract

Learning latent dynamics is central to assessing the current states and forecasting the future trajectories of high-dimensional time series. For locally-stationary latent dynamics $\mathcal{F}(\mathbf{z}_{<t}; \mathbf{c})$ with latent dynamics state $\mathbf{z}_t$ and environment variable $\mathbf{c}$ , prior identifiability results have largely focused on $\mathbf{z}_t$ when conditioned on predefined label $u$ of dynamics environments. This leaves two limitations: reliance on predefined labels that hinder generalization to unseen environments, and a limited understanding of the identifiability of $\mathcal{F}$ and $\mathbf{c}$ which -- although offering important structural properties for the identifiability of $\mathbf{z}_t$ -- are learned jointly with $\mathbf{z}_t$ . We address these limitations with Meta-iLaD, a novel identifiable latent dynamics framework attained by meta-learning across dynamics environments. Meta-iLaD introduces a novel conditional prior of $\mathbf{c}$ , modeled as a feedforward meta-learner to rapidly extract $\mathbf{c}$ from few-shot examples. Meta-iLaD further establishes identifiability for $\mathbf{z}_t$ , $\mathbf{c}$ , and $\mathcal{F}$ simultaneously, for a general formulation of $\mathcal{F}(\mathbf{z}_{< t};\mathbf{c})$ without restricting the dimension of $\mathbf{c}$ or how it modulates $\mathcal{F}$ . On synthetic and real data, we provide strong empirical evidence that 1) conditioning on few-shot examples enables generalization to out-of-distribution environments, and 2) identifiability for $\mathbf{c}$ and $\mathcal{F}$ is critical for accurate forecasting beyond reconstructing observed trajectories. Code is available at https://github.com/GoForIt-YYB/Meta-iLaD.

Figures & tables

Figure 1: Overview of a latent dynamics system with latent dynamics state $\mathbf{z}_{t}$ , environment variable $\mathbf{c}$ , and dynamics function $\mathcal{F}$ (a), its use in reconstruction (b) vs . prediction (c) settings, and Meta-iLaD which establishes $\mathbf{z}_{t}=\mathcal{F}(\mathbf{z}_{<t};\mathbf{c})$ with identifiable $\mathbf{z}_{t},\mathbf{c}$ and $\mathcal{F}$ (d).

Dataset	Method	Reconstruction			Prediction			OOD Generalization
Dataset	Method	MSE $\downarrow$	MCC( $\mathbf{z}$ ) $\uparrow$	MCC( $\mathbf{c}$ ) $\uparrow$	MSE $\downarrow$	MCC( $\mathbf{z}$ ) $\uparrow$	MCC( $\mathbf{c}$ ) $\uparrow$	MSE $\downarrow$	MCC( $\mathbf{z}$ ) $\uparrow$	MCC( $\mathbf{c}$ ) $\uparrow$
Pendulum	Meta-iLaD	3.58(0.27)e-1	1.00(0.00)	0.98(0.01)	3.56(0.35)e-1	1.00(0.00)	0.98(0.01)	5.89(0.29)e-1	1.00(0.00)	0.97(0.00)
	ODE2VAE	/	/	/	7.26(0.10)e0	0.88(0.01)	/	7.97(0.13)e0	0.86(0.01)	/
	MoNODE	3.67(0.06)e0	0.23(0.06)	0.24(0.07)	4.62(0.26)e0	0.23(0.06)	0.30(0.11)	5.83(0.81)e0	0.19(0.07)	0.26(0.05)
	TDRL	3.98(0.27)e-1	0.97(0.02)	0.14(0.01)	/	/	/	/	/	/
	IDF	3.78(0.34)e-1	0.98(0.01)	0.04(0.01)	1.26(0.47)e1	0.78(0.00)	0.02(0.01)	/	/	/

Table 1: Quantitative results on Pendulum and CartPole.

Figure 2: Scatter plots of estimated vs . true latent states $\mathbf{z}_{t}$ (blue) and environment variable $\mathbf{c}$ (red). Left: Pendulum; Right: CartPole.

Figure 3: Examples of IDF vs . Meta-iLaD in reconstruction (t=0:75) and prediction (t=75:125) on an in-distribution subject.

Models	ID		OOD
Models	0-75	75-125	0-75	75-125
ODE2VAE	2.56(0.00)e4	2.55(0.00)e4	2.74(0.00)e4	2.73(0.00)e4
MONODE	1.40(0.23)e1	3.87(0.46)e1	5.34(0.43)e1	1.08(0.04)e2
IDF	6.12(0.42)e0	3.21(0.18)e1	7.65(1.09)e1	1.18(0.06)e2
Meta-iLaD	2.80(0.49)e0	1.95(0.11)e1	3.29(1.23)e1	7.87(1.25)e1

Table 2: Reconstruction (0-75) and prediction (75-125) MSEs on MoCAP, tested on in-distribution (ID) and OOD subjects.

	Prediction		OOD Generalization
	MSE(xt)	MCC(c)	MSE(xt)	MCC(c)
Context Samples	3.49(0.40)e-1	0.98(0.00)	6.16(0.08)e-1	0.96(0.00)
Query Samples	5.06(0.50)e-1	0.91(0.00)	8.88(2.75)e-1	0.88(0.01)
Predefined Index	3.45(0.22)e-1	0.97(0.00)	/	/
True Param Value	3.23(0.10)e-1	0.99(0.00)	5.06(0.20)e-1	0.98(0.00)

Table 3: Ablation of conditioning

\mathbf{c}

on few-shot context samples (representing Meta-iLaD), query sample itself (representing an unconditional prior for

\mathbf{c}

), predefined indices of environments (representing a realization of iVAE in our model), and true environment variables (representing the best scenario).

$\mathcal{L}_{\text{TC}}$	$\mathcal{L}_{\text{flow}}$	Prediction
$\mathcal{L}_{\text{TC}}$	$\mathcal{L}_{\text{flow}}$	MSE(xt)	MCC(zt)	MCC(c)
✗	✗	3.87(0.30)e0	0.20(0.10)	0.86(0.06)
✓	✗	3.58(0.27)e0	0.26(0.08)	0.95(0.01)
✗	✓	4.82(0.22)e-1	1.00(0.00)	0.88(0.02)
✓	✓	3.49(0.40)e-1	1.00(0.00)	0.98(0.00)

Table 4: Ablation of the presence/absence of TC regularization

\mathcal{L}_{\text{TC}}

and warm-up training strategy (tied to

\mathcal{L}_{\text{flow}}

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: The general overview of the datasets

Double Pendulum	Reconstruction			Prediction			OOD Generalization
Double Pendulum	MSE $\downarrow$	MCC( $\mathbf{z}$ ) $\uparrow$	MCC( $\mathbf{c}$ ) $\uparrow$	MSE $\downarrow$	MCC( $\mathbf{z}$ ) $\uparrow$	MCC( $\mathbf{c}$ ) $\uparrow$	MSE $\downarrow$	MCC( $\mathbf{z}$ ) $\uparrow$	MCC( $\mathbf{c}$ ) $\uparrow$
Meta-iLaD	6.01(0.03)e-1	0.98(0.01)	0.98(0.01)	6.06(0.02)e-1	0.98(0.01)	0.98(0.01)	9.18(0.02)e-1	0.98(0.01)	0.98(0.01)
ODE2VAE	/	/	/	8.88(0.07)e0	0.60(0.03)	/	9.94(0.38)e0	0.60(0.07)	/
MoNODE	1.51(0.24)e0	0.85(0.01)	0.02(0.00)	5.43(0.16)e0	0.70(0.02)	0.12(0.01)	6.54(0.30)e0	0.75(0.07)	0.08(0.06)
TDRL	8.41(0.04)e-1	0.90(0.04)	0.13(0.02)	/	/	/	/	/	/
IDF	8.04(0.03)e-1	0.94(0.00)	0.04(0.02)	1.61(0.18)e0	0.94(0.00)	0.08(0.04)	/	/	/

Appendix

Table 5: Quantitative results on Double Pendulum.

Figure 5: Scatter plots of estimated versus true values for latent angles $\theta_{1},\theta_{2}$ (blue) and environment parameter $G$ (red) on Double Pendulum during reconstruction. Meta-iLaD successfully identifies both latent states and environment variable (clear linear relationships), while IDF identifies latent states but fails to recover $G$ (noise pattern).

Figure 6: Scatter plot of estimated versus true environment parameters on CartPole. Blue points represent in-distribution range $[-2.5,2.5]$ , red points represent OOD ranges $[-4.0,-2.5]\cup[2.5,4.0]$ . Meta-iLaD maintains identifiability across both ID and OOD regions.

Figure 7: Test trajectory predictions for IDF and Meta-iLaD methods. At test time, first 75 time points are given as input, and the prediction horizon is 50 steps (frames 76-125). The figure shows comparisons across multiple joints (Joint 3, 13, 27, 44, 46, 47), where the gray shaded region indicates the input range, the black solid line represents the ground truth trajectory, the blue dashed line shows IDF predictions, and the red dash-dot line shows Meta-iLaD predictions.

Figure 8: Scatter plots of estimated latent environment dimensions $\{\hat{c}_{1},\hat{c}_{2},\hat{c}_{3},\hat{c}_{4}\}$ (sorted by MCC in descending order) versus the true environment parameter on CartPole dataset. Rows correspond to different estimated dimensions $\hat{d}\in\{2,3,4\}$ with true dimension $d=1$ . Only $\hat{c}_{1}$ exhibits identifiable relationship with the true parameter (clear monotonic trend), while all remaining dimensions contain only noise, confirming Corollary 1.

Figure 9: Results on real double pendulum data (left), including MCC on latent $\mathbf{z}_{t}$ ’s (middle) and examples on reconstructing (NCTRL) or forecasting (Meta-iLaD) $\mathbf{x}_{t}$ ’s (right).

VAE Encoder (for $\mathbf{z}_{t}$ )	VAE Decoder (for $\mathbf{z}_{t}$ )
Input: $\mathbb{R}^{L\times H\times W}$ , window size $k=3$	Input: $\mathbb{R}^{d_{z}/2}$
Conv2d( $k$ , 32, 4, stride=2) + ReLU	FC( $d_{z}/2$ , $128\times\frac{H}{8}\times\frac{W}{8}$ ), Unflatten
Conv2d(32, 64, 4, stride=2) + ReLU	ConvTranspose2d(128, 64, 4, stride=2) + ReLU
Conv2d(64, 128, 4, stride=2) + ReLU	ConvTranspose2d(64, 32, 4, stride=2) + ReLU
Flatten	ConvTranspose2d(32, 1, 4, stride=2)
FC to $\mu_{z}$ , $\sigma_{z}$ (dim= $d_{z}$ )	Output: $\mathbb{R}^{H\times W}$

Appendix

Table 6: Architecture details of Meta-iLaD components.

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Department of Statistics & Data Science University of California, Los Angeles

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Open paper AskPDF arXiv

Identifying linear time-invariant (LTI) dynamical systems is challenging when trajectories are short, noisy, or high-dimensional. Traditional system identification typically treats each system independently and cannot exploit shared structure across related systems. We propose PBML-LTI, a PAC-Bayesian meta-learning framework for few-shot LTI system identification that learns a transferable prior over task-specific dynamics while preserving task heterogeneity. Each task corresponds to an unknown LTI system, and the meta-learner uses training trajectories to learn a data-dependent prior over transition matrices. For a new system with limited data, PBML-LTI performs Bayesian adaptation under this prior to obtain a task-specific posterior, providing accurate estimates and principled uncertainty quantification. A key challenge is temporal dependence, since LTI trajectories violate the i.i.d. assumptions underlying most PAC-Bayes meta-learning analyses. We address this with a martingale PAC-Bayes analysis for dependent trajectory losses and derive a support-query predictive-risk bound that motivates a fit-KL meta-training objective. The bound clarifies the roles of empirical fit, posterior complexity, and prior quality in few-shot adaptation under sequential dependence. We further derive corollaries for transition-matrix recovery and multi-step trajectory prediction, connecting uncertainty-aware meta-identification with finite-sample guarantees for dependent dynamical data.

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