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

# Identifiable Latent Dynamics via Meta-Learning of Dynamics Environments
Source: [https://arxiv.org/html/2610.27180](https://arxiv.org/html/2610.27180)
Yubo YeAffiliation:Department of Computing and Information Sciences, Rochester Institute of Technology, Rochester, New York, USACorrespondence to:[yy8339@rit\.edu](mailto:[email protected])Sweekar PiyaXiajun JiangAffiliation:Department of Computer Science, Rowan University, Glassboro, New Jersey, USALinwei WangAffiliation:Department of Computing and Information Sciences, Rochester Institute of Technology, Rochester, New York, USA

###### Abstract

Learninglatent dynamicsis central to assessing the current states and forecasting the future trajectories of high\-dimensional time series\. For locally\-stationary latent dynamicsℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)with latent dynamics state𝐳t\\mathbf\{z\}\_\{t\}and environment variable𝐜\\mathbf\{c\}, prior identifiability results have largely focused on𝐳t\\mathbf\{z\}\_\{t\}when conditioned on predefined labeluuof dynamics environments\. This leaves two limitations: reliance on predefined labels that hinder generalization to unseen environments, and a limited understanding of the identifiability ofℱ\\mathcal\{F\}and𝐜\\mathbf\{c\}which—although offering important structural properties for the identifiability of𝐳t\\mathbf\{z\}\_\{t\}—are learned jointly with𝐳t\\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 of𝐜\\mathbf\{c\}, modeled as a feedforward meta\-learner to rapidly extract𝐜\\mathbf\{c\}from few\-shot examples\. Meta\-iLaD further establishes identifiability for𝐳t\\mathbf\{z\}\_\{t\},𝐜\\mathbf\{c\}, andℱ\\mathcal\{F\}simultaneously, for a general formulation ofℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)without restricting the dimension of𝐜\\mathbf\{c\}or how it modulatesℱ\\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 for𝐜\\mathbf\{c\}andℱ\\mathcal\{F\}is critical for accurate forecasting beyond reconstructing observed trajectories\. Code is available at[https://github\.com/GoForIt\-YYB/Meta\-iLaD](https://github.com/GoForIt-YYB/Meta-iLaD)\.

###### Keywords:

Machine Learning, ICML

## 1Introduction

Many domains such as healthcare are observing increasingly high\-dimensional \(e\.g\.,image or multi\-modal\) data over time\. It is important to extract the latent dynamics that governs the observed data, not only for assessing the current states of the system, but also for forecasting its future trajectories\. An important approach is to model the dynamics in a low\-dimensional latent space as illustrated in Fig\.[1](https://arxiv.org/html/2610.27180#S1.F1)\(a\): the observed𝐱t\\mathbf\{x\}\_\{t\}’s is generated from thelatent dynamics state𝐳t\\mathbf\{z\}\_\{t\}via anemission function𝐱t=𝐠⁡\(𝐳t\)\\mathbf\{x\}\_\{t\}=\\mathbf\{g\}\(\\mathbf\{z\}\_\{t\}\); the temporal transition of𝐳t\\mathbf\{z\}\_\{t\}is governed by alatent dynamics function𝐳t=ℱ⁡\(𝐳<t,𝐜\)\\mathbf\{z\}\_\{t\}=\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\), where adynamics environment variable𝐜\\mathbf\{c\}allows non\-stationaryℱ\\mathcal\{F\}\(but can be omitted otherwise\)\. The goal oflatent dynamics modelingis to learnℱ\\mathcal\{F\}and𝐠\\mathbf\{g\}from observed𝐱t\\mathbf\{x\}\_\{t\}’swithoutdirect supervisions on𝐳t\\mathbf\{z\}\_\{t\}’s\.

While significant progress has been made in this research direction\([Krishnan et al\., 2017](https://arxiv.org/html/2610.27180#bib.bib20);[Fraccaro et al\., 2017](https://arxiv.org/html/2610.27180#bib.bib6);[Yıldız et al\., 2019](https://arxiv.org/html/2610.27180#bib.bib32);[Botev et al\., 2021](https://arxiv.org/html/2610.27180#bib.bib4)\), we are interested in a specific question: even whenp⁡\(𝐱^t\)p\(\\hat\{\\mathbf\{x\}\}\_\{t\}\)is perfectly optimized to be equivalent to the observedp⁡\(𝐱t\)p\(\\mathbf\{x\}\_\{t\}\), what can we say about its underlying𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\},𝐜^\\hat\{\\mathbf\{c\}\}, andℱ^\\hat\{\\mathcal\{F\}\}in relation to their respective truth? This question ofidentifiability, as outlined in Fig\.[1](https://arxiv.org/html/2610.27180#S1.F1)\(a\), exists at three levels: identifiability of the latent dynamics state𝐳t\\mathbf\{z\}\_\{t\}, identifiability of the latent dynamics functionℱ\\mathcal\{F\}, and identifiability of the dynamics environment variable𝐜\\mathbf\{c\}\.

![Refer to caption](https://arxiv.org/html/2610.27180v1/Fig/Figure_26.png)Figure 1:Overview of a latent dynamics system with latent dynamics state𝐳t\\mathbf\{z\}\_\{t\}, environment variable𝐜\\mathbf\{c\}, and dynamics functionℱ\\mathcal\{F\}\(a\), its use in reconstruction \(b\)vs\. prediction \(c\) settings, and Meta\-iLaD which establishes𝐳t=ℱ⁡\(𝐳<t,𝐜\)\\mathbf\{z\}\_\{t\}=\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)with identifiable𝐳t,𝐜\\mathbf\{z\}\_\{t\},\\mathbf\{c\}andℱ\\mathcal\{F\}\(d\)\.Identifiability of latent dynamics states:Identifiability of the latent state𝐳t\\mathbf\{z\}\_\{t\}has been the primary focus of existing works in identifiable latent dynamics\. Successes are mainly built on the theory of nonlinear independent component analysis \(ICA\)\([Hyvärinen et al\., 2019](https://arxiv.org/html/2610.27180#bib.bib12);[Khemakhem et al\., 2020a](https://arxiv.org/html/2610.27180#bib.bib15)\), across three types of latent dynamics systems: stationaryℱ⁡\(𝐳<t\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\}\)\([Hyvärinen & Morioka, 2017](https://arxiv.org/html/2610.27180#bib.bib11);[Klindt et al\., 2021](https://arxiv.org/html/2610.27180#bib.bib19)\), non\-stationaryℱ⁡\(𝐳<t,𝐜t\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\_\{t\}\)where𝐜t\\mathbf\{c\}\_\{t\}follows a hidden Markov model\([Song et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib24);[Li et al\., 2024](https://arxiv.org/html/2610.27180#bib.bib21);[Hälvä & Hyvärinen, 2020](https://arxiv.org/html/2610.27180#bib.bib7);[Hälvä et al\., 2021](https://arxiv.org/html/2610.27180#bib.bib8);[Balsells\-Rodas et al\., 2024](https://arxiv.org/html/2610.27180#bib.bib2)\), and locally\-stationaryℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)where𝐜\\mathbf\{c\}characterizes the underlying dynamics environments\([Khemakhem et al\., 2020a](https://arxiv.org/html/2610.27180#bib.bib15);[Yao et al\., 2022a](https://arxiv.org/html/2610.27180#bib.bib30);[Yao et al\., 2022b](https://arxiv.org/html/2610.27180#bib.bib31);[Hızlı et al\., 2025](https://arxiv.org/html/2610.27180#bib.bib9)\)\. Our primary interest is in the last category of locally\-stationary dynamicsℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\), where two critical gaps remain\.

First, existing works inℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)establish the identifiability of𝐳t\\mathbf\{z\}\_\{t\}’s by conditioningℱ\\mathcal\{F\}on observed labelsuuof the dynamics environments\. Unless true parameters characterizing such environments are available,uuneeds to be predefined as discrete environment labels\. This raises fundamental questions regarding 1\) generalization to out\-of\-distribution \(OOD\) environments not included in the training labels, and 2\) applicability when such labeluuis absent\.

Second, whileℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)is leveraged as important structural properties to establish the identifiability of𝐳t\\mathbf\{z\}\_\{t\}, the identifiability ofℱ\\mathcal\{F\}and𝐜\\mathbf\{c\}themselves has only been examined in one recent work\([Hızlı et al\., 2025](https://arxiv.org/html/2610.27180#bib.bib9)\)even though they are learned simultaneously with𝐳t\\mathbf\{z\}\_\{t\}\. This is largely inconsequential in thereconstructionsetting considered in most existing works where, as illustrated in Fig\.[1](https://arxiv.org/html/2610.27180#S1.F1)\(b\), one observes a sequence of𝐱0:T\\mathbf\{x\}\_\{0:T\}’s and extract latent𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\}’s to reconstruct𝐱^0:T\\hat\{\\mathbf\{x\}\}\_\{0:T\}\. There is however a distinct yet equally importantpredictionsetting where, as illustrated in Fig\.[1](https://arxiv.org/html/2610.27180#S1.F1)\(c\), one observes𝐱0:l\\mathbf\{x\}\_\{0:l\}and uses learned latent dynamics functionℱ^​\(𝐳^<t,𝐜^\)\\hat\{\\mathcal\{F\}\}\(\\hat\{\\mathbf\{z\}\}\_\{<t\};\\hat\{\\mathbf\{c\}\}\)to predict future𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\}and𝐱^t\\hat\{\\mathbf\{x\}\}\_\{t\}fort\>lt\>l\. Because future latent states are evolved directly throughℱ^​\(𝐳^<t,𝐜^\)\\hat\{\\mathcal\{F\}\}\(\\hat\{\\mathbf\{z\}\}\_\{<t\};\\hat\{\\mathbf\{c\}\}\)in this setting, the identifiability ofℱ\\mathcal\{F\}and𝐜\\mathbf\{c\}becomes more important yet less studied\.

Identifiability of latent dynamics environments & functions:In the general literature of latent dynamics modeling, increasing works have taken interest in inferring the latent environment variable𝐜\\mathbf\{c\}underlying locally\-stationaryℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)\([Auzina et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib1);[Wang et al\., 2022](https://arxiv.org/html/2610.27180#bib.bib27);[Kirchmeyer et al\., 2022](https://arxiv.org/html/2610.27180#bib.bib18);[Jiang et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib13)\)\. These works however do not investigate or establish the identifiability for𝐜\\mathbf\{c\}, nor its relation to the identifiability of dynamics functionℱ\\mathcal\{F\}\.

In the literature of identifiable latent dynamics, the identifiability of latent environment variable𝐜\\mathbf\{c\}has been mainly studied in the category of non\-stationaryℱ⁡\(𝐳<t,𝐜t\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\_\{t\}\), assuming𝐜t\\mathbf\{c\}\_\{t\}to be the hidden state of an HMM process\([Song et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib24);[Li et al\., 2024](https://arxiv.org/html/2610.27180#bib.bib21);[Hälvä & Hyvärinen, 2020](https://arxiv.org/html/2610.27180#bib.bib7);[Hälvä et al\., 2021](https://arxiv.org/html/2610.27180#bib.bib8);[Balsells\-Rodas et al\., 2024](https://arxiv.org/html/2610.27180#bib.bib2)\)\. For locally\-stationaryℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\), a recent work\([Hızlı et al\., 2025](https://arxiv.org/html/2610.27180#bib.bib9)\)took the first step through by introducing a process noise𝐬t\\mathbf\{s\}\_\{t\}and assuming it to be factorized when conditioned on predefined environment labeluu: withℱ⁡\(𝐳<t,𝐬t\|u\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\},\\mathbf\{s\}\_\{t\}\|u\), identifiability is established simultaneously for𝐳t\\mathbf\{z\}\_\{t\},𝐬t\\mathbf\{s\}\_\{t\}, andℱ\\mathcal\{F\}\. While an important first step, the use of process noise𝐬t\\mathbf\{s\}\_\{t\}represents a specific assumption of how latent dynamics is modulated by its environments\. It is associated with several restrictive conditions,e\.g\.,𝐬t\\mathbf\{s\}\_\{t\}must match the dimensionality of𝐳t\\mathbf\{z\}\_\{t\}and have adimension\-wiseinfluence onℱ\\mathcal\{F\}\(each dimension ofℱ\\mathcal\{F\}is influenced by a single dimension noise variable in𝐬t\\mathbf\{s\}\_\{t\}\)\.

Contribution:We introduce Meta\-iLaD, a novel framework for identifiable latent dynamicsℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)via meta\-learning across dynamics environments\. As outlined in Fig\.[1](https://arxiv.org/html/2610.27180#S1.F1)\(d\), instead of conditioning on predefined environment labels, Meta\-iLaD conditionsℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)on few\-shot context examples from a dynamics environment, from which the latent environment variable𝐜\\mathbf\{c\}is inferred\. It advances identifiability research with the following contributions:

- •Theoretically, we establish the first simultaneous identifiability results for latent state𝐳t\\mathbf\{z\}\_\{t\}, latent environment variable𝐜\\mathbf\{c\}\(including its dimensionality\), and latent dynamics functionℱ\\mathcal\{F\}for a general formulation ofℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\), where the latent environment variable𝐜\\mathbf\{c\}is not limited to a specific dimension or form, but can modulateℱ\\mathcal\{F\}in a general and interpretable fashion,e\.g\., as parameters characterizing the dynamics environments\. Furthermore, this is the first identifiableℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)that is conditioned on few\-shot context samples rather than predefined environment labels\.
- •Methodologically, we introduce a novel formulation of the conditional prior of𝐜\\mathbf\{c\}, as a feedforward meta\-learner that rapidly extracts𝐜\\mathbf\{c\}from few\-shot context examples from an environment\. This conditional latent environment variable then adapts the latentℱ\\mathcal\{F\}to evolve latent dynamics states𝐳t\\mathbf\{z\}\_\{t\}’s to emit to𝐱t\\mathbf\{x\}\_\{t\}’s specific to that environment, establishing meta\-learning as a novel solution to attain identifiable latent dynamics\.
- •Experimentally, we provide strong empirical evidence for the two key theoretical contributions of Meta\-iLaD\. First, compared to predefined environment labelsuu’s, we show that conditioning𝐜\\mathbf\{c\}on few\-shot examples enables Meta\-iLaD to generalize to OOD environments as well as apply to a broader variety of systems where environment labelsuuare difficult to define\. Second, moving beyond reconstruction\-focused evaluation settings commonly adopted in prior work, we provide evidence that identifiability of the environment variable𝐜\\mathbf\{c\}and dynamics functionℱ\\mathcal\{F\}are critical for predicting beyond observed trajectories\.

## 2Related Works

### 2\.1Identifiable Latent Dynamics Models

Recent advances in nonlinear ICA theory have established that identifiability of a latent variable model can be attained by constructing its conditional independence given auxiliary information\([Hyvärinen & Morioka, 2016](https://arxiv.org/html/2610.27180#bib.bib10);[Hyvärinen & Morioka, 2017](https://arxiv.org/html/2610.27180#bib.bib11);[Hyvärinen et al\., 2019](https://arxiv.org/html/2610.27180#bib.bib12);[Khemakhem et al\., 2020a](https://arxiv.org/html/2610.27180#bib.bib15)\)\. This theoretical foundation has enabled rapid developments in identifiable latent dynamics models, primarily focused on identifiability of𝐳t\\mathbf\{z\}\_\{t\}for three types of latent dynamics\.

Identifiability of latent states\.Forstationary dynamicsℱ⁡\(𝐳<t\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\}\), identifiability of𝐳t\\mathbf\{z\}\_\{t\}has been established by its temporal dependency\([Hyvärinen & Morioka, 2017](https://arxiv.org/html/2610.27180#bib.bib11);[Klindt et al\., 2021](https://arxiv.org/html/2610.27180#bib.bib19)\)\. Fornon\-stationary dynamicsℱ⁡\(𝐳<t,𝐜t\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\_\{t\}\), identifiability of𝐳t\\mathbf\{z\}\_\{t\}has been established when𝐜t\\mathbf\{c\}\_\{t\}represents the hidden state of a first\-order Markov process \(𝐜t∼Markov Chain​\(𝐀\)\\mathbf\{c\}\_\{t\}\\sim\\text\{Markov Chain\}\(\\mathbf\{A\}\)\)\. Different forms ofℱ\\mathcal\{F\}have been considered, including no additional temporal transition beyond the HMM\([Hälvä & Hyvärinen, 2020](https://arxiv.org/html/2610.27180#bib.bib7);[Balsells\-Rodas et al\., 2024](https://arxiv.org/html/2610.27180#bib.bib2)\), linear transitions\([Hälvä et al\., 2021](https://arxiv.org/html/2610.27180#bib.bib8)\), and non\-parametric transitions\([Song et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib24)\)\. Forlocally\-stationary dynamics𝐳t=ℱ⁡\(𝐳<t,𝐜\)\\mathbf\{z\}\_\{t\}=\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)where𝐜\\mathbf\{c\}governs a time segment, identifiability of𝐳t\\mathbf\{z\}\_\{t\}has been established for linear\([Yao et al\., 2022b](https://arxiv.org/html/2610.27180#bib.bib31)\), non\-parametric\([Yao et al\., 2022a](https://arxiv.org/html/2610.27180#bib.bib30);[Yao et al\., 2022b](https://arxiv.org/html/2610.27180#bib.bib31)\), and nonlinear forms ofℱ\\mathcal\{F\}\([Hızlı et al\., 2025](https://arxiv.org/html/2610.27180#bib.bib9)\)\. Critically, all these approaches conditionsℱ\\mathcal\{F\}on a predefined labeluuof the dynamics environment\([Khemakhem et al\., 2020a](https://arxiv.org/html/2610.27180#bib.bib15);[Yao et al\., 2022a](https://arxiv.org/html/2610.27180#bib.bib30);[Yao et al\., 2022b](https://arxiv.org/html/2610.27180#bib.bib31);[Hızlı et al\., 2025](https://arxiv.org/html/2610.27180#bib.bib9)\)\.

Identifiability of dynamics functions\.Most existing works focus solely on identifiability of𝐳t\\mathbf\{z\}\_\{t\}\. A recent exception\([Hızlı et al\., 2025](https://arxiv.org/html/2610.27180#bib.bib9)\)establishes identifiability for both𝐳t\\mathbf\{z\}\_\{t\}andℱ\\mathcal\{F\}by introducing a conditionally\-factorized process noise𝐬t\\mathbf\{s\}\_\{t\}withp⁡\(𝐬t\|u\)p\(\\mathbf\{s\}\_\{t\}\|u\)\. This assumes the environment to modulate each dimension ofℱ\\mathcal\{F\}via a scalar noise, requiring𝐬t\\mathbf\{s\}\_\{t\}to match the dimension of𝐳t\\mathbf\{z\}\_\{t\}and still relies on predefined labelsuu\.

Position of our work\.We advance locally\-stationary latent dynamicsℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)in two key aspects\. First, we extend identifiability to the latent environment variable𝐜\\mathbf\{c\}and dynamics functionℱ\\mathcal\{F\}for a general formulation ofℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)\. It goes beyond the assumptions of dimension\-wise modulation ofℱ\\mathcal\{F\}and does not restrict the dimensionality of𝐜\\mathbf\{c\}\. Second, while the general formulation ofℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)exists\([Yao et al\., 2022a](https://arxiv.org/html/2610.27180#bib.bib30)\), we condition it on few\-shot data examples instead of predefined environment labels, and show that this allows generalization to OOD environments as well as systems without available environment labels\.

### 2\.2Non\-Identifiable Latent Dynamics Models

Outside the identifiability literature, significant progress has been made in learning latent dynamics for forecasting high\-dimensional time series\([Chung et al\., 2015](https://arxiv.org/html/2610.27180#bib.bib5);[Krishnan et al\., 2017](https://arxiv.org/html/2610.27180#bib.bib20);[Karl et al\., 2017](https://arxiv.org/html/2610.27180#bib.bib14);[Fraccaro et al\., 2017](https://arxiv.org/html/2610.27180#bib.bib6);[Becker\-Ehmck et al\., 2019](https://arxiv.org/html/2610.27180#bib.bib3);[Linderman et al\., 2017](https://arxiv.org/html/2610.27180#bib.bib22);[Rubanova et al\., 2019](https://arxiv.org/html/2610.27180#bib.bib23);[Yıldız et al\., 2019](https://arxiv.org/html/2610.27180#bib.bib32);[Botev et al\., 2021](https://arxiv.org/html/2610.27180#bib.bib4)\)\. These methods prioritize predictive performance but do not establish identifiability for the learned latent states or dynamics functions\.

Environment\-adaptive dynamics\.There has been a rising interest in inferring latent environment variables to adapt latent dynamics functions\([Auzina et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib1);[Wang et al\., 2022](https://arxiv.org/html/2610.27180#bib.bib27);[Jiang et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib13);[Kirchmeyer et al\., 2022](https://arxiv.org/html/2610.27180#bib.bib18)\)\. While these approaches demonstrate improved forecasting by capturing environment\-specific dynamics, they do not discuss the identifiability of the latent environment variable or its relationship to the recoverability of the underlying dynamics\.

Position of our work\.In contrast, Meta\-iLaD establishes identifiability for both the latent environment variable𝐜\\mathbf\{c\}and the dynamics functionℱ\\mathcal\{F\}, introduces meta\-learning as a principled solution to attain this identifiability, and provides empirical evidence that such identifiability is critical for accurate forecasting performance beyond reconstruction\.

## 3Identifiable Latent Dynamics via Meta\-iLaD

Considernn\-dimensional time\-series data𝐱t∈ℝn\\mathbf\{x\}\_\{t\}\\in\\mathbb\{R\}^\{n\}at timett, generated frommm\-dimensional latent variable𝐳t∈ℝm\\mathbf\{z\}\_\{t\}\\in\\mathbb\{R\}^\{m\}via an invertible mixing function𝐠\\mathbf\{g\}\. We assume that the dynamics of𝐳t\\mathbf\{z\}\_\{t\}is governed by the transition functionℱ\\mathcal\{F\}and the environment variable𝐜∈ℝd\\mathbf\{c\}\\in\\mathbb\{R\}^\{d\}:

𝐳t=ℱ⁡\(𝐳t−1,𝐜\);𝐱t=𝐠⁡\(𝐳t\)\+ϵ,0≤t≤T\\mathbf\{z\}\_\{t\}=\\mathcal\{F\}\(\\mathbf\{z\}\_\{t\-1\};\\mathbf\{c\}\);\\quad\\mathbf\{x\}\_\{t\}=\\mathbf\{g\}\(\\mathbf\{z\}\_\{t\}\)\+\\epsilon,\\quad 0\\leq t\\leq T\(1\)whereϵ∼pϵ​\(ϵ\)\\epsilon\\sim p\_\{\\epsilon\}\(\\epsilon\)is an independent noise\. Whileℱ\\mathcal\{F\}can be described by a variety of functions, we consider a neural ordinary differential equation \(NODE\)𝐟θ\\mathbf\{f\}\_\{\\theta\}:

d​zi,td​t=fi,θ​\(𝐳t,𝐜\)⇔d​𝐳td​t=𝐟θ​\(𝐳t,𝐜\),\\frac\{dz\_\{i,t\}\}\{dt\}=f\_\{i,\\theta\}\(\\mathbf\{z\}\_\{t\};\\mathbf\{c\}\)\\Leftrightarrow\\frac\{d\\mathbf\{z\}\_\{t\}\}\{dt\}=\\mathbf\{f\}\_\{\\theta\}\(\\mathbf\{z\}\_\{t\};\\mathbf\{c\}\),\(2\)wherezi,tz\_\{i,t\}is theii\-th component in𝐳t\\mathbf\{z\}\_\{t\}\.𝐟θ\\mathbf\{f\}\_\{\\theta\}is parameterized byθ\\thetaand locally stationary dependent on𝐜\\mathbf\{c\}\. An unconditional generative model for this system is:

pΘ\(𝐱0:T\)=∫𝐜∫𝐳0:T\\displaystyle p\_\{\\Theta\}\(\\mathbf\{x\}\_\{0:T\}\)=\\int\_\{\\mathbf\{c\}\}\\int\_\{\\mathbf\{z\}\_\{0:T\}\}p⁡\(𝐜\)​p​\(𝐳0\)​∏t=1Tpθ​\(𝐳t\|𝐳t−1,𝐜\)\\displaystyle p\(\\mathbf\{c\}\)p\(\\mathbf\{z\}\_\{0\}\)\\prod\\nolimits\_\{t=1\}^\{T\}p\_\{\\theta\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{c\}\)\(3\)×∏t=0Tp𝐠\(𝐱t\|𝐳t\)d𝐳0:Td𝐜,\\displaystyle\\times\\prod\\nolimits\_\{t=0\}^\{T\}p\_\{\\mathbf\{g\}\}\(\\mathbf\{x\}\_\{t\}\|\\mathbf\{z\}\_\{t\}\)d\\mathbf\{z\}\_\{0:T\}d\\mathbf\{c\},whereΘ=\{θ,𝐠\}\\Theta=\\\{\\theta,\\mathbf\{g\}\\\}andp𝐠​\(𝐱t\|𝐳t\)=pϵ​\(𝐱t−𝐠⁡\(𝐳t\)\)p\_\{\\mathbf\{g\}\}\(\\mathbf\{x\}\_\{t\}\|\\mathbf\{z\}\_\{t\}\)=p\_\{\\epsilon\}\(\\mathbf\{x\}\_\{t\}\-\\mathbf\{g\}\(\\mathbf\{z\}\_\{t\}\)\)\.

Unfortunately, with this unconditional generative model, neither the environment variable𝐜\\mathbf\{c\}nor the latent variables𝐳t\\mathbf\{z\}\_\{t\}can be uniquely recovered from observations𝐱0:T\\mathbf\{x\}\_\{0:T\}alone\. To address this identifiability issue, we construct a conditional generative modelpΦ\(𝐱0:T\|𝐮\)p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:T\}\|\\mathbf\{u\}\), where𝐮\\mathbf\{u\}consists ofkk\-shot samples that share the same environment variable𝐜\\mathbf\{c\}as𝐱0:T\\mathbf\{x\}\_\{0:T\}but are otherwise distinct\. For clarity, in Section[3\.1](https://arxiv.org/html/2610.27180#S3.SS1), we define𝐮≔𝒟s\\mathbf\{u\}\\coloneqq\\mathcal\{D\}^\{s\}and refer to the samples in𝒟s\\mathcal\{D\}^\{s\}ascontextsamples \(for inferring𝐜\\mathbf\{c\}\); we also define𝐱0:T≔𝐱q0:T\\mathbf\{x\}\_\{0:T\}\\coloneqq\\mathbf\{x\}^\{q\}\_\{0:T\}and refer to it as thequery sample\(to be generated\)\. The conditional densitypΦ\(𝐱0:T\|𝐮\)≔pΦ\(𝐱0:Tq\|𝒟s\)p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:T\}\|\\mathbf\{u\}\)\\coloneqq p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:T\}^\{q\}\|\\mathcal\{D\}^\{s\}\)thus defines the marginal likelihood of a query sample𝐱0:Tq\\mathbf\{x\}\_\{0:T\}^\{q\}conditioned on distinct context samples from the same environment\.

### 3\.1Construction of Identifiable Latent Dynamics

Latent environment variable𝐜\\mathbf\{c\}:Given context samples𝒟s\\mathcal\{D\}^\{s\}from the same dynamics environment, we define the conditional density for its latent environment variable𝐜\\mathbf\{c\}as:

pζ​\(𝐜\|𝒟s\)\\displaystyle p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\)=∏i=1d𝒩⁡\(ci\|μi,ζ​\(𝒟s\),σi,ζ​\(𝒟s\)\)\\displaystyle=\\prod\\nolimits\_\{i=1\}^\{d\}\\mathcal\{N\}\(c\_\{i\}\|\\mu\_\{i,\\zeta\}\(\\mathcal\{D\}^\{s\}\),\\sigma\_\{i,\\zeta\}\(\\mathcal\{D\}^\{s\}\)\)\(4\)=𝒩⁡\(𝐜\|𝝁ζ​\(𝒟s\),𝝈ζ​\(𝒟s\)\),\\displaystyle=\\mathcal\{N\}\(\\mathbf\{c\}\|\\boldsymbol\{\\mu\}\_\{\\zeta\}\(\\mathcal\{D\}^\{s\}\),\\boldsymbol\{\\sigma\}\_\{\\zeta\}\(\\mathcal\{D\}^\{s\}\)\),where thedd\-dimensional𝐜\\mathbf\{c\}is assumed to be conditionally factorized and each elementci∈𝐜c\_\{i\}\\in\\mathbf\{c\}is Gaussian\. The multi\-variate mean𝝁ζ​\(𝒟s\)\\boldsymbol\{\\mu\}\_\{\\zeta\}\(\\mathcal\{D\}^\{s\}\)and variance𝝈ζ​\(𝒟s\)\\boldsymbol\{\\sigma\}\_\{\\zeta\}\(\\mathcal\{D\}^\{s\}\)are realized as:

𝝁ζ​\(𝒟s\)\\displaystyle\\boldsymbol\{\\mu\}\_\{\\zeta\}\(\\mathcal\{D\}^\{s\}\)=1\|𝒟s\|∑𝐱0:Ts∈𝒟s𝒉μ,ζ\(𝐱0:Ts\),\\displaystyle=\\frac\{1\}\{\|\\mathcal\{D\}^\{s\}\|\}\\sum\\nolimits\_\{\\mathbf\{x\}\_\{0:T\}^\{s\}\\in\\mathcal\{D\}^\{s\}\}\\boldsymbol\{h\}\_\{\\mu,\\zeta\}\(\\mathbf\{x\}\_\{0:T\}^\{s\}\),\(5\)𝝈ζ​\(𝒟s\)\\displaystyle\\boldsymbol\{\\sigma\}\_\{\\zeta\}\(\\mathcal\{D\}^\{s\}\)=1\|𝒟s\|∑𝐱0:Ts∈𝒟s𝒉σ,ζ\(𝐱s0:T\),\\displaystyle=\\frac\{1\}\{\|\\mathcal\{D\}^\{s\}\|\}\\sum\\nolimits\_\{\\mathbf\{x\}\_\{0:T\}^\{s\}\\in\\mathcal\{D\}^\{s\}\}\\boldsymbol\{h\}\_\{\\sigma,\\zeta\}\(\\mathbf\{x\}^\{s\}\_\{0:T\}\),where𝒉μ,ζ\\boldsymbol\{h\}\_\{\\mu,\\zeta\}and𝒉σ,ζ\\boldsymbol\{h\}\_\{\\sigma,\\zeta\}represent the two heads of an encoding network parameterized byζ\\zeta, and the averaging function extracts the embedding shared by thekk\-shot context samples\.

Latent dynamics state𝐳t\\mathbf\{z\}\_\{t\}:When conditioned on𝐜\\mathbf\{c\}, we define the transition probability of𝐳t\\mathbf\{z\}\_\{t\}as:

pθ​\(𝐳t\|𝐳t−1,𝐜\)=δ⁡\(𝐳t−ℱθ​\(𝐳t−1,𝐜\)\),ℱθ​\(𝐳t−1,𝐜\)=𝐳t−1\+∫s=t−1t𝐟θ​\(𝐳s,𝐜\)​ds\.\\begin\{split\}p\_\{\\theta\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{c\}\)&=\\delta\(\\mathbf\{z\}\_\{t\}\-\\mathcal\{F\}\_\{\\theta\}\(\\mathbf\{z\}\_\{t\-1\},\\mathbf\{c\}\)\),\\\\ \\mathcal\{F\}\_\{\\theta\}\(\\mathbf\{z\}\_\{t\-1\},\\mathbf\{c\}\)&=\\mathbf\{z\}\_\{t\-1\}\+\\int\_\{s=t\-1\}^\{t\}\\mathbf\{f\}\_\{\\theta\}\(\\mathbf\{z\}\_\{s\};\\mathbf\{c\}\)ds\.\\end\{split\}In other words, we assume the stochasticity inℱ\\mathcal\{F\}to be explained by the environment variable𝐜\\mathbf\{c\}and𝐳0\\mathbf\{z\}\_\{0\}\. Furthermore, when conditioned on𝐜\\mathbf\{c\}and𝐳t−1\\mathbf\{z\}\_\{t\-1\},𝐳t\\mathbf\{z\}\_\{t\}is assumed factorized aspθ​\(𝐳t\|𝐳t−1,𝐜\)=∏i=1mpθ​\(zi,t\|𝐳t−1,𝐜\)p\_\{\\theta\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{c\}\)=\\prod\_\{i=1\}^\{m\}p\_\{\\theta\}\(z\_\{i,t\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{c\}\)\.

To marginalize out𝐳t−1\\mathbf\{z\}\_\{t\-1\}frompθ​\(𝐳t\|𝐳t−1,𝐜\)p\_\{\\theta\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{c\}\),pθ​\(𝐳t\|𝐜\)p\_\{\\theta\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)can also be derived fromp⁡\(𝐳0\)p\(\\mathbf\{z\}\_\{0\}\)through the NODE flow in Equation \([2](https://arxiv.org/html/2610.27180#S3.E2)\) via the instantaneous change of variables:

log⁡pθ​\(𝐳t\|𝐜\)=log⁡p⁡\(𝐳0\)−∫0ttr​\(J𝐟θ​\(𝐳τ,𝐜\)\)​𝑑τ,\\log p\_\{\\theta\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)=\\log p\(\\mathbf\{z\}\_\{0\}\)\-\\int\_\{0\}^\{t\}\\text\{tr\}\\left\(J\_\{\\mathbf\{f\}\_\{\\theta\}\}\(\\mathbf\{z\}\_\{\\tau\},\\mathbf\{c\}\)\\right\)d\\tau,\(6\)whereJ𝐟θ​\(𝐳t,𝐜\)=∂𝐟θ​\(𝐳t,𝐜\)∂𝐳tJ\_\{\\mathbf\{f\}\_\{\\theta\}\}\(\\mathbf\{z\}\_\{t\},\\mathbf\{c\}\)=\\frac\{\\partial\\mathbf\{f\}\_\{\\theta\}\(\\mathbf\{z\}\_\{t\},\\mathbf\{c\}\)\}\{\\partial\\mathbf\{z\}\_\{t\}\}denotes the Jacobian matrix\.

Conditional generative models:For a query sample𝐱0:Tq\\mathbf\{x\}\_\{0:T\}^\{q\}from the same environment as the context samples𝒟s\\mathcal\{D\}^\{s\}, we now extend the unconditonal generative model in Equation \([3](https://arxiv.org/html/2610.27180#S3.E3)\) to a conditionalpΦ\(𝐱0:Tq\|𝒟s\)p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:T\}^\{q\}\|\\mathcal\{D\}^\{s\}\)withΦ=\(Θ,ζ\)\\Phi=\(\\Theta,\\zeta\)as:

\\displaystylepΦ\(𝐱0:Tq\|𝒟s\)=∫𝐜∫𝐳0pΘ\(𝐱0:Tq\|𝐜,𝐳0\)pζ\(𝐜\|𝒟s\)p\(𝐳0\)d𝐜d𝐳0\\displaystyle p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:T\}^\{q\}\|\\mathcal\{D\}^\{s\}\)=\\int\_\{\\mathbf\{c\}\}\\int\_\{\\mathbf\{z\}\_\{0\}\}p\_\{\\Theta\}\(\\mathbf\{x\}\_\{0:T\}^\{q\}\|\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\)p\(\\mathbf\{z\}\_\{0\}\)d\\mathbf\{c\}d\\mathbf\{z\}\_\{0\}=∫𝐜∫𝐳0pg​\(𝐱0\|𝐳0\)​∏t=1Tpg​\(𝐱t\|𝐳t\)\|𝐳t=ℱθ​\(𝐳t−1,𝐜\)\\displaystyle=\\int\_\{\\mathbf\{c\}\}\\int\_\{\\mathbf\{z\}\_\{0\}\}p\_\{g\}\(\\mathbf\{x\}\_\{0\}\|\\mathbf\{z\}\_\{0\}\)\\prod\\nolimits\_\{t=1\}^\{T\}p\_\{g\}\(\\mathbf\{x\}\_\{t\}\|\\mathbf\{z\}\_\{t\}\)\|\_\{\\mathbf\{z\}\_\{t\}=\\mathcal\{F\}\_\{\\theta\}\(\\mathbf\{z\}\_\{t\-1\};\\mathbf\{c\}\)\}×pζ​\(𝐜\|𝒟s\)​p​\(𝐳0\)​d​𝐜​d​𝐳0\.\\displaystyle\\times p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\)p\(\\mathbf\{z\}\_\{0\}\)d\\mathbf\{c\}d\\mathbf\{z\}\_\{0\}\.

### 3\.2Identifiability Theory

For simplicity in notation, in this section, we again denote𝐱0:Tq\\mathbf\{x\}\_\{0:T\}^\{q\}as𝐱0:T\\mathbf\{x\}\_\{0:T\}and𝒟s\\mathcal\{D\}^\{s\}as𝐮\\mathbf\{u\}\. We define:

𝐱t\\displaystyle\\mathbf\{x\}\_\{t\}=𝐠⁡\(𝐳0\+∫s=0t𝐟θ​\(𝐳s,𝐜\)​𝑑s\)\+ϵt\\displaystyle=\\mathbf\{g\}\\left\(\\mathbf\{z\}\_\{0\}\+\\int\_\{s=0\}^\{t\}\\mathbf\{f\}\_\{\\theta\}\(\\mathbf\{z\}\_\{s\},\\mathbf\{c\}\)ds\\right\)\+\\epsilon\_\{t\}=F𝐠,θ,t​\(𝐳0,𝐜\)\+ϵt,\\displaystyle=F\_\{\\mathbf\{g\},\\theta,t\}\(\\mathbf\{z\}\_\{0\},\\mathbf\{c\}\)\+\\epsilon\_\{t\},\(7\)which gives𝐱0:T=𝐅𝐠,θ\(𝐳0,𝐜\)\+ϵ\\mathbf\{x\}\_\{0:T\}=\\mathbf\{F\}\_\{\\mathbf\{g\},\\theta\}\(\\mathbf\{z\}\_\{0\},\\mathbf\{c\}\)\+\\boldsymbol\{\\epsilon\}, where𝐅𝐠,θ​\(𝐳0,𝐜\)=\[F𝐠,θ,0​\(𝐳0,𝐜\),…,F𝐠,θ,T​\(𝐳0,𝐜\)\]T\\mathbf\{F\}\_\{\\mathbf\{g\},\\theta\}\(\\mathbf\{z\}\_\{0\},\\mathbf\{c\}\)=\[F\_\{\\mathbf\{g\},\\theta,0\}\(\\mathbf\{z\}\_\{0\},\\mathbf\{c\}\),\\ldots,F\_\{\\mathbf\{g\},\\theta,T\}\(\\mathbf\{z\}\_\{0\},\\mathbf\{c\}\)\]^\{T\}andϵ=\[ϵ0,…,ϵT\]T\\boldsymbol\{\\epsilon\}=\[\\epsilon\_\{0\},\\ldots,\\epsilon\_\{T\}\]^\{T\}\.

For the priorpζ​\(𝐜\|𝐮\)p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{u\}\), we define sufficient statistics𝐓⁡\(𝐜\)=\(𝐓1​\(c1\),…,𝐓d​\(cd\)\)\\mathbf\{T\}\(\\mathbf\{c\}\)=\(\\mathbf\{T\}\_\{1\}\(c\_\{1\}\),\\ldots,\\mathbf\{T\}\_\{d\}\(c\_\{d\}\)\)with𝐓i​\(ci\)=\(ci,ci2\)\\mathbf\{T\}\_\{i\}\(c\_\{i\}\)=\(c\_\{i\},c\_\{i\}^\{2\}\), and natural parameters𝝀ζ​\(𝐮\)=\(𝝀ζ​1​\(𝐮\),…,𝝀ζ​d​\(𝐮\)\)\\boldsymbol\{\\lambda\}\_\{\\zeta\}\(\\mathbf\{u\}\)=\(\\boldsymbol\{\\lambda\}\_\{\\zeta 1\}\(\\mathbf\{u\}\),\\ldots,\\boldsymbol\{\\lambda\}\_\{\\zeta d\}\(\\mathbf\{u\}\)\)where

𝝀ζ​i\(𝐮\)=\(𝝁ζi\(𝐮\)/𝝈ζi2\(𝐮\),−1/2𝝈ζi2\(𝐮\)\)\.\\boldsymbol\{\\lambda\}\_\{\\zeta i\}\(\\mathbf\{u\}\)=\\left\(\\boldsymbol\{\\mu\}\_\{\\zeta\_\{i\}\}\(\\mathbf\{u\}\)/\{\\boldsymbol\{\\sigma\}^\{2\}\_\{\\zeta\_\{i\}\}\(\\mathbf\{u\}\)\},\-1/\{2\\boldsymbol\{\\sigma\}^\{2\}\_\{\\zeta\_\{i\}\}\(\\mathbf\{u\}\)\}\\right\)\.\(8\)
#### 3\.2\.1Identifiability of Environment Variable𝐜\\mathbf\{c\}

Definition 1:Let𝐱0:T\\mathbf\{x\}\_\{0:T\}be an observed time series generated by Equations \([4](https://arxiv.org/html/2610.27180#S3.E4)\-[3\.1](https://arxiv.org/html/2610.27180#S3.Ex3)\) with parameterΦ=\(𝐠,θ,ζ\)\\Phi=\(\\mathbf\{g\},\\theta,\\zeta\)and latent environment variable𝐜\\mathbf\{c\}\. A model with parameterΦ^=\(𝐠^,θ^,ζ^\)\\hat\{\\Phi\}=\(\\hat\{\\mathbf\{g\}\},\\hat\{\\theta\},\\hat\{\\zeta\}\)and latent variable𝐜^\\hat\{\\mathbf\{c\}\}is observationally equivalent toΦ\\PhiifpΦ^\(𝐱0:T\)p\_\{\\hat\{\\Phi\}\}\(\\mathbf\{x\}\_\{0:T\}\)matchespΦ\(𝐱0:T\)p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:T\}\)everywhere\. Let∼\\simbe an equivalence relation onΦ\\Phi\(see Definition 2\), we say that \([3\.1](https://arxiv.org/html/2610.27180#S3.Ex3)\) is∼\\simidentifiable if

pΦ\(𝐱0:T\|𝐮\)=pΦ^\(𝐱0:T\|𝐮\)⇒Φ∼Φ^\.p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:T\}\|\\mathbf\{u\}\)=p\_\{\\hat\{\\Phi\}\}\(\\mathbf\{x\}\_\{0:T\}\|\\mathbf\{u\}\)\\Rightarrow\\Phi\\sim\\hat\{\\Phi\}\.\(9\)
Definition 2:Let∼\\simbe the equivalence relation onΦ\\Phias:

\(𝐠,θ,ζ\)∼\(𝐠^,θ^,ζ^\)⇔∃A,𝐛,P\|𝐜=A⋅P⁡\(𝐜^\)\+𝐛,\(\\mathbf\{g\},\\theta,\\zeta\)\\sim\(\\hat\{\\mathbf\{g\}\},\\hat\{\\theta\},\\hat\{\\zeta\}\)\\Leftrightarrow\\exists A,\\mathbf\{b\},P\\mid\\mathbf\{c\}=A\\cdot P\(\\hat\{\\mathbf\{c\}\}\)\+\\mathbf\{b\},\(10\)whereA=diag​\(a1,…,ad\)A=\\text\{diag\}\(a\_\{1\},\\ldots,a\_\{d\}\)withai≠0a\_\{i\}\\neq 0is a diagonal scaling matrix,𝐛∈ℝd\\mathbf\{b\}\\in\\mathbb\{R\}^\{d\}is a translation vector, andPPis a permutation matrix\. This means that the true latent environment variable𝐜\\mathbf\{c\}is recovered up to permutation, scaling, and translation as the variable in its estimated latent space\.

Theorem 1:Assume that we observe data generated according to Equations \([4](https://arxiv.org/html/2610.27180#S3.E4)\-[3\.1](https://arxiv.org/html/2610.27180#S3.Ex3)\) with parameterΦ=\(𝐠,θ,ζ\)\\Phi=\(\\mathbf\{g\},\\theta,\\zeta\)and latent variable𝐜\\mathbf\{c\}\. Assume the following holds:

1. 1\.The set\{𝐱0:T∈𝒳T\+1\|φϵ\(𝐱0:T\)=0\}\\\{\\mathbf\{x\}\_\{0:T\}\\in\\mathcal\{X\}^\{T\+1\}\|\\varphi\_\{\\epsilon\}\(\\mathbf\{x\}\_\{0:T\}\)=0\\\}has measure zero, whereφϵ\\varphi\_\{\\epsilon\}is the characteristic function of the densitypϵp\_\{\\epsilon\}\.
2. 2\.The mixing function𝐅𝐠,θ\\mathbf\{F\}\_\{\\mathbf\{g\},\\theta\}is injective\.
3. 3\.The sufficient statisticsTi​jT\_\{ij\}are differentiable almost everywhere, and\(Ti​j\)1≤j≤2\(T\_\{ij\}\)\_\{1\\leq j\\leq 2\}are linearly independent on any subset of𝒳T\+1\\mathcal\{X\}^\{T\+1\}of measure greater than zero\.
4. 4\.There exist at least2​d\+12d\+1distinct𝐮0,…,𝐮2​d\\mathbf\{u\}\_\{0\},\.\.\.,\\mathbf\{u\}\_\{2d\}such that the2​d×2​d2d\\times 2dmatrix below is invertible 𝐋=\(𝝀ζ​\(𝐮1\)−𝝀ζ​\(𝐮0\),…,𝝀ζ​\(𝐮2​d\)−𝝀ζ​\(𝐮0\)\),\\mathbf\{L\}=\(\\boldsymbol\{\\lambda\}\_\{\\zeta\}\(\\mathbf\{u\}\_\{1\}\)\-\\boldsymbol\{\\lambda\}\_\{\\zeta\}\(\\mathbf\{u\}\_\{0\}\),\.\.\.,\\boldsymbol\{\\lambda\}\_\{\\zeta\}\(\\mathbf\{u\}\_\{2d\}\)\-\\boldsymbol\{\\lambda\}\_\{\\zeta\}\(\\mathbf\{u\}\_\{0\}\)\),\(11\)which requiressufficient variabilityin the latent variable𝐜\\mathbf\{c\}, i\.e\., the dynamics environments, to be observed\.

then the parametersΦ=\(𝐠,θ,ζ\)\\Phi=\(\\mathbf\{g\},\\theta,\\zeta\)are∼\\sim\-identifiable\.

Corollary 1:When the true dimensionddof𝐜\\mathbf\{c\}is unknown, setting the dimension of𝐜^\\hat\{\\mathbf\{c\}\}tod^≥d\\hat\{d\}\\geq densures that the original latent variable𝐜\\mathbf\{c\}is recovered up to an affine transformation and permutation within a subset of the estimated latent space, while the remaining dimensions encode only noise\.

#### 3\.2\.2Identifiability of Latent State𝐳t\\mathbf\{z\}\_\{t\}

Theorem 2:Suppose there exists an invertible function𝐠^−1\\hat\{\\mathbf\{g\}\}^\{\-1\}that𝐳^t=𝐠^−1​\(𝐱t\)\\hat\{\\mathbf\{z\}\}\_\{t\}=\\hat\{\\mathbf\{g\}\}^\{\-1\}\(\\mathbf\{x\}\_\{t\}\)\. Assume𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\}is factorized when conditioned on𝐳^t−1\\hat\{\\mathbf\{z\}\}\_\{t\-1\}and𝐮\\mathbf\{u\}\. Letηk​t​\(𝐮\)=log⁡p⁡\(zk​t\|𝐳t−1,𝐮\)\\eta\_\{kt\}\(\\mathbf\{u\}\)=\\log p\(z\_\{kt\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{u\}\)and assumeηk​t​\(𝐮\)\\eta\_\{kt\}\(\\mathbf\{u\}\)is twice differentiable inzk​tz\_\{kt\}\. Let∇2ηk​t:=∂2ηk​t∂zk​t2\\nabla^\{2\}\\eta\_\{kt\}:=\\frac\{\\partial^\{2\}\\eta\_\{kt\}\}\{\\partial z\_\{kt\}^\{2\}\}and∇ηk​t:=∂ηk​t∂zk​t\\nabla\\eta\_\{kt\}:=\\frac\{\\partial\\eta\_\{kt\}\}\{\\partial z\_\{kt\}\}\. If there exists at least2​m\+12m\+1unique environments𝐮0,𝐮1,…,𝐮2​m\\mathbf\{u\}\_\{0\},\\mathbf\{u\}\_\{1\},\\ldots,\\mathbf\{u\}\_\{2m\}such that the2​m2m\-dimensional function vectors𝐬k​t\\mathbf\{s\}\_\{kt\}and𝐬̊k​t\\mathring\{\\mathbf\{s\}\}\_\{kt\}defined below are linearly independent withkk=1:m, for each value of𝐳t\\mathbf\{z\}\_\{t\}:

𝐬k​t\\displaystyle\\mathbf\{s\}\_\{kt\}=\(​∇2ηk​t​\(𝐮1\)−∇2ηk​t​\(𝐮0\),…,\\displaystyle=\\mathord\{\\bigl\(\}\\nabla^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{1\}\)\-\\nabla^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{0\}\),\\ldots,∇2ηk​t​\(𝐮2​m\)−∇2ηk​t​\(𝐮2​m−1\)​\)T,\\displaystyle\\nabla^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{2m\}\)\-\\nabla^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{2m\-1\}\)\\mathord\{\\bigr\)\}^\{T\},𝐬̊k​t\\displaystyle\\mathring\{\\mathbf\{s\}\}\_\{kt\}=\(∇ηk​t\(𝐮1\)−∇ηk​t\(𝐮0\),…,\\displaystyle=\\mathord\{\\bigl\(\}\\nabla\\eta\_\{kt\}\(\\mathbf\{u\}\_\{1\}\)\-\\nabla\\eta\_\{kt\}\(\\mathbf\{u\}\_\{0\}\),\\ldots,∇ηk​t​\(𝐮2​m\)−∇ηk​t​\(𝐮2​m−1\)​\)T,\\displaystyle\\nabla\\eta\_\{kt\}\(\\mathbf\{u\}\_\{2m\}\)\-\\nabla\\eta\_\{kt\}\(\\mathbf\{u\}\_\{2m\-1\}\)\\mathord\{\\bigr\)\}^\{T\},which requiressufficient variabilityin the first\- and second\-order derivatives ofη⁡\(𝐮\)\\eta\(\\mathbf\{u\}\)to be observed on𝐳t\\mathbf\{z\}\_\{t\}, then𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\}is a permuted invertible component\-wise transformation of𝐳t\\mathbf\{z\}\_\{t\}:

𝐠^−1​\(𝐱t\)=C∘P∘𝐠−1​\(𝐱t\)⇔𝐳^t=C∘P⁡\(𝐳t\),\\hat\{\\mathbf\{g\}\}^\{\-1\}\(\\mathbf\{x\}\_\{t\}\)=C\\circ P\\circ\\mathbf\{g\}^\{\-1\}\(\\mathbf\{x\}\_\{t\}\)\\Leftrightarrow\\hat\{\\mathbf\{z\}\}\_\{t\}=C\\circ P\(\\mathbf\{z\}\_\{t\}\),\(12\)whereCCis a component\-wise invertible transformation andPPis a permutation\.

Corollary 2: \(Identifiability of latent dynamics function\)Given that Theorems 1 and 2 are established, there exist invertible functions𝐡\\mathbf\{h\}and𝐤\\mathbf\{k\}, each a composition of permutations and component\-wise transformations, such that

ℱ^=𝐡−1∘ℱ∘𝐤\.\\hat\{\\mathcal\{F\}\}=\\mathbf\{h\}^\{\-1\}\\circ\\mathcal\{F\}\\circ\\mathbf\{k\}\.\(13\)Consequently, the learned transition functionℱ^\\hat\{\\mathcal\{F\}\}is equivalent to the true transition functionℱ\\mathcal\{F\}up to compositions of permutations and component\-wise transformations\.

Detailed proofs can be found in Appendix[A](https://arxiv.org/html/2610.27180#A1)\.

## 4Practical Implementation of Meta\-iLaD

Variational inference:To maximize the conditional likelihood in Equations \([4](https://arxiv.org/html/2610.27180#S3.E4)\)\-\([3\.1](https://arxiv.org/html/2610.27180#S3.Ex3)\), we approximate the posterior conditional density of𝐜\\mathbf\{c\}asqζ\(𝐜\|𝐱0:Tq∪𝒟s\)q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}\_\{0:T\}^\{q\}\\cup\\mathcal\{D\}^\{s\}\), sharing the same networks as its conditional prior\. We approximate the posterior density of𝐳0\\mathbf\{z\}\_\{0\}withqϕ\(𝐳0\|𝐱0:lq\)=𝒩\(𝐳0\|𝝁ϕ\(𝐱0:lq\),𝝈ϕ\(𝐱0:lq\)\)q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)=\\mathcal\{N\}\(\\mathbf\{z\}\_\{0\}\|\\boldsymbol\{\\mu\}\_\{\\phi\}\(\\mathbf\{x\}\_\{0:l\}^\{q\}\),\\boldsymbol\{\\sigma\}\_\{\\phi\}\(\\mathbf\{x\}\_\{0:l\}^\{q\}\)\)parameterized byϕ\\phiandl<<Tl<<T\.

Consider data𝒟\\mathcal\{D\}that includes time series of𝐱0:T\\mathbf\{x\}\_\{0:T\}’s generated fromMMenvironments:𝒟=\{𝒟m\}m=1M\\mathcal\{D\}=\\left\\\{\\mathcal\{D\}\_\{m\}\\right\\\}\_\{m=1\}^\{M\}\. Variational inference over Equations \([4](https://arxiv.org/html/2610.27180#S3.E4)\)\-\([3\.1](https://arxiv.org/html/2610.27180#S3.Ex3)\) involves maximizing the following evidence lower\-bound \(ELBO\) for the conditional likelihoodpΦ\(𝐱0:Tq\|𝒟s\)p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:T\}^\{q\}\|\\mathcal\{D\}^\{s\}\)of all𝐱0:Tq\\mathbf\{x\}\_\{0:T\}^\{q\}and𝒟m\\mathcal\{D\}\_\{m\}’s:

ℒELBO=\\displaystyle\\mathcal\{L\}\_\{\\text\{ELBO\}\}=∑𝒟m∈𝒟∑𝐱0:Tq∈𝒟m\{\\displaystyle\\sum\\nolimits\_\{\\mathcal\{D\}\_\{m\}\\in\\mathcal\{D\}\}\\sum\\nolimits\_\{\\mathbf\{x\}\_\{0:T\}^\{q\}\\in\\mathcal\{D\}\_\{m\}\}\\Big\\\{\(14\)𝔼qζ\(𝐜\|𝐱0:Tq∪𝒟s\),qϕ\(𝐳0\|𝐱0:lq\)\[logpΘ\(𝐱0:Tq\|𝐜,𝐳0\)\]\\displaystyle\\mathbb\{E\}\_\{q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}\_\{0:T\}^\{q\}\\cup\\mathcal\{D\}^\{s\}\),q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)\}\[\\log p\_\{\\Theta\}\(\\mathbf\{x\}\_\{0:T\}^\{q\}\|\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)\]−KL\(qζ\(𝐜\|𝐱q0:T∪𝒟s\)\|\|pζ\(𝐜\|𝒟s\)\)\\displaystyle\-\\text\{KL\}\(q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}^\{q\}\_\{0:T\}\\cup\\mathcal\{D\}^\{s\}\)\|\|p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\)\)−KL\(qϕ\(𝐳0\|𝐱0:lq\)\|\|p\(𝐳0\)\)\},\\displaystyle\-\\text\{KL\}\(q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)\|\|p\(\\mathbf\{z\}\_\{0\}\)\)\\Big\\\},for which the derivation is included in Appendix[B](https://arxiv.org/html/2610.27180#A2)\. The last two terms in Equation \([14](https://arxiv.org/html/2610.27180#S4.E14)\) regularize the posterior density of𝐜\\mathbf\{c\}and𝐳0\\mathbf\{z\}\_\{0\}with their respective priors\. The first term in Equation \([14](https://arxiv.org/html/2610.27180#S4.E14)\) describes the likelihood of each query sample𝐱0:Tq∈𝒟m\\mathbf\{x\}\_\{0:T\}^\{q\}\\in\\mathcal\{D\}\_\{m\}when its generation is conditioned on the latent environment variable𝐜\\mathbf\{c\}inferred from a disjoint set of context samples𝒟s⊂𝒟m\\mathcal\{D\}^\{s\}\\subset\\mathcal\{D\}\_\{m\}, along with𝐳0\\mathbf\{z\}\_\{0\}inferred from initialll\+ 1 frames of𝐱0:Tq\\mathbf\{x\}\_\{0:T\}^\{q\}itself\. In other words, the likelihood ofpg​\(𝐱t\|𝐳t\)p\_\{g\}\(\\mathbf\{x\}\_\{t\}\|\\mathbf\{z\}\_\{t\}\)at eachttfort≥1t\\geq 1are evaluated at a densityqℱ​\(𝐳t\|𝐜\)q\_\{\\mathcal\{F\}\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)following the NODE flow as defined in Equation \([6](https://arxiv.org/html/2610.27180#S3.E6)\) with the density of𝐳0\\mathbf\{z\}\_\{0\}now defined byqϕ\(𝐳0\|𝐱0:lq\)q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)as:

logqℱ\(𝐳t\|𝐜\)=logqϕ\(𝐳0\|𝐱0:lq\)−∫0ttr\(J𝐟θ\(𝐳τ,𝐜\)\)dτ\.\\small\{\\log q\_\{\\mathcal\{F\}\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)=\\log q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}^\{q\}\_\{0:l\}\)\-\\int\_\{0\}^\{t\}\\text\{tr\}\\left\(J\_\{\\mathbf\{f\}\_\{\\theta\}\}\(\\mathbf\{z\}\_\{\\tau\},\\mathbf\{c\}\)\\right\)d\\tau\.\}\(15\)
Relation to meta\-learning:Equation \([14](https://arxiv.org/html/2610.27180#S4.E14)\) represents a Bayesian meta\-learning objective:pζ​\(𝐜\|𝒟s\)p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\)andpθ​\(𝐳t\|𝐳t−1,𝐜\)p\_\{\\theta\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{c\}\)are feedforward meta\-models to extract𝐜\\mathbf\{c\}fromkk\-shot context samples and to adapt latent dynamics; the conditional likelihood of all query samples𝐱0:Tq\\mathbf\{x\}\_\{0:T\}^\{q\}across all environments accumulates the meta\-loss to optimize the meta\-models\. The practical optimization of Equation \([14](https://arxiv.org/html/2610.27180#S4.E14)\) involves two encoders:qζ\(𝐜\|𝐱0:Tq∪𝒟s\)q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}\_\{0:T\}^\{q\}\\cup\\mathcal\{D\}^\{s\}\)for the latent environment variable𝐜\\mathbf\{c\}, andqϕ\(𝐳0\|𝐱0:lq\)q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)for initiating the ODE flow\-based densityqℱ​\(𝐳t\|𝐜\)q\_\{\\mathcal\{F\}\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)as defined in Equation \([15](https://arxiv.org/html/2610.27180#S4.E15)\)\. Below we discuss additional regularization strategies to improve the optimization of each encoder\.

Disentanglement of𝐜\\mathbf\{c\}:As defined in Equation \([4](https://arxiv.org/html/2610.27180#S3.E4)\), the latent environment𝐜\\mathbf\{c\}is assumed to be conditionally factorized among itsdddimensions\. To enforce this constraint, we follow FactorVAE\([Kim & Mnih, 2018](https://arxiv.org/html/2610.27180#bib.bib17)\)to minimize its total correlation \(TC\) byℒTC=𝔼qζ​\[log⁡qζq¯ζ\]\\mathcal\{L\}\_\{\\text\{TC\}\}=\\mathbb\{E\}\_\{q\_\{\\zeta\}\}\\big\[\\log\\frac\{q\_\{\\zeta\}\}\{\\bar\{q\}\_\{\\zeta\}\}\\big\]whereq¯ζ≔∏j=1dqζ​\(cj\)\\bar\{q\}\_\{\\zeta\}\\coloneq\\prod\_\{j=1\}^\{d\}q\_\{\\zeta\}\(c\_\{j\}\), approximated by a discriminator trained to distinguish between samples fromqζq\_\{\\zeta\}andq¯ζ\\bar\{q\}\_\{\\zeta\}\.

Warm\-up training strategy:The optimization ofqϕ\(𝐳0\|𝐱0:lq\)q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)in Equation \([14](https://arxiv.org/html/2610.27180#S4.E14)\) in theory receives loss signals from the likelihood at all time framestt=00:TTthroughqℱ​\(𝐳t\|𝐜\)q\_\{\\mathcal\{F\}\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)as defined in Equation \([15](https://arxiv.org/html/2610.27180#S4.E15)\)\. In practice, because the calculation ofqℱ​\(𝐳t\|𝐜\)q\_\{\\mathcal\{F\}\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)att≥1t\\geq 1involves iterative ODE solver through multiple time steps, back propagation of the gradient toqϕq\_\{\\phi\}could be difficult\. To mitigate this optimization challenge, we adopt awarm\-uptraining strategy to first obtain a reasonable optimization ofqϕq\_\{\\phi\}via areconstructionloss \(where𝐱q0:T\\mathbf\{x\}^\{q\}\_\{0:T\}is reconstructed fromqϕ\(𝐳t\|𝐱t:t\+lq\)q\_\{\\phi\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{x\}\_\{t:t\+l\}^\{q\}\)\), before optimizing theforecastingmeta\-objective \(where𝐱q0:T\\mathbf\{x\}^\{q\}\_\{0:T\}is calculated fromqℱ​\(𝐳t\|𝐜\)q\_\{\\mathcal\{F\}\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)as defined in Equations \([14](https://arxiv.org/html/2610.27180#S4.E14)\)\-\([15](https://arxiv.org/html/2610.27180#S4.E15)\)\)\. For transition, we add a regularization term between the two densitiesqϕ\(𝐳t\|𝐱t:t\+lq\)q\_\{\\phi\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{x\}\_\{t:t\+l\}^\{q\}\)andqℱ​\(𝐳t\|𝐜\)q\_\{\\mathcal\{F\}\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)att≥1t\\geq 1:ℒFlow=𝔼qζ,qϕ\(𝐳0\|𝐱0:lq\)\[∑t=1TKL\[qϕ\(𝐳t\|𝐱t:t\+lq\)∥qℱ\(𝐳t\|𝐜\)\]\]\\mathcal\{L\}\_\{\\text\{Flow\}\}=\\mathbb\{E\}\_\{q\_\{\\zeta\},q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)\}\\Big\[\\sum\\nolimits\_\{t=1\}^\{T\}\\text\{KL\}\[q\_\{\\phi\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{x\}\_\{t:t\+l\}^\{q\}\)\\\|q\_\{\\mathcal\{F\}\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{c\}\)\]\\Big\]\.

Unified loss function:The two regularization lossesℒTC\\mathcal\{L\}\_\{\\text\{TC\}\}andℒFlow\\mathcal\{L\}\_\{\\text\{Flow\}\}are added toℒELBO\\mathcal\{L\}\_\{\\text\{ELBO\}\}with their respective hyperparamtersλTC\\lambda\_\{\\text\{TC\}\}andλFlow\\lambda\_\{\\text\{Flow\}\}as:

ℒTotal=−ℒELBO\+λTC​ℒTC\+λFlow​ℒFlow\\mathcal\{L\}\_\{\\text\{Total\}\}=\-\\mathcal\{L\}\_\{\\text\{ELBO\}\}\+\\lambda\_\{\\text\{TC\}\}\\mathcal\{L\}\_\{\\text\{TC\}\}\+\\lambda\_\{\\text\{Flow\}\}\\mathcal\{L\}\_\{\\text\{Flow\}\}\(16\)whereλFlow\\lambda\_\{\\text\{Flow\}\}is annealed during warm\-up\. The number of warm\-up epochs is determined by the training loss\. The effect of these two regularizations is ablated in Section[5\.3](https://arxiv.org/html/2610.27180#S5.SS3)\.

## 5Experiments & Results

Table 1:Quantitative results on Pendulum and CartPole\.![Refer to caption](https://arxiv.org/html/2610.27180v1/Fig/Figure_ep1.png)Figure 2:Scatter plots of estimatedvs\. true latent states𝐳t\\mathbf\{z\}\_\{t\}\(blue\) and environment variable𝐜\\mathbf\{c\}\(red\)\. Left: Pendulum; Right: CartPole\.### 5\.1Experiments on Synthetic Physics Systems

Datasets:For a comprehensive identifiability assessment, we first considered three benchmark physics systems with known ground truth dynamics: forced damped Pendulum\([Takeishi & Kalousis, 2021](https://arxiv.org/html/2610.27180#bib.bib25)\), CartPole\([Song et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib24)\), and Double Pendulum\([Wehenkel et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib28)\)\. For each system, the dynamics environments are characterized by governing physical parameters \(e\.g\., damping coefficient, mass, length\), which we randomly sampled to generate image sequences with varying initial conditions\. This controlled setting enabled rigorous quantitative evaluation of identifiability at both the latent state and environment variable levels\. Additional data details are provided in Appendix[C\.1](https://arxiv.org/html/2610.27180#A3.SS1)\.

Baselines:We evaluated Meta\-iLaD against two categories of locally\-stationary latent dynamics modelsℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)\. Foridentifiablemodels, we included TDRL\([Yao et al\., 2022a](https://arxiv.org/html/2610.27180#bib.bib30)\)and our own implementation of IDF \(due to absence of public versions\)\([Hızlı et al\., 2025](https://arxiv.org/html/2610.27180#bib.bib9)\): as previously noted, both baselines leverage predefined environment labels to condition the latent dynamics, with which IDF further established the identifiability of a latent process noise𝐬t\\mathbf\{s\}\_\{t\}andℱ\\mathcal\{F\}\.

For modelswithout established identifiability, we included: 1\) ODE2VAE which does not infer latent environment variables\([Yıldız et al\., 2019](https://arxiv.org/html/2610.27180#bib.bib32)\), and 2\) MoNODE which infers environment variables from the query sample itself\([Auzina et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib1)\)—note that the subtle difference in inferring𝐜\\mathbf\{c\}from context samples \(Meta\-iLaD\)vs\. the query sample itself \(MoNODE\) makes a fundamental difference in the definition of a conditional prior ofp⁡\(𝐜\|𝒟s\)p\(\\mathbf\{c\}\|\\mathcal\{D\}\_\{s\}\)vs\. unconditional priorp⁡\(𝐜\)p\(\\mathbf\{c\}\), which is the key to our identifiability results\.

Evaluation scenarios:We evaluated all models under three progressively challenging settings\. First, inreconstruction, the full test sequence was observed by the model to assess the recovery of𝐳t\\mathbf\{z\}\_\{t\}and𝐜\\mathbf\{c\}—ODE2VAE by design does not apply in this setting\. Second, inprediction, the model observed only the firstllframes of a test sequence and predicts future trajectories from the estimated𝐳0\\mathbf\{z\}\_\{0\}to evaluate whether the learnedℱ\\mathcal\{F\}and𝐜\\mathbf\{c\}captures the true underlying dynamics function—𝐜\\mathbf\{c\}came from context samples in Meta\-iLaD, the environment label in IDF, and a previous query sample in MoNODE; TDRL by design does not apply in this setting\. Finally, inOOD generalization, prediction was performed on dynamics environments with parameters outside the training distribution, TDRL and IDF cannot apply here due to their use of predefined environment labels\.

Metrics:The performance of all models in the above scenarios were evaluated at both the data and latent space\.Data spacequality was measured by mean squared errors \(MSE\) between model outputs and ground truth𝐱t\\mathbf\{x\}\_\{t\}\.Identifiabilityquality of latent variables was measured by mean correlation coefficients \(MCC\) with known ground\-truth, adopting the strong\-MCC metric in\([Khemakhem et al\., 2020a](https://arxiv.org/html/2610.27180#bib.bib15);[Khemakhem et al\., 2020b](https://arxiv.org/html/2610.27180#bib.bib16)\)a value closer to 1 indicating better identifiability\.

Reconstruction & prediction results:Table[1](https://arxiv.org/html/2610.27180#S5.T1)summarizes the full quantitative results on Pendulum and CartPole\. Fig\.[2](https://arxiv.org/html/2610.27180#S5.F2)provides MCC visuals on these two datasets for the three identifiable models during reconstruction\. Additional results are provided in Appendix[C\.2](https://arxiv.org/html/2610.27180#A3.SS2)\.

As shown, for ODE2VAE and MoNODE \(without identifiability established\), poor MCC\(𝐳t\\mathbf\{z\}\_\{t\}\) and weaker MSE\(𝐱t\\mathbf\{x\}\_\{t\}\) were observed in both reconstruction and prediction settings\. Although MoNODE inferred latent environment variable𝐜\\mathbf\{c\}, MCC\(𝐜\\mathbf\{c\}\) was low in reconstructions and predictions\.

In comparison, the three identifiable models showed much stronger MCC\(𝐳t\\mathbf\{z\}\_\{t\}\) and thus strong MSE\(𝐱t\\mathbf\{x\}\_\{t\}\) during reconstruction\. Interestingly, while this strong performance was underlined by an equally strong MCC\(𝐜\\mathbf\{c\}\) in Meta\-iLaD, TDRL and IDF delivered either poor or inconsistent MCC\(𝐜\\mathbf\{c\}\) during reconstruction\. Similar performance was observed during prediction, where Meta\-iLaD delivered significantly stronger MCC\(𝐜\\mathbf\{c\}\) than IDF\. Unlike the reconstruction setting, this difference in MCC\(𝐜\\mathbf\{c\}\) directly translated to a significant difference in MCC\(𝐳t\\mathbf\{z\}\_\{t\}\) and MSE\(𝐱t\\mathbf\{x\}\_\{t\}\) between the two models\. The low MCC\(𝐜\\mathbf\{c\}\) for IDF suggest that the assumption of process noises in IDF restricted its use to represent more general environment variables, such as parameters of the dynamics equations in these experiments\.

OOD results:The two identifiable baselines \(TDRL and IDF\) were not applied to OOD settings due to their use of predefined environment labels\. ODE2VAE, while applicable, delivered limited performance potentially due to its non\-adaptable latent dynamics\. MoNODE with its inference of latent environment variable improved this performance at the data space, but only to a limited extent potentially due to the lack of identifiability\. In comparison, Meta\-iLaD delivered consistently strong results across all metrics\.

![Refer to caption](https://arxiv.org/html/2610.27180v1/Fig/Figure_ep2.png)Figure 3:Examples of IDFvs\. Meta\-iLaD in reconstruction \(t=0:75\) and prediction \(t=75:125\) on an in\-distribution subject\.
### 5\.2Experiments on Real\-World CMU Data

Datasets:We further evaluated on a subset of the CMU MoCAP dataset consisting of 62 walking sequences from 5 different subjects\. Each subject was considered a distinct dynamics environment\. Following\([Wang et al\., 2008](https://arxiv.org/html/2610.27180#bib.bib26)\), we preprocessed the data to obtain 50\-dimensional observation sequences\. During testing, models receive the first 75 time steps as input and predict the full 125\-step horizon\. We used all five subjects for in\-distribution \(ID\) and left one subject out for OOD\. Additional data details are in Appendix[C\.1](https://arxiv.org/html/2610.27180#A3.SS1)\.

Baselines & Metrics:We included baselines capable of prediction\. Due to the lack of true latent variables, model performance was measured by MSE\(𝐱t\)\(\\mathbf\{x\}\_\{t\}\)when reconstructing the first 75 time steps and predicting the next 50 steps\.

Results:As summarized in Table[2](https://arxiv.org/html/2610.27180#S5.T2), the two identifiable models showed stronger performance inID setting, with Meta\-iLaD delivering the most accurate reconstructions and predictions\. Fig\.[3](https://arxiv.org/html/2610.27180#S5.F3)provides visual examples of this difference, noting the improvements of Meta\-iLaD when predicting over longer horizons\. All baseline models showed significant deterioration when applied to theOOD setting: the significant drop of performance for IDF, including reconstruction, suggested the limitations of utilizing predefined environment labels\. The performance of Meta\-iLaD remained the most stable, especially in prediction\. Additional results on MoCAP are included in Appendix[C\.3](https://arxiv.org/html/2610.27180#A3.SS3)\.

### 5\.3Ablation Studies

All ablation experiments were performed on the Pendulum system with ID and OOD environments\.

Meta\-Learning via context samples:A core innovation of Meta\-iLaD is conditioning𝐜\\mathbf\{c\}on few\-shot context samples, rather than predefined labels \(e\.g\., TDRL and IDF\) or inferred from the query sample itself \(e\.g\., MoNODE\)\. While the comparison with relevant baselines already validated this design choice, here we further ablated its importance when controling all other differences\. We compared Meta\-iLaD with 1\) a variant that infers𝐜\\mathbf\{c\}from the query sample and 2\) a variant that conditions on predefined environment labelsuu—this last variant can be viewed as an realization of iVAE\([Khemakhem et al\., 2020a](https://arxiv.org/html/2610.27180#bib.bib15)\)in our model: we considered both true environment parameter values asuuas the best\-case performance, and discrete labels ofuuas its practical variant\.

As summarized in Table[3](https://arxiv.org/html/2610.27180#S5.T3), inferring𝐜\\mathbf\{c\}from the query samples resulted in limited identifiability, especially in OOD settings\. The use of predefined label of environments delivered comparable performance to Meta\-iLaD, but could not be generalized to OOD environments\. Meta\-iLaD delivered consistent performance to the best\-case scenario using true environment variables to condition𝐜\\mathbf\{c\}, across test settings\.

Table 2:Reconstruction \(0\-75\) and prediction \(75\-125\) MSEs on MoCAP, tested on in\-distribution \(ID\) and OOD subjects\.Table 3:Ablation of conditioning𝐜\\mathbf\{c\}on few\-shot context samples \(representing Meta\-iLaD\), query sample itself \(representing an unconditional prior for𝐜\\mathbf\{c\}\), predefined indices of environments \(representing a realization of iVAE in our model\), and true environment variables \(representing the best scenario\)\.TC regularization & Warm\-up:Table[4](https://arxiv.org/html/2610.27180#S5.T4)summarizes ablation results on the effect of the two training strategies we used to improve the optimization of Meta\-iLaD:ℒTC\\mathcal\{L\}\_\{\\text\{TC\}\}to enforce the conditionally factorized prior ofpζ​\(𝐜\|𝒟s\)p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\), and the warm\-up strategy to help with the optimization ofqϕ\(𝐳t\|𝐱t:t\+lq\)q\_\{\\phi\}\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{x\}^\{q\}\_\{t:t\+l\}\)\. As shown, the presence/absence ofℒTC\\mathcal\{L\}\_\{\\text\{TC\}\}has a significant effect on the MCC\(𝐜\\mathbf\{c\}\) metrics, while the presence/absence of warm\-up has a significant effect on the MCC\(𝐳t\\mathbf\{z\}\_\{t\}\) metrics, supporting their role as intended\.

Table 4:Ablation of the presence/absence of TC regularizationℒTC\\mathcal\{L\}\_\{\\text\{TC\}\}and warm\-up training strategy \(tied toℒflow\\mathcal\{L\}\_\{\\text\{flow\}\}\.Additional results:In Appendix[D](https://arxiv.org/html/2610.27180#A4), we provide empirical evidence for Corollary 1, for Meta\-iLaD’s ability to recover the true dimension of𝐜\\mathbf\{c\}\. In Appendix[E](https://arxiv.org/html/2610.27180#A5), we provide real\-data application scenarios for Meta\-iLaD when knowledge about dynamics environments is not able: based on the assumption of local stationarity, Meta\-iLaD can use preceding time windows as context samples to support a query window–this is not possible with any existing identifiableℱ⁡\(𝐳<t,𝐜\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\)\.

## 6Conclusion and Discussions

We present Meta\-iLaD to theoretically and empirically show that 1\) conditioning latent dynamics on context samples enabled OOD generalizations, and 2\) identifiability of latent environment variable𝐜\\mathbf\{c\}and dynamicsℱ\\mathcal\{F\}, while inconsequential in reconstructions, has a significant impact on the latent dynamics’ forecasting capabilities\.Limitations:Meta\-iLaD requires access to few\-shot context samples from each dynamics environment to infer𝐜\\mathbf\{c\}\. In many real\-world scenarios, obtaining multiple samples from the same environment may be challenging,e\.g\., collecting multiple patient trajectories under identical clinical conditions\. Additionally, the use of NODEs introduces computational overhead, and our identifiability guarantees require sufficient environment diversity which may not always be available in practice\.

## Impact Statement

This paper presents theoretical works that prove the simultaneous identifiability of latent dynamics and environments via meta\-learning\. In the context of scientific modeling, this enhances the interpretability of black\-box models, allowing researchers to map learned parameters to physical constants with greater confidence\. There are also many potential societal consequences of our work, none of which we feel must be specifically highlighted here\.

## Acknowledgements

This work is funded by Award NO: R01HL145590 by National Institutes of Health \(NIH\) / NATIONAL HEART, LUNG, AND BLOOD INSTITUTE \(NHLBI\)

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## Appendix AProof of Identifiability Theory

### A\.1Proof of Theorem 1

Prior to commencing the proof of Theorem 1, we shall introduce the exponential family distribution and the strong exponential family distribution, along with a pertinent lemma concerning the strong exponential family distribution\. These preliminaries establish the requisite foundation for our subsequent proof of Theorem 1\.

Definition 3 \(Exponential family\)A univariate exponential family is a set of distributions whose probability density function can be written as

p⁡\(x\)=Q⁡\(x\)​Z​\(𝜽\)​e⟨𝐓⁡\(x\),𝜽⟩p\(x\)=Q\(x\)Z\(\\boldsymbol\{\\theta\}\)e^\{\\langle\\mathbf\{T\}\(x\),\\boldsymbol\{\\theta\}\\rangle\}\(17\)where𝐓:ℝ→ℝk\\mathbf\{T\}:\\mathbb\{R\}\\to\\mathbb\{R\}^\{k\}is called the sufficient statistic,𝜽∈ℝk\\boldsymbol\{\\theta\}\\in\\mathbb\{R\}^\{k\}is the natural parameter,Q:ℝ→ℝQ:\\mathbb\{R\}\\to\\mathbb\{R\}the base measure andZ⁡\(𝜽\)Z\(\\boldsymbol\{\\theta\}\)the normalization constant\. The dimensionk∈ℕ∖\{0\}k\\in\\mathbb\{N\}\\setminus\\\{0\\\}of the parameter is always considered to be minimal, meaning that we can’t rewrite the densityppto have the form Equation[17](https://arxiv.org/html/2610.27180#A1.E17)with a smallerk′<kk^\{\\prime\}<k\.We callkkthe size ofpp\.

Definition 4 \(Strongly exponential distributions\)We say that an exponential family distribution is strongly exponential if for any subset𝒳\\mathcal\{X\}ofℝ\\mathbb\{R\}the following is true:

\(∃𝜽∈ℝk∣∀x∈𝒳,⟨𝐓\(x\),𝜽⟩=const\)⟹\(l\(𝒳\)=0or𝜽=0\)\(\\exists\\boldsymbol\{\\theta\}\\in\\mathbb\{R\}^\{k\}\\mid\\forall x\\in\\mathcal\{X\},\\langle\\mathbf\{T\}\(x\),\\boldsymbol\{\\theta\}\\rangle=const\)\\Longrightarrow\(l\(\\mathcal\{X\}\)=0\\text\{ or \}\\boldsymbol\{\\theta\}=0\)\(18\)wherellis the Lebesgue measure\.

Lemma 1Consider a strongly exponential distribution of sizek≥2k\\geq 2with sufficient statistic𝐓⁡\(x\)=\(T1​\(x\),…,Tk​\(x\)\)\\mathbf\{T\}\(x\)=\(T\_\{1\}\(x\),\\ldots,T\_\{k\}\(x\)\)\.Further assume that𝐓\\mathbf\{T\}isdifferentiable almost everywhere\.Then there existkkdistinct valuesx1x\_\{1\}toxkx\_\{k\}such that\(𝐓′​\(x1\),…,𝐓′​\(xk\)\)\(\\mathbf\{T\}^\{\\prime\}\(x\_\{1\}\),\\ldots,\\mathbf\{T\}^\{\\prime\}\(x\_\{k\}\)\)are linearly independent inℝk\\mathbb\{R\}^\{k\}\.

Proof:Suppose that for any choice of suchkkpoints, the family\(𝐓′​\(x1\),…,𝐓′​\(xk\)\)\(\\mathbf\{T\}^\{\\prime\}\(x\_\{1\}\),\\ldots,\\mathbf\{T\}^\{\\prime\}\(x\_\{k\}\)\)is never linearly independent\. That means that𝐓′​\(ℝ\)\\mathbf\{T\}^\{\\prime\}\(\\mathbb\{R\}\)is included in a subspace ofℝk\\mathbb\{R\}^\{k\}of dimension at mostk−1k\-1\. Let𝜽\\boldsymbol\{\\theta\}a non zero vector that is orthogonal to𝐓′​\(ℝ\)\\mathbf\{T\}^\{\\prime\}\(\\mathbb\{R\}\)\. Then for allx∈ℝx\\in\\mathbb\{R\}, we have⟨𝐓′​\(x\),𝜽⟩=0\\langle\\mathbf\{T\}^\{\\prime\}\(x\),\\boldsymbol\{\\theta\}\\rangle=0\. By integrating we find that⟨𝐓⁡\(x\),𝜽⟩=const\\langle\\mathbf\{T\}\(x\),\\boldsymbol\{\\theta\}\\rangle=\\text\{const\}\. Since this is true for allx∈ℝx\\in\\mathbb\{R\}and for a𝜽≠0\\boldsymbol\{\\theta\}\\neq 0, we conclude that the distribution is not strongly exponential, which contradicts our hypothesis\.

Theorem 1Assume that we observe data sampled from a generative model defined according to Equation \([3\.1](https://arxiv.org/html/2610.27180#S3.Ex3)\) with parameterΦ=\(𝐠,θ,ζ\)\\Phi=\(\\mathbf\{g\},\\theta,\\zeta\)and latent variable𝐜\\mathbf\{c\}\. Assume the following holds:

1. iThe set\{𝐱0:T∈𝒳T\+1\|φϵ\(𝐱0:T\)=0\}\\left\\\{\\mathbf\{x\}\_\{0:T\}\\in\\mathcal\{X\}^\{T\+1\}\|\\varphi\_\{\\epsilon\}\(\\mathbf\{x\}\_\{0:T\}\)=0\\right\\\}has measure zero, whereφϵ\\varphi\_\{\\epsilon\}is the characteristic function of the densitypϵp\_\{\\epsilon\}\.
2. iiThe mixing function𝐅𝐠,θ\\mathbf\{F\}\_\{\\mathbf\{g\},\\theta\}is injective\.
3. iiiThe sufficient statisticsTi​jT\_\{ij\}are differentiable almost everywhere, and\(Ti​j\)1≤j≤2\(T\_\{ij\}\)\_\{1\\leq j\\leq 2\}are linearly independent on any subset of𝒳T\+1\\mathcal\{X\}^\{T\+1\}of measure greater than zero\.
4. ivThere exist2​d\+12d\+1distinct points𝐮0,…,𝐮2​d\\mathbf\{u\}\_\{0\},\.\.\.,\\mathbf\{u\}\_\{2d\}such that the matrix 𝐋=\(𝝀ζ​\(𝐮1\)−𝝀ζ​\(𝐮0\),…,𝝀ζ​\(𝐮2​d\)−𝝀ζ​\(𝐮0\)\)\\mathbf\{L\}=\(\\boldsymbol\{\\lambda\}\_\{\\zeta\}\(\\mathbf\{u\}\_\{1\}\)\-\\boldsymbol\{\\lambda\}\_\{\\zeta\}\(\\mathbf\{u\}\_\{0\}\),\.\.\.,\\boldsymbol\{\\lambda\}\_\{\\zeta\}\(\\mathbf\{u\}\_\{2d\}\)\-\\boldsymbol\{\\lambda\}\_\{\\zeta\}\(\\mathbf\{u\}\_\{0\}\)\)\(19\)of size2​d×2​d2d\\times 2dis invertible\.

then the parametersΦ=\(𝐠,θ,ζ\)\\Phi=\(\\mathbf\{g\},\\theta,\\zeta\)are∼\\sim\-identifiable\.

Proof:Suppose we have two sets of parametersΦ=\(𝐠,θ,ζ\)\\Phi=\(\\mathbf\{g\},\\theta,\\zeta\)andΦ^=\(𝐠^,θ^,ζ^\)\\hat\{\\Phi\}=\(\\hat\{\\mathbf\{g\}\},\\hat\{\\theta\},\\hat\{\\zeta\}\)such that:

pΦ\(𝐱0:T\|𝐮\)\\displaystyle p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:T\}\|\\mathbf\{u\}\)=pΦ^\(𝐱0:T\|𝐮\)\\displaystyle=p\_\{\\hat\{\\Phi\}\}\(\\mathbf\{x\}\_\{0:T\}\|\\mathbf\{u\}\)\(20\)∫p𝐠,θ\(𝐱0:T\|𝐜,𝐳0\)pζ\(𝐜\|𝐮\)p\(𝐳0\)d𝐜d𝐳0\\displaystyle\\int p\_\{\\mathbf\{g\},\\theta\}\(\\mathbf\{x\}\_\{0:T\}\|\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{u\}\)p\(\\mathbf\{z\}\_\{0\}\)d\\mathbf\{c\}d\\mathbf\{z\}\_\{0\}=∫p𝐠^,θ^\(𝐱0:T\|𝐜,𝐳0\)pζ^\(𝐜\|𝐮\)p\(𝐳0\)d𝐜d𝐳0\\displaystyle=\\int p\_\{\\hat\{\\mathbf\{g\}\},\\hat\{\\theta\}\}\(\\mathbf\{x\}\_\{0:T\}\|\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)p\_\{\\hat\{\\zeta\}\}\(\\mathbf\{c\}\|\\mathbf\{u\}\)p\(\\mathbf\{z\}\_\{0\}\)d\\mathbf\{c\}d\\mathbf\{z\}\_\{0\}\(21\)To simplify without loss of generality, we rewrite the above equation as:

∫p\(𝐱0:T\|𝐜,𝐳0\)p\(𝐜\|𝐮\)p\(𝐳0\)d𝐜d𝐳0\\displaystyle\\int p\(\\mathbf\{x\}\_\{0:T\}\|\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)p\(\\mathbf\{c\}\|\\mathbf\{u\}\)p\(\\mathbf\{z\}\_\{0\}\)d\\mathbf\{c\}d\\mathbf\{z\}\_\{0\}=∫p\(𝐱0:T\|𝐜^,𝐳^0\)p\(𝐜^\|𝐮\)p\(𝐳^0\)d𝐜^d𝐳^0\\displaystyle=\\int p\(\\mathbf\{x\}\_\{0:T\}\|\\hat\{\\mathbf\{c\}\},\\hat\{\\mathbf\{z\}\}\_\{0\}\)p\(\\hat\{\\mathbf\{c\}\}\|\\mathbf\{u\}\)p\(\\hat\{\\mathbf\{z\}\}\_\{0\}\)d\\hat\{\\mathbf\{c\}\}d\\hat\{\\mathbf\{z\}\}\_\{0\}\(22\)∫δ\(𝐱0:T−𝐅𝐠,θ\(𝐜,𝐳0\)\)p\(𝐜\|𝐮\)p\(𝐳0\)d𝐜d𝐳0\\displaystyle\\int\\delta\(\\mathbf\{x\}\_\{0:T\}\-\\mathbf\{F\}\_\{\\mathbf\{g\},\\theta\}\(\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)\)p\(\\mathbf\{c\}\|\\mathbf\{u\}\)p\(\\mathbf\{z\}\_\{0\}\)d\\mathbf\{c\}d\\mathbf\{z\}\_\{0\}=∫δ\(𝐱0:T−𝐅𝐠^,θ^\(𝐜^,𝐳^0\)\)p\(𝐜^\|𝐮\)p\(𝐳^0\)d𝐜^d𝐳^0\\displaystyle=\\int\\delta\(\\mathbf\{x\}\_\{0:T\}\-\\mathbf\{F\}\_\{\\hat\{\\mathbf\{g\}\},\\hat\{\\theta\}\}\(\\hat\{\\mathbf\{c\}\},\\hat\{\\mathbf\{z\}\}\_\{0\}\)\)p\(\\hat\{\\mathbf\{c\}\}\|\\mathbf\{u\}\)p\(\\hat\{\\mathbf\{z\}\}\_\{0\}\)d\\hat\{\\mathbf\{c\}\}d\\hat\{\\mathbf\{z\}\}\_\{0\}\(23\)p\(𝐜\|𝐮\)p\(𝐳0\)\|𝐉𝐅𝐠,θ−1\(𝐱0:T\)\|\\displaystyle p\(\\mathbf\{c\}\|\\mathbf\{u\}\)p\(\\mathbf\{z\}\_\{0\}\)\|\\mathbf\{J\}\_\{\\mathbf\{F\}^\{\-1\}\_\{\\mathbf\{g\},\\theta\}\}\(\\mathbf\{x\}\_\{0:T\}\)\|=p\(𝐜^\|𝐮\)p\(𝐳^0\)\|𝐉𝐅𝐠^,θ^−1\(𝐱0:T\)\|\\displaystyle=p\(\\hat\{\\mathbf\{c\}\}\|\\mathbf\{u\}\)p\(\\hat\{\\mathbf\{z\}\}\_\{0\}\)\|\\mathbf\{J\}\_\{\\mathbf\{F\}^\{\-1\}\_\{\\hat\{\\mathbf\{g\}\},\\hat\{\\theta\}\}\}\(\\mathbf\{x\}\_\{0:T\}\)\|\(24\)Equation[24](https://arxiv.org/html/2610.27180#A1.E24)can also be derived from Equation[22](https://arxiv.org/html/2610.27180#A1.E22)with a change of variables\.

Now consider two different values of𝐮\\mathbf\{u\}:𝐮i\\mathbf\{u\}\_\{i\},𝐮0\\mathbf\{u\}\_\{0\}

logp\(𝐜\|𝐮i\)p\(𝐳0\)\|detJ𝐅𝐠,θ−1\(𝐱0:τ\)\|−logp\(𝐜\|𝐮0\)p\(𝐳0\)\|detJ𝐅𝐠,θ−1\(𝐱0:τ\)\|\\displaystyle\\log p\(\\mathbf\{c\}\|\\mathbf\{u\}\_\{i\}\)p\(\\mathbf\{z\}\_\{0\}\)\|\\det J\_\{\\mathbf\{F\}\_\{\\mathbf\{g\},\\theta\}^\{\-1\}\}\(\\mathbf\{x\}\_\{0:\\tau\}\)\|\-\\log p\(\\mathbf\{c\}\|\\mathbf\{u\}\_\{0\}\)p\(\\mathbf\{z\}\_\{0\}\)\|\\det J\_\{\\mathbf\{F\}\_\{\\mathbf\{g\},\\theta\}^\{\-1\}\}\(\\mathbf\{x\}\_\{0:\\tau\}\)\|=log⁡Z⁡\(𝐮0\)Z⁡\(𝐮i\)\+⟨𝐓⁡\(𝐜\),𝝀⁡\(𝐮i\)−𝝀⁡\(𝐮0\)⟩=log⁡Z⁡\(𝐮0\)Z⁡\(𝐮i\)\+⟨𝐓⁡\(𝐜\),𝝀¯​\(𝐮i\)⟩\\displaystyle=\\log\\frac\{Z\(\\mathbf\{u\}\_\{0\}\)\}\{Z\(\\mathbf\{u\}\_\{i\}\)\}\+\\langle\\mathbf\{T\}\(\\mathbf\{c\}\),\\boldsymbol\{\\lambda\}\(\\mathbf\{u\}\_\{i\}\)\-\\boldsymbol\{\\lambda\}\(\\mathbf\{u\}\_\{0\}\)\\rangle=\\log\\frac\{Z\(\\mathbf\{u\}\_\{0\}\)\}\{Z\(\\mathbf\{u\}\_\{i\}\)\}\+\\langle\\mathbf\{T\}\(\\mathbf\{c\}\),\\bar\{\\boldsymbol\{\\lambda\}\}\(\\mathbf\{u\}\_\{i\}\)\\rangle\(25\)Considering the same forp\(𝐜^\|𝐮\)p\(𝐳^0\)\|𝐉𝐅𝐠^,θ^\(𝐱¯0:τ\)\|p\(\\hat\{\\mathbf\{c\}\}\|\\mathbf\{u\}\)p\(\\hat\{\\mathbf\{z\}\}\_\{0\}\)\|\\mathbf\{J\}\_\{\\mathbf\{F\}\_\{\\hat\{\\mathbf\{g\}\},\\hat\{\\theta\}\}\}\(\\bar\{\\mathbf\{x\}\}\_\{0:\\tau\}\)\|, we get:

log⁡Z⁡\(𝐮0\)Z⁡\(𝐮i\)\+⟨𝐓⁡\(𝐜\),𝝀¯​\(𝐮i\)⟩=log⁡Z^​\(𝐮0\)Z^​\(𝐮i\)\+⟨𝐓⁡\(𝐜^\),𝝀¯​\(𝐮i\)⟩\\displaystyle\\log\\frac\{Z\(\\mathbf\{u\}\_\{0\}\)\}\{Z\(\\mathbf\{u\}\_\{i\}\)\}\+\\langle\\mathbf\{T\}\(\\mathbf\{c\}\),\\bar\{\\boldsymbol\{\\lambda\}\}\(\\mathbf\{u\}\_\{i\}\)\\rangle=\\log\\frac\{\\hat\{Z\}\(\\mathbf\{u\}\_\{0\}\)\}\{\\hat\{Z\}\(\\mathbf\{u\}\_\{i\}\)\}\+\\langle\\mathbf\{T\}\(\\hat\{\\mathbf\{c\}\}\),\\bar\{\\boldsymbol\{\\lambda\}\}\(\\mathbf\{u\}\_\{i\}\)\\rangle\(26\)⟨𝐓⁡\(𝐜\),𝝀¯​\(𝐮i\)⟩=⟨𝐓⁡\(𝐜^\),𝝀¯​\(𝐮i\)⟩\+log⁡Z^​\(𝐮0\)​Z​\(𝐮i\)Z⁡\(𝐮i\)​Z​\(𝐮0\)=⟨𝐓⁡\(𝐜^\),𝝀¯​\(𝐮i\)⟩\+bi\\displaystyle\\langle\\mathbf\{T\}\(\\mathbf\{c\}\),\\bar\{\\boldsymbol\{\\lambda\}\}\(\\mathbf\{u\}\_\{i\}\)\\rangle=\\langle\\mathbf\{T\}\(\\hat\{\\mathbf\{c\}\}\),\\bar\{\\boldsymbol\{\\lambda\}\}\(\\mathbf\{u\}\_\{i\}\)\\rangle\+\\log\\frac\{\\hat\{Z\}\(\\mathbf\{u\}\_\{0\}\)Z\(\\mathbf\{u\}\_\{i\}\)\}\{Z\(\\mathbf\{u\}\_\{i\}\)Z\(\\mathbf\{u\}\_\{0\}\)\}=\\langle\\mathbf\{T\}\(\\hat\{\\mathbf\{c\}\}\),\\bar\{\\boldsymbol\{\\lambda\}\}\(\\mathbf\{u\}\_\{i\}\)\\rangle\+b\_\{i\}\(27\)Based on assumption \(iii\), for2​d\+12d\+1distinct points𝐮0,…,𝐮2​d\\mathbf\{u\}^\{0\},\.\.\.,\\mathbf\{u\}^\{2d\}, we get

𝐋T​𝐓​\(𝐜\)=𝐋^T​𝐓​\(𝐜^\)\+𝐛\\mathbf\{L\}^\{T\}\\mathbf\{T\}\(\\mathbf\{c\}\)=\\hat\{\\mathbf\{L\}\}^\{T\}\\mathbf\{T\}\(\\hat\{\\mathbf\{c\}\}\)\+\\mathbf\{b\}\(28\)𝐓⁡\(𝐜\)=𝐀¯​𝐓​\(𝐜^\)\+𝐛¯,𝐀¯=𝐋−T​𝐋^​and​𝐛¯=𝐋−T​𝐛\\mathbf\{T\}\(\\mathbf\{c\}\)=\\bar\{\\mathbf\{A\}\}\\mathbf\{T\}\(\\hat\{\\mathbf\{c\}\}\)\+\\bar\{\\mathbf\{b\}\},\\quad\\bar\{\\mathbf\{A\}\}=\\mathbf\{L\}^\{\-T\}\\hat\{\\mathbf\{L\}\}\\text\{ and \}\\bar\{\\mathbf\{b\}\}=\\mathbf\{L\}^\{\-T\}\\mathbf\{b\}\(29\)𝐓⁡\(𝐜\)=𝐀¯​𝐓​\(𝐜^\)\+𝐛¯\\mathbf\{T\}\(\\mathbf\{c\}\)=\\bar\{\\mathbf\{A\}\}\\mathbf\{T\}\(\\hat\{\\mathbf\{c\}\}\)\+\\bar\{\\mathbf\{b\}\}\(30\)Based on assumption \(ii\) and definition 2, we knowp⁡\(𝐜\|𝐮\)p\(\\mathbf\{c\}\|\\mathbf\{u\}\)is strongly exponential\. SinceJ𝐓​\(𝐜\)J\_\{\\mathbf\{T\}\}\(\\mathbf\{c\}\)exists and is an2​d×d2d\\times dmatrix of rankdd, which implies the rank of𝐀¯\\bar\{\\mathbf\{A\}\}isdd\. Based on Lemma 1, for eachi∈\[1,…,d\]i\\in\[1,\\dots,d\], define𝐓i​\(ci\)=\(Ti,1​\(ci\),Ti,2​\(ci\)\)\\mathbf\{T\}\_\{i\}\(c\_\{i\}\)=\(T\_\{i,1\}\(c\_\{i\}\),T\_\{i,2\}\(c\_\{i\}\)\), there exist22pointsci1,ci2c\_\{i\}^\{1\},c\_\{i\}^\{2\}that\(𝐓i​\(ci1\),𝐓i​\(ci2\)\)\(\\mathbf\{T\}\_\{i\}\(c\_\{i\}^\{1\}\),\\mathbf\{T\}\_\{i\}\(c\_\{i\}^\{2\}\)\)are linearly independent\. Collect those points into22vectors\(𝐜1,𝐜2\)\(\\mathbf\{c\}^\{1\},\\mathbf\{c\}^\{2\}\), and concatenate the22Jacobians the matrix𝐐=\(𝐉𝐓​\(𝐜1\),𝐉𝐓​\(𝐜2\)\)\\mathbf\{Q\}=\(\\mathbf\{J\}\_\{\\mathbf\{T\}\}\(\\mathbf\{c\}^\{1\}\),\\mathbf\{J\}\_\{\\mathbf\{T\}\}\(\\mathbf\{c\}^\{2\}\)\)\. Then the matrix𝐐\\mathbf\{Q\}is invertible \(through a combination of Lemma 1 and the fact that each component ofTTunivariate\)\. Then define,

𝐐^=\(𝐉𝐓⁡\(𝐅𝐠,θ−1,𝐜∘𝐅𝐠^,θ^\)​\(𝐜1\),𝐉𝐓⁡\(𝐅𝐠,θ−1,𝐜∘𝐅𝐠^,θ^\)​\(𝐜2\)\)\\hat\{\\mathbf\{Q\}\}=\\left\(\\mathbf\{J\}\_\{\\mathbf\{T\}\\left\(\\mathbf\{F\}\_\{\\mathbf\{g\},\\theta\}^\{\-1,\\mathbf\{c\}\}\\circ\\mathbf\{F\}\_\{\\hat\{\\mathbf\{g\}\},\\hat\{\\theta\}\}\\right\)\}\(\\mathbf\{c\}^\{1\}\),\\mathbf\{J\}\_\{\\mathbf\{T\}\\left\(\\mathbf\{F\}\_\{\\mathbf\{g\},\\theta\}^\{\-1,\\mathbf\{c\}\}\\circ\\mathbf\{F\}\_\{\\hat\{\\mathbf\{g\}\},\\hat\{\\theta\}\}\\right\)\}\(\\mathbf\{c\}^\{2\}\)\\right\)\(31\)we get𝐐=𝐀¯​𝐐^\\mathbf\{Q\}=\\bar\{\\mathbf\{A\}\}\\hat\{\\mathbf\{Q\}\}\. The invertibility of𝐐\\mathbf\{Q\}implies the invertibility of𝐀¯\\bar\{\\mathbf\{A\}\}and𝐐^\\hat\{\\mathbf\{Q\}\}\.

Then since𝐓i​\(ci\)=\(ci,ci2\)\\mathbf\{T\}\_\{i\}\(c\_\{i\}\)=\(c\_\{i\},c\_\{i\}^\{2\}\), the relationship between the latent spaces becomes

\(𝐜𝐜2\)=𝐀¯​\(𝐜^𝐜^2\)\+𝐛¯\\begin\{pmatrix\}\\mathbf\{c\}\\\\ \\mathbf\{c\}^\{2\}\\end\{pmatrix\}=\\bar\{\\mathbf\{A\}\}\\begin\{pmatrix\}\\hat\{\\mathbf\{c\}\}\\\\ \\hat\{\\mathbf\{c\}\}^\{2\}\\end\{pmatrix\}\+\\bar\{\\mathbf\{b\}\}\(32\)where the squaring is applied element\-wise\. We can write𝐀¯\\bar\{\\mathbf\{A\}\}and𝐛¯\\bar\{\\mathbf\{b\}\}in block matrix form as

𝐀¯=\(𝐀\(1\)𝐀\(2\)𝐀\(3\)𝐀\(4\)\),𝐛¯=\(𝐛\(1\)𝐛\(2\)\)\\bar\{\\mathbf\{A\}\}=\\begin\{pmatrix\}\\mathbf\{A\}^\{\(1\)\}&\\mathbf\{A\}^\{\(2\)\}\\\\ \\mathbf\{A\}^\{\(3\)\}&\\mathbf\{A\}^\{\(4\)\}\\end\{pmatrix\},\\quad\\bar\{\\mathbf\{b\}\}=\\begin\{pmatrix\}\\mathbf\{b\}^\{\(1\)\}\\\\ \\mathbf\{b\}^\{\(2\)\}\\end\{pmatrix\}\(33\)Then

𝐜\\displaystyle\\mathbf\{c\}=𝐀\(1\)​𝐜^\+𝐀\(2\)​𝐜^2\+𝐛\(1\)\\displaystyle=\\mathbf\{A\}^\{\(1\)\}\\hat\{\\mathbf\{c\}\}\+\\mathbf\{A\}^\{\(2\)\}\\hat\{\\mathbf\{c\}\}^\{2\}\+\\mathbf\{b\}^\{\(1\)\}\(34\)𝐜2\\displaystyle\\mathbf\{c\}^\{2\}=𝐀\(3\)​𝐜^\+𝐀\(4\)​𝐜^2\+𝐛\(2\)\\displaystyle=\\mathbf\{A\}^\{\(3\)\}\\hat\{\\mathbf\{c\}\}\+\\mathbf\{A\}^\{\(4\)\}\\hat\{\\mathbf\{c\}\}^\{2\}\+\\mathbf\{b\}^\{\(2\)\}\(35\)
So we can write for each dimensioniiof𝐜\\mathbf\{c\}

ci\\displaystyle c\_\{i\}=∑jAi​j\(1\)​c^j\+∑jAi​j\(2\)​c^j2\+bi\(1\)\\displaystyle=\\sum\_\{j\}A\_\{ij\}^\{\(1\)\}\\hat\{c\}\_\{j\}\+\\sum\_\{j\}A\_\{ij\}^\{\(2\)\}\\hat\{c\}\_\{j\}^\{2\}\+b\_\{i\}^\{\(1\)\}\(36\)ci2\\displaystyle c\_\{i\}^\{2\}=∑jAi​j\(3\)​c^j\+∑jAi​j\(4\)​c^j2\+bi\(2\)\\displaystyle=\\sum\_\{j\}A\_\{ij\}^\{\(3\)\}\\hat\{c\}\_\{j\}\+\\sum\_\{j\}A\_\{ij\}^\{\(4\)\}\\hat\{c\}\_\{j\}^\{2\}\+b\_\{i\}^\{\(2\)\}\(37\)
In order to compare the equations, we square the second term of \([36](https://arxiv.org/html/2610.27180#A1.E36)\) on the right hand side, involvingc^j2\\hat\{c\}\_\{j\}^\{2\}

\(∑jAi​j\(2\)​c^j2\)2=∑j\(Ai​j\(2\)\)2​c^j4\+∑j≠j′Ai​j\(2\)​Ai​j′\(2\)​c^j2​c^j′2\\left\(\\sum\_\{j\}A\_\{ij\}^\{\(2\)\}\\hat\{c\}\_\{j\}^\{2\}\\right\)^\{2\}=\\sum\_\{j\}\\left\(A\_\{ij\}^\{\(2\)\}\\right\)^\{2\}\\hat\{c\}\_\{j\}^\{4\}\+\\sum\_\{j\\neq j^\{\\prime\}\}A\_\{ij\}^\{\(2\)\}A\_\{ij^\{\\prime\}\}^\{\(2\)\}\\hat\{c\}\_\{j\}^\{2\}\\hat\{c\}\_\{j^\{\\prime\}\}^\{2\}\(38\)The first term withc^j4\\hat\{c\}\_\{j\}^\{4\}matches no term in Equation[37](https://arxiv.org/html/2610.27180#A1.E37), so we have to setAi​j\(2\)=0A\_\{ij\}^\{\(2\)\}=0for alliiandjj\. This simplifies the earlier equation:

ci=∑jAi​j\(1\)​c^j\+bi\(1\)c\_\{i\}=\\sum\_\{j\}A\_\{ij\}^\{\(1\)\}\\hat\{c\}\_\{j\}\+b\_\{i\}^\{\(1\)\}\(39\)Then the square of the first term on the right hand side involves terms withc^j​c^j′\\hat\{c\}\_\{j\}\\hat\{c\}\_\{j^\{\\prime\}\}cross terms:

\(∑jAi​j\(1\)​c^j\)2=∑j\(Ai​j\(1\)\)2​c^j2\+∑j≠j′Ai​j\(1\)​Ai​j′\(1\)​c^j​c^j′\\left\(\\sum\_\{j\}A\_\{ij\}^\{\(1\)\}\\hat\{c\}\_\{j\}\\right\)^\{2\}=\\sum\_\{j\}\\left\(A\_\{ij\}^\{\(1\)\}\\right\)^\{2\}\\hat\{c\}\_\{j\}^\{2\}\+\\sum\_\{j\\neq j^\{\\prime\}\}A\_\{ij\}^\{\(1\)\}A\_\{ij^\{\\prime\}\}^\{\(1\)\}\\hat\{c\}\_\{j\}\\hat\{c\}\_\{j^\{\\prime\}\}\(40\)
So we have to setAi​j\(1\)​Ai​j′\(1\)=0A\_\{ij\}^\{\(1\)\}A\_\{ij^\{\\prime\}\}^\{\(1\)\}=0for allj≠j′j\\neq j^\{\\prime\}\. This means that theii\-th row of𝐀\(1\)\\mathbf\{A\}^\{\(1\)\}can have at most one nonzero entry\. It must also have at least one nonzero entry, since if the row were all zero, a row of𝐀¯\\bar\{\\mathbf\{A\}\}would be all zero \(since𝐀\(2\)=0\\mathbf\{A\}^\{\(2\)\}=0\), but𝐀¯\\bar\{\\mathbf\{A\}\}has full rank\. Hence each row of𝐀\(1\)\\mathbf\{A\}^\{\(1\)\}has exactly one nonzero entry and no two rows of𝐀\(1\)\\mathbf\{A\}^\{\(1\)\}have their nonzero entries in the same column \(since𝐀¯\\bar\{\\mathbf\{A\}\}has full rank\)\. Therefore, we can write:

ci=Ai​j\(1\)​c^j\+bi\(1\)c\_\{i\}=A\_\{ij\}^\{\(1\)\}\\hat\{c\}\_\{j\}\+b\_\{i\}^\{\(1\)\}\(41\)Since each row of𝐀\(1\)\\mathbf\{A\}^\{\(1\)\}has exactly one nonzero entry in distinct columns, we can write𝐀\(1\)=A⋅P\\mathbf\{A\}^\{\(1\)\}=A\\cdot PwhereA=diag​\(a1,…,ad\)A=\\text\{diag\}\(a\_\{1\},\\ldots,a\_\{d\}\)withai≠0a\_\{i\}\\neq 0andPPis a permutation matrix\. In vector form:

𝐜=A⋅P⁡\(𝐜^\)\+𝐛\\mathbf\{c\}=A\\cdot P\(\\hat\{\\mathbf\{c\}\}\)\+\\mathbf\{b\}\(42\)where𝐛=\(b1\(1\),…,bd\(1\)\)⊤\\mathbf\{b\}=\(b\_\{1\}^\{\(1\)\},\\ldots,b\_\{d\}^\{\(1\)\}\)^\{\\top\}\. This establishes the equivalence relation in Definition 2\.□\\square

### A\.2Proof of Corollary 1

Corollary 1When the true dimensionddof the latent variable𝐜\\mathbf\{c\}is unknown, setting the model dimension tod^≥d\\hat\{d\}\\geq densures that the decoder recovers the original latent variables𝐜\\mathbf\{c\}up to an affine transformation and permutation within a subset of the estimated latent space, while the remaining dimensions encode only noise\.

Proof:Following the proof of Theorem 1, we obtain the relationship:

𝐜=𝐀​𝐜^\+𝐛\\mathbf\{c\}=\\mathbf\{A\}\\hat\{\\mathbf\{c\}\}\+\\mathbf\{b\}\(43\)whereAAis ad×d^d\\times\\hat\{d\}matrix and each row ofAAhas exactly one nonzero entry\. Therefore, we can write:

ci=Ai​j​c^j\+bic\_\{i\}=A\_\{ij\}\\hat\{c\}\_\{j\}\+b\_\{i\}\(44\)This establishes that each generating latent environment variablecic\_\{i\}is linearly related to some latent variablec^j\\hat\{c\}\_\{j\}of the estimating model\. This estimated latent variable is uniquely associated withcic\_\{i\}, and any estimated latent variables not associated with a generating latent variablecic\_\{i\}\(in the cased^\>d\\hat\{d\}\>d\) encode no information about the generating latent space\.

Consequently, the model has decoded the original latent variables𝐜\\mathbf\{c\}up to an affine transformation and permutation as a subset of variables in its estimated latent space and has encoded no information \(only noise\) into the remaining latent variables\.□\\square

### A\.3Proof of Theorem 2

Theorem 2Suppose there exists an invertible function𝐠^−1\\hat\{\\mathbf\{g\}\}^\{\-1\}that𝐳^t=𝐠^−1​\(𝐱t\)\\hat\{\\mathbf\{z\}\}\_\{t\}=\\hat\{\\mathbf\{g\}\}^\{\-1\}\(\\mathbf\{x\}\_\{t\}\)and the conditional distributionp⁡\(zk​t\|𝐳t−1,𝐮\)p\(z\_\{kt\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{u\}\)changes acrossMMvalue of the environment variables𝐮\\mathbf\{u\}, denoted by𝐮1,𝐮2,…,𝐮M\\mathbf\{u\}\_\{1\},\\mathbf\{u\}\_\{2\},\\ldots,\\mathbf\{u\}\_\{M\}\. Assume the components of𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\}are also mutually independent conditional on𝐳^t−1\\hat\{\\mathbf\{z\}\}\_\{t\-1\}and𝐮\\mathbf\{u\}\. Letηk​t​\(𝐮\)=log⁡p⁡\(zk​t\|𝐳t−1,𝐮\)\\eta\_\{kt\}\(\\mathbf\{u\}\)=\\log p\(z\_\{kt\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{u\}\)and assumeηk​t​\(𝐮\)\\eta\_\{kt\}\(\\mathbf\{u\}\)is twice differentiable inzk​tz\_\{kt\}and is differentiable inzl,t−1,l=1,2,…,nz\_\{l,t\-1\},l=1,2,\\ldots,n\. Also let

𝐬k​t\\displaystyle\\mathbf\{s\}\_\{kt\}=\(∂2ηk​t​\(𝐮2\)∂zk​t2−∂2ηk​t​\(𝐮1\)∂zk​t2,…,∂2ηk​t​\(𝐮M\)∂zk​t2−∂2ηk​t​\(𝐮M−1\)∂zk​t2\)T\\displaystyle=\\left\(\\frac\{\\partial^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{2\}\)\}\{\\partial z\_\{kt\}^\{2\}\}\-\\frac\{\\partial^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{1\}\)\}\{\\partial z\_\{kt\}^\{2\}\},\\ldots,\\frac\{\\partial^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{M\}\)\}\{\\partial z\_\{kt\}^\{2\}\}\-\\frac\{\\partial^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{M\-1\}\)\}\{\\partial z\_\{kt\}^\{2\}\}\\right\)^\{T\}\(45\)𝐬̊k​t\\displaystyle\\mathring\{\\mathbf\{s\}\}\_\{kt\}=\(∂ηk​t​\(𝐮2\)∂zk​t−∂ηk​t​\(𝐮1\)∂zk​t,…,∂ηk​t​\(𝐮M\)∂zk​t−∂ηk​t​\(𝐮M−1\)∂zk​t\)T\\displaystyle=\\left\(\\frac\{\\partial\\eta\_\{kt\}\(\\mathbf\{u\}\_\{2\}\)\}\{\\partial z\_\{kt\}\}\-\\frac\{\\partial\\eta\_\{kt\}\(\\mathbf\{u\}\_\{1\}\)\}\{\\partial z\_\{kt\}\},\\ldots,\\frac\{\\partial\\eta\_\{kt\}\(\\mathbf\{u\}\_\{M\}\)\}\{\\partial z\_\{kt\}\}\-\\frac\{\\partial\\eta\_\{kt\}\(\\mathbf\{u\}\_\{M\-1\}\)\}\{\\partial z\_\{kt\}\}\\right\)^\{T\}\(46\)If for each value of𝐳t\\mathbf\{z\}\_\{t\}, the2​m2mfunction vectors𝐬k​t\\mathbf\{s\}\_\{kt\}and𝐬̊k​t\\mathring\{\\mathbf\{s\}\}\_\{kt\}, withk=1,2,…,mk=1,2,\\ldots,m, are linearly independent, then𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\}is a permuted invertible component\-wise transformation of𝐳t\\mathbf\{z\}\_\{t\}, that

𝐠^−1​\(𝐱t\)=C∘P∘𝐠−1​\(𝐱t\)⇔𝐳^t=C∘P⁡\(𝐳t\)\\hat\{\\mathbf\{g\}\}^\{\-1\}\(\\mathbf\{x\}\_\{t\}\)=C\\circ P\\circ\\mathbf\{g\}^\{\-1\}\(\\mathbf\{x\}\_\{t\}\)\\Leftrightarrow\\hat\{\\mathbf\{z\}\}\_\{t\}=C\\circ P\(\\mathbf\{z\}\_\{t\}\)\(47\)whereCCis a component\-wise invertible transformation andPPis a permutation\.

Proof:Using𝐠\\mathbf\{g\}and𝐠^\\hat\{\\mathbf\{g\}\}, we can relate the ground\-truth and estimated latent states𝐳t\\mathbf\{z\}\_\{t\}and𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\}to each other:

𝐳t=𝐠−1​\(𝐱t\)=𝐠−1∘𝐠^​\(𝐳^t\)=𝐡⁡\(𝐳^t\)\\mathbf\{z\}\_\{t\}=\\mathbf\{g\}^\{\-1\}\(\\mathbf\{x\}\_\{t\}\)=\\mathbf\{g\}^\{\-1\}\\circ\\hat\{\\mathbf\{g\}\}\(\\hat\{\\mathbf\{z\}\}\_\{t\}\)=\\mathbf\{h\}\(\\hat\{\\mathbf\{z\}\}\_\{t\}\)\(48\)where𝐡=𝐠−1∘𝐠^\\mathbf\{h\}=\\mathbf\{g\}^\{\-1\}\\circ\\hat\{\\mathbf\{g\}\}\. Using𝐳t=𝐡⁡\(𝐳^t\)\\mathbf\{z\}\_\{t\}=\\mathbf\{h\}\(\\hat\{\\mathbf\{z\}\}\_\{t\}\), we perform change of variables on the conditional latent densitylog⁡p⁡\(𝐳^t\|𝐳^t−1,𝐮\)\\log p\(\\hat\{\\mathbf\{z\}\}\_\{t\}\|\\hat\{\\mathbf\{z\}\}\_\{t\-1\},\\mathbf\{u\}\)as follows:

log⁡p⁡\(𝐳^t\|𝐳^t−1,𝐮\)=log⁡p⁡\(𝐳t\|𝐳t−1,𝐮\)\+log⁡\|𝐇t\|\\log p\(\\hat\{\\mathbf\{z\}\}\_\{t\}\|\\hat\{\\mathbf\{z\}\}\_\{t\-1\},\\mathbf\{u\}\)=\\log p\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{u\}\)\+\\log\|\\mathbf\{H\}\_\{t\}\|\(49\)where𝐇t=𝐉𝐡​\(𝐳^t\)\\mathbf\{H\}\_\{t\}=\\mathbf\{J\}\_\{\\mathbf\{h\}\}\(\\hat\{\\mathbf\{z\}\}\_\{t\}\)is the Jacobian matrix of𝐡\\mathbf\{h\}evaluated at𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\}\. We make use of the fact that the components of𝐳t\\mathbf\{z\}\_\{t\}are mutually independent conditional on𝐳^t−1\\hat\{\\mathbf\{z\}\}\_\{t\-1\}and𝐮\\mathbf\{u\}, then for anyi≠ji\\neq j,z^i​t\\hat\{z\}\_\{it\}andz^j​t\\hat\{z\}\_\{jt\}are conditionally independent given𝐳^t−1∪\{𝐳^t∖\{z^i​t,z^j​t\}\}\\hat\{\\mathbf\{z\}\}\_\{t\-1\}\\cup\\\{\\hat\{\\mathbf\{z\}\}\_\{t\}\\setminus\\\{\\hat\{z\}\_\{it\},\\hat\{z\}\_\{jt\}\\\}\\\}, such that:

∂2log⁡p⁡\(𝐳^t\|𝐳^t−1,𝐮\)∂z^i​t​∂z^j​t=0\\frac\{\\partial^\{2\}\\log p\(\\hat\{\\mathbf\{z\}\}\_\{t\}\|\\hat\{\\mathbf\{z\}\}\_\{t\-1\},\\mathbf\{u\}\)\}\{\\partial\\hat\{z\}\_\{it\}\\partial\\hat\{z\}\_\{jt\}\}=0\(50\)Letlog⁡p⁡\(𝐳t\|𝐳t−1,𝐮\)=∑k=1mηk​t​\(𝐮\)\\log p\(\\mathbf\{z\}\_\{t\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{u\}\)=\\sum\_\{k=1\}^\{m\}\\eta\_\{kt\}\(\\mathbf\{u\}\),ηk​t​\(𝐮\)=log⁡p⁡\(zk​t\|𝐳t−1,𝐮\)\\eta\_\{kt\}\(\\mathbf\{u\}\)=\\log p\(z\_\{kt\}\|\\mathbf\{z\}\_\{t\-1\},\\mathbf\{u\}\), then

∂2log⁡p⁡\(𝐳^t\|𝐳^t−1,𝐮\)∂z^i​t​∂z^j​t=∑k=1m\(∂2ηk​t​\(𝐮\)∂zk​t2​𝐇k​i​t​𝐇k​j​t\+∂ηk​t​\(𝐮\)∂zk​t​∂𝐇k​i​t∂z^j​t\)−∂2log⁡\|𝐇t\|∂z^i​t​∂z^j​t=0\\frac\{\\partial^\{2\}\\log p\(\\hat\{\\mathbf\{z\}\}\_\{t\}\|\\hat\{\\mathbf\{z\}\}\_\{t\-1\},\\mathbf\{u\}\)\}\{\\partial\\hat\{z\}\_\{it\}\\partial\\hat\{z\}\_\{jt\}\}=\\sum\_\{k=1\}^\{m\}\\left\(\\frac\{\\partial^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\)\}\{\\partial z\_\{kt\}^\{2\}\}\\mathbf\{H\}\_\{kit\}\\mathbf\{H\}\_\{kjt\}\+\\frac\{\\partial\\eta\_\{kt\}\(\\mathbf\{u\}\)\}\{\\partial z\_\{kt\}\}\\frac\{\\partial\\mathbf\{H\}\_\{kit\}\}\{\\partial\\hat\{z\}\_\{jt\}\}\\right\)\-\\frac\{\\partial^\{2\}\\log\|\\mathbf\{H\}\_\{t\}\|\}\{\\partial\\hat\{z\}\_\{it\}\\partial\\hat\{z\}\_\{jt\}\}=0\(51\)Using different values for𝐮\\mathbf\{u\}take the difference of this equation across them gives

∂2log⁡p⁡\(𝐳^t\|𝐳^t−1,𝐮r\+1\)∂z^i​t​∂z^j​t−∂2log⁡p⁡\(𝐳^t\|𝐳^t−1,𝐮r\)∂z^i​t​∂z^j​t=∑k=1m\[\(∂2ηk​t​\(𝐮r\+1\)∂zk​t2−∂2ηk​t​\(𝐮r\)∂zk​t2\)​𝐇k​i​t​𝐇k​j​t\+\(∂ηk​t​\(𝐮r\+1\)∂zk​t−∂ηk​t​\(𝐮r\)∂zk​t\)​∂𝐇k​i​t∂z^j​t\]=0\\begin\{split\}&\\frac\{\\partial^\{2\}\\log p\(\\hat\{\\mathbf\{z\}\}\_\{t\}\|\\hat\{\\mathbf\{z\}\}\_\{t\-1\},\\mathbf\{u\}\_\{r\+1\}\)\}\{\\partial\\hat\{z\}\_\{it\}\\partial\\hat\{z\}\_\{jt\}\}\-\\frac\{\\partial^\{2\}\\log p\(\\hat\{\\mathbf\{z\}\}\_\{t\}\|\\hat\{\\mathbf\{z\}\}\_\{t\-1\},\\mathbf\{u\}\_\{r\}\)\}\{\\partial\\hat\{z\}\_\{it\}\\partial\\hat\{z\}\_\{jt\}\}\\\\ &=\\sum\_\{k=1\}^\{m\}\\left\[\\left\(\\frac\{\\partial^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{r\+1\}\)\}\{\\partial z\_\{kt\}^\{2\}\}\-\\frac\{\\partial^\{2\}\\eta\_\{kt\}\(\\mathbf\{u\}\_\{r\}\)\}\{\\partial z\_\{kt\}^\{2\}\}\\right\)\\mathbf\{H\}\_\{kit\}\\mathbf\{H\}\_\{kjt\}\+\\left\(\\frac\{\\partial\\eta\_\{kt\}\(\\mathbf\{u\}\_\{r\+1\}\)\}\{\\partial z\_\{kt\}\}\-\\frac\{\\partial\\eta\_\{kt\}\(\\mathbf\{u\}\_\{r\}\)\}\{\\partial z\_\{kt\}\}\\right\)\\frac\{\\partial\\mathbf\{H\}\_\{kit\}\}\{\\partial\\hat\{z\}\_\{jt\}\}\\right\]=0\\end\{split\}\(52\)Therefore, if𝐬k​t\\mathbf\{s\}\_\{kt\}and𝐬̊k​t\\mathring\{\\mathbf\{s\}\}\_\{kt\}, fork=1,2,…,mk=1,2,\\ldots,m, are linearly independent,𝐇k​i​t​𝐇k​j​t\\mathbf\{H\}\_\{kit\}\\mathbf\{H\}\_\{kjt\}has to be zero for allkkandi≠ji\\neq j\. That is, in each row of𝐇\\mathbf\{H\}, there is only one non\-zero entry\. Since𝐡\\mathbf\{h\}is invertible, then𝐳^t\\hat\{\\mathbf\{z\}\}\_\{t\}must be a permuted invertible component\-wise transformation of𝐳t\\mathbf\{z\}\_\{t\}\.

### A\.4Proof of Corollary 2

Corollary 2Given that Theorems 1 and 2 are established, there exist two invertible functions𝐡\\mathbf\{h\}and𝐤\\mathbf\{k\}, each being a composition of permutations and component\-wise transformations, such that

ℱ^=𝐡−1∘ℱ∘𝐤\\hat\{\\mathcal\{F\}\}=\\mathbf\{h\}^\{\-1\}\\circ\\mathcal\{F\}\\circ\\mathbf\{k\}\(53\)Consequently, the learned transition functionℱ^\\hat\{\\mathcal\{F\}\}is equivalent to the true transition functionℱ\\mathcal\{F\}up to compositions of permutations and component\-wise transformations\.

Proof:Based on Theorem 1 and 2, we can get

𝐳t=𝐡⁡\(𝐳^t\),𝐜=𝐝⁡\(𝐜^\)\\mathbf\{z\}\_\{t\}=\\mathbf\{h\}\(\\hat\{\\mathbf\{z\}\}\_\{t\}\),\\mathbf\{c\}=\\mathbf\{d\}\(\\hat\{\\mathbf\{c\}\}\)\(54\)where both functions𝐡\\mathbf\{h\}and𝐝\\mathbf\{d\}are invertible and compositions of permutations and component\-wise transformations\. We define

\[𝐳t𝐜\]=\[𝐡⁡\(𝐳^t\)𝐝⁡\(𝐜^\)\]=𝐤​\[𝐳^t𝐜^\]\\begin\{bmatrix\}\\mathbf\{z\}\_\{t\}\\\\ \\mathbf\{c\}\\end\{bmatrix\}=\\begin\{bmatrix\}\\mathbf\{h\}\(\\hat\{\\mathbf\{z\}\}\_\{t\}\)\\\\ \\mathbf\{d\}\(\\hat\{\\mathbf\{c\}\}\)\\end\{bmatrix\}=\\mathbf\{k\}\\begin\{bmatrix\}\\hat\{\\mathbf\{z\}\}\_\{t\}\\\\ \\hat\{\\mathbf\{c\}\}\\end\{bmatrix\}\(55\)then𝐤\\mathbf\{k\}must also be an invertible function which is a composition of permutations and component\-wise transformations\. Then,

𝐳^t\+1=ℱ⁡\(\[𝐳^t𝐜^\]\)=𝐡−1∘ℱ⁡\(\[𝐳t𝐜\]\)=𝐡−1∘ℱ∘𝐤⁡\(\[𝐳^t𝐜^\]\)\\hat\{\\mathbf\{z\}\}\_\{t\+1\}=\\mathcal\{F\}\\left\(\\begin\{bmatrix\}\\hat\{\\mathbf\{z\}\}\_\{t\}\\\\ \\hat\{\\mathbf\{c\}\}\\end\{bmatrix\}\\right\)=\\mathbf\{h\}^\{\-1\}\\circ\\mathcal\{F\}\\left\(\\begin\{bmatrix\}\\mathbf\{z\}\_\{t\}\\\\ \\mathbf\{c\}\\end\{bmatrix\}\\right\)=\\mathbf\{h\}^\{\-1\}\\circ\\mathcal\{F\}\\circ\\mathbf\{k\}\\left\(\\begin\{bmatrix\}\\hat\{\\mathbf\{z\}\}\_\{t\}\\\\ \\hat\{\\mathbf\{c\}\}\\end\{bmatrix\}\\right\)\(56\)So that

ℱ^=𝐡−1∘ℱ∘𝐤\\hat\{\\mathcal\{F\}\}=\\mathbf\{h\}^\{\-1\}\\circ\\mathcal\{F\}\\circ\\mathbf\{k\}\(57\)Therefore, the learned transition functionℱ^\\hat\{\\mathcal\{F\}\}is equivalent to the true transition functionℱ\\mathcal\{F\}up to compositions of permutations and component\-wise transformations\.

## Appendix BDerivation Of ELBO

∑𝒟m∈𝒟∑x0:τq∈𝒟mlogpΦ\(𝐱0:τq\|𝒟s\)\\displaystyle\\sum\_\{\\mathcal\{D\}\_\{m\}\\in\\mathcal\{D\}\}\\sum\_\{x\_\{0:\\tau\}^\{q\}\\in\\mathcal\{D\}\_\{m\}\}\\log p\_\{\\Phi\}\(\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\|\\mathcal\{D\}^\{s\}\)=∑𝒟m∈𝒟∑x0:τq∈𝒟mlog∬pθ,g\(𝐱0:τq\|𝐜,𝐳0\)pζ\(𝐜\|𝒟s\)p\(𝐳0\)d𝐜d𝐳0\\displaystyle=\\sum\_\{\\mathcal\{D\}\_\{m\}\\in\\mathcal\{D\}\}\\sum\_\{x\_\{0:\\tau\}^\{q\}\\in\\mathcal\{D\}\_\{m\}\}\\log\\iint p\_\{\\theta,g\}\(\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\|\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\)p\(\\mathbf\{z\}\_\{0\}\)d\\mathbf\{c\}d\\mathbf\{z\}\_\{0\}=∑𝒟m∈𝒟∑x0:τq∈𝒟mlog∬pθ,g\(𝐱0:τq\|𝐜,𝐳0\)p⁡\(𝐳0\)qϕ\(𝐳0\|𝐱0:lq\)qϕ\(𝐳0\|𝐱0:lq\)\\displaystyle=\\sum\_\{\\mathcal\{D\}\_\{m\}\\in\\mathcal\{D\}\}\\sum\_\{x\_\{0:\\tau\}^\{q\}\\in\\mathcal\{D\}\_\{m\}\}\\log\\iint p\_\{\\theta,g\}\(\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\|\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)\\frac\{p\(\\mathbf\{z\}\_\{0\}\)\}\{q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)\}q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)⋅pζ​\(𝐜\|𝒟s\)qζ\(𝐜\|𝐱0:τq∪𝒟s\)qζ\(𝐜\|𝐱0:τq∪𝒟s\)d𝐜d𝐳0\\displaystyle\\cdot\\frac\{p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\)\}\{q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\\cup\\mathcal\{D\}^\{s\}\)\}q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\\cup\\mathcal\{D\}^\{s\}\)d\\mathbf\{c\}d\\mathbf\{z\}\_\{0\}≥∑𝒟m∈𝒟∑x0:τq∈𝒟m∬\[logpθ,g\(𝐱0:τq\|𝐜,𝐳0\)−logqζ\(𝐜\|𝐱0:τq∪𝒟s\)pζ​\(𝐜\|𝒟s\)−logqϕ\(𝐳0\|𝐱0:lq\)p⁡\(𝐳0\)\]\\displaystyle\\geq\\sum\_\{\\mathcal\{D\}\_\{m\}\\in\\mathcal\{D\}\}\\sum\_\{x\_\{0:\\tau\}^\{q\}\\in\\mathcal\{D\}\_\{m\}\}\\iint\\left\[\\log p\_\{\\theta,g\}\(\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\|\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)\-\\log\\frac\{q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\\cup\\mathcal\{D\}^\{s\}\)\}\{p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\)\}\-\\log\\frac\{q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)\}\{p\(\\mathbf\{z\}\_\{0\}\)\}\\right\]⋅qζ\(𝐜\|𝐱0:τq∪𝒟s\)qϕ\(𝐳0\|𝐱0:lq\)d𝐜d𝐳0\\displaystyle\\cdot q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\\cup\\mathcal\{D\}^\{s\}\)q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)d\\mathbf\{c\}d\\mathbf\{z\}\_\{0\}=∑𝒟m∈𝒟∑x0:τq∈𝒟m𝔼qζ\(𝐜\|𝐱0:τq∪𝒟s\),qϕ\(𝐳0\|𝐱0:lq\)\[logpθ,g\(𝐱0:τq\|𝐜,𝐳0\)\]\\displaystyle=\\sum\_\{\\mathcal\{D\}\_\{m\}\\in\\mathcal\{D\}\}\\sum\_\{x\_\{0:\\tau\}^\{q\}\\in\\mathcal\{D\}\_\{m\}\}\\mathbb\{E\}\_\{q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\\cup\\mathcal\{D\}^\{s\}\),q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)\}\\left\[\\log p\_\{\\theta,g\}\(\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\|\\mathbf\{c\},\\mathbf\{z\}\_\{0\}\)\\right\]−KL\[qζ\(𝐜\|𝐱0:τq∪𝒟s\)∥pζ\(𝐜\|𝒟s\)\]−KL\[qϕ\(𝐳0\|𝐱0:lq\)∥p\(𝐳0\)\]\\displaystyle\-\\text\{KL\}\\left\[q\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathbf\{x\}\_\{0:\\tau\}^\{q\}\\cup\\mathcal\{D\}^\{s\}\)\\\|p\_\{\\zeta\}\(\\mathbf\{c\}\|\\mathcal\{D\}^\{s\}\)\\right\]\-\\text\{KL\}\\left\[q\_\{\\phi\}\(\\mathbf\{z\}\_\{0\}\|\\mathbf\{x\}\_\{0:l\}^\{q\}\)\\\|p\(\\mathbf\{z\}\_\{0\}\)\\right\]

## Appendix CAdditional Details for Experiments in Section 5\.1 & 5\.2

### C\.1Data descriptions

For each dataset, we generate about 10k samples for both training and testing\. A general overview of the datastes is as follows and their schematic/examples in Fig\.[4](https://arxiv.org/html/2610.27180#A3.F4)

![Refer to caption](https://arxiv.org/html/2610.27180v1/Fig/Fig_Data.png)

Figure 4:The general overview of the datasets#### C\.1\.1Forced Damped Pendulum

d2​φd​t2=−ω2​sin⁡φ−α​d​φd​t\+A​cos⁡\(2​π​t\)\\frac\{d^\{2\}\\varphi\}\{dt^\{2\}\}=\-\\omega^\{2\}\\sin\\varphi\-\\alpha\\frac\{d\\varphi\}\{dt\}\+A\\cos\(2\\pi t\)\(58\)We randomly sampleφ⁡\(t=0\)∈\[−1\.0,1\.0\]\\varphi\(t=0\)\\in\[\-1\.0,1\.0\],φ˙​\(t=0\)∈\[−0\.5,0\.5\]\\dot\{\\varphi\}\(t=0\)\\in\[\-0\.5,0\.5\],ω∈\[2\.5,3\.5\]\\omega\\in\[2\.5,3\.5\]andα∈\[0\.5,0\.8\]\\alpha\\in\[0\.5,0\.8\], and fixA=3\.0A=3\.0to generate data\. For OOD, we considerω∈\[2\.0,2\.5\]\\omega\\in\[2\.0,2\.5\]andα∈\[0\.8,1\.0\]\\alpha\\in\[0\.8,1\.0\]

#### C\.1\.2CartPole

\[x¨φ¨\]=\[−g​sin⁡φ​cos⁡φ−\(F\+l​φ˙2​sin⁡φ\)cos2⁡φ−\(m~\+1\)\(m~\+1\)​g​sin⁡φ\+cos⁡φ⁡\(F\+m~​l​φ˙2​sin⁡φ\)l​cos2⁡φ−\(m~\+1\)​l\]\\begin\{bmatrix\}\\ddot\{x\}\\\\ \\ddot\{\\varphi\}\\end\{bmatrix\}=\\begin\{bmatrix\}\\frac\{\-g\\sin\\varphi\\cos\\varphi\-\(F\+l\\dot\{\\varphi\}^\{2\}\\sin\\varphi\)\}\{\\cos^\{2\}\\varphi\-\(\\tilde\{m\}\+1\)\}\\\\ \\frac\{\(\\tilde\{m\}\+1\)g\\sin\\varphi\+\\cos\\varphi\(F\+\\tilde\{m\}l\\dot\{\\varphi\}^\{2\}\\sin\\varphi\)\}\{l\\cos^\{2\}\\varphi\-\(\\tilde\{m\}\+1\)l\}\\end\{bmatrix\}\(59\)We randomly samplex⁡\(t=0\)∈\[−1\.0,1\.0\]x\(t=0\)\\in\[\-1\.0,1\.0\],φ⁡\(t=0\)∈\[−1\.0,1\.0\]\\varphi\(t=0\)\\in\[\-1\.0,1\.0\]andF∈\[−2\.5,2\.5\]F\\in\[\-2\.5,2\.5\], then fixx˙​\(t=0\)=0\\dot\{x\}\(t=0\)=0,φ˙​\(t=0\)=0\\dot\{\\varphi\}\(t=0\)=0,l=3\.0l=3\.0,m~=5\.0​\(m2=1\.0\)\\tilde\{m\}=5\.0\(m\_\{2\}=1\.0\),g=9\.8g=9\.8to generate data\. For OOD, we considerF∈\[−4\.0,−2\.5\]∪\[2\.5,4\.0\]F\\in\[\-4\.0,\-2\.5\]\\cup\[2\.5,4\.0\]\.

#### C\.1\.3Double Pendulum

\[φ¨1φ¨2\]=\[−g\(m~\+1\)sinφ1−gsin\(θ1−2θ2\)−2sin\(φ1−φ2\)\(φ2˙2l2\+φ1˙2l1cos\(φ1−φ2\)\)l1​\(2​m~\+1−cos⁡\(2​θ1−2​θ2\)\)2sin\(φ1−φ2\)\(φ1˙2l1\(m~\+1\)\+g\(m~\+1\)cosφ1\+φ2˙2l2cos\(φ1−φ2\)\)l2​\(2​m~\+1−cos⁡\(2​θ1−2​θ2\)\)\]\\begin\{bmatrix\}\\ddot\{\\varphi\}\_\{1\}\\\\ \\ddot\{\\varphi\}\_\{2\}\\end\{bmatrix\}=\\begin\{bmatrix\}\\frac\{\-g\(\\tilde\{m\}\+1\)\\sin\\varphi\_\{1\}\-g\\sin\(\\theta\_\{1\}\-2\\theta\_\{2\}\)\-2\\sin\(\\varphi\_\{1\}\-\\varphi\_\{2\}\)\(\\dot\{\\varphi\_\{2\}\}^\{2\}l\_\{2\}\+\\dot\{\\varphi\_\{1\}\}^\{2\}l\_\{1\}\\cos\(\\varphi\_\{1\}\-\\varphi\_\{2\}\)\)\}\{l\_\{1\}\(2\\tilde\{m\}\+1\-\\cos\(2\\theta\_\{1\}\-2\\theta\_\{2\}\)\)\}\\\\ \\frac\{2\\sin\(\\varphi\_\{1\}\-\\varphi\_\{2\}\)\(\\dot\{\\varphi\_\{1\}\}^\{2\}l\_\{1\}\(\\tilde\{m\}\+1\)\+g\(\\tilde\{m\}\+1\)\\cos\\varphi\_\{1\}\+\\dot\{\\varphi\_\{2\}\}^\{2\}l\_\{2\}\\cos\(\\varphi\_\{1\}\-\\varphi\_\{2\}\)\)\}\{l\_\{2\}\(2\\tilde\{m\}\+1\-\\cos\(2\\theta\_\{1\}\-2\\theta\_\{2\}\)\)\}\\end\{bmatrix\}\(60\)We randomly sampleφ1​\(t=0\)∈\[−1\.0,1\.0\]\\varphi\_\{1\}\(t=0\)\\in\[\-1\.0,1\.0\],φ2​\(t=0\)∈\[−1\.0,1\.0\]\\varphi\_\{2\}\(t=0\)\\in\[\-1\.0,1\.0\],g∈\[5\.0,10\.0\]g\\in\[5\.0,10\.0\]and , then fixφ˙1​\(t=0\)=0\\dot\{\\varphi\}\_\{1\}\(t=0\)=0,m~=2\.0\\tilde\{m\}=2\.0,φ˙2​\(t=0\)=0\\dot\{\\varphi\}\_\{2\}\(t=0\)=0,l1=1\.0l\_\{1\}=1\.0andl2=2\.5l\_\{2\}=2\.5to generate data\. For OOD, we considerg∈\[10\.0,12\.5\]g\\in\[10\.0,12\.5\]

#### C\.1\.4MoCap

The walking sequences we consider are35\_33\.amc,39\_08\.amc,16\_16\.amc,35\_06\.amc,07\_02\.amc,35\_10\.amc,35\_14\.amc,07\_06\.amc,35\_13\.amc,39\_06\.amc,16\_21\.amc,39\_07\.amc,08\_01\.amc,35\_31\.amc,07\_01\.amc,35\_07\.amc,08\_10\.amc,16\_15\.amc,39\_01\.amc,08\_02\.amc,35\_16\.amc,39\_14\.amc,08\_03\.amc,39\_13\.amc,07\_09\.amc,16\_58\.amc,39\_04\.amc,16\_31\.amc,35\_32\.amc,35\_34\.amc,16\_22\.amc,35\_08\.amc,35\_01\.amc,35\_12\.amc,38\_02\.amc,39\_03\.amc,35\_11\.amc,39\_02\.amc,39\_10\.amc,16\_47\.amc,07\_03\.amc,08\_06\.amc,39\_12\.amc,07\_08\.amc,35\_29\.amc,07\_10\.amc,38\_01\.amc,35\_09\.amc,35\_05\.amc,35\_02\.amc,39\_05\.amc,35\_28\.amc,35\_04\.amc,08\_08\.amc,08\_09\.amc,35\_03\.amc,39\_09\.amc,16\_32\.amc,35\_30\.amc,07\_07\.amc,35\_15\.amc,07\_11\.amc\. The number of training, validation, and test samples in MOCAP\-Multiple \(ID\) and MOCAP\-SHIFT \(OOD\) splits are 52\-5\-5 and 49\-5\-8, respectively\. Since sequences are already dense, we skip every other data point\. For ease of implementation, we take the last 250 time points, leading to sequences of lengthT=125T=125\.

### C\.2Additional Experimental Results on Synthetic Physics Systems

#### C\.2\.1Experimental Results on Double Pendulum systems

We provide additional results on the Double Pendulum system with two\-dimensional latent state𝐳t=\[θ1,θ2\]\\mathbf\{z\}\_\{t\}=\[\\theta\_\{1\},\\theta\_\{2\}\]\(angles of the two pendulum arms\) and environment variableGG\(gravitational constant\)\.

Table 5 shows that Meta\-iLaD achieves consistently strong performance across all settings with near\-perfect MCC scores\. In contrast, IDF shows significant degradation in prediction \(MSE increases from 8\.04e\-1 to 1\.61e0\) with poor MCC\(c\)=0\.08, indicating failure to identify the environment variable\. Figure 5 visualizes this through scatter plots: Meta\-iLaD \(bottom row\) exhibits clear linear relationships for both angles and parameterGG, while IDF \(top row\) achieves good identifiability for latent angles but fails to recoverGG\(random noise pattern\)\. This directly explains IDF’s poor prediction performance—without properly identifyingGG, the learned dynamics function cannot accurately forecast future trajectories\.

Table 5:Quantitative results on Double Pendulum\.![Refer to caption](https://arxiv.org/html/2610.27180v1/Fig/Figure_C1.png)Figure 5:Scatter plots of estimated versus true values for latent anglesθ1,θ2\\theta\_\{1\},\\theta\_\{2\}\(blue\) and environment parameterGG\(red\) 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 recoverGG\(noise pattern\)\.
#### C\.2\.2OOD Generalization on CartPole

Figure[6](https://arxiv.org/html/2610.27180#A3.F6)demonstrates Meta\-iLaD’s ability to identify environment parameters in out\-of\-distribution regions on the CartPole dataset\. The model was trained on in\-distribution parameter range\[−2\.5,2\.5\]\[\-2\.5,2\.5\]\(blue points\) and tested on OOD ranges\[−4\.0,−2\.5\]∪\[2\.5,4\.0\]\[\-4\.0,\-2\.5\]\\cup\[2\.5,4\.0\]\(red points\)\. The scatter plot shows that Meta\-iLaD maintains a clear linear relationship between estimated and true parameters across both ID and OOD regions, indicating successful identifiability beyond the training distribution\. This ability to correctly identify environment variables in OOD regions directly explains Meta\-iLaD’s strong OOD generalization performance reported in Table 1\.

![Refer to caption](https://arxiv.org/html/2610.27180v1/Fig/Figure_C2.png)Figure 6:Scatter plot of estimated versus true environment parameters on CartPole\. Blue points represent in\-distribution range\[−2\.5,2\.5\]\[\-2\.5,2\.5\], red points represent OOD ranges\[−4\.0,−2\.5\]∪\[2\.5,4\.0\]\[\-4\.0,\-2\.5\]\\cup\[2\.5,4\.0\]\. Meta\-iLaD maintains identifiability across both ID and OOD regions\.

### C\.3Additional Experimental Results on Real\-World CMU Data

Figure[7](https://arxiv.org/html/2610.27180#A3.F7)presents qualitative comparisons of trajectory predictions between IDF and Meta\-iLaD on the CMU MoCAP dataset\. We visualize predictions across six representative joint angles \(Joint 3, 13, 27, 44, 46, 47\) for an in\-distribution test subject\. The gray shaded region indicates the test\-time input range \(frames 0\-75\), while the unshaded region shows the prediction horizon \(frames 76\-125\)\.

As shown in the figure, Meta\-iLaD \(red dash\-dot line\) consistently tracks the ground truth trajectory \(black solid line\) more accurately than IDF \(blue dashed line\) across all joints\. The performance gap becomes particularly evident in the prediction phase \(t\>\>75\), where IDF predictions exhibit significant drift and fail to capture the periodic patterns of the walking motion\. In contrast, Meta\-iLaD maintains better alignment with the true dynamics, successfully predicting the amplitude and phase of joint movements\. These qualitative results align with the quantitative findings in Table 2, demonstrating that identifiability of the environment variable𝐜\\mathbf\{c\}and dynamics functionFFis critical for accurate long\-term forecasting\.

![Refer to caption](https://arxiv.org/html/2610.27180v1/Fig/Figure_C.png)

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\.

## Appendix DEmpirical Validation of Corollary 1

Corollary 1 states that when the true dimensionddof the latent environment variable𝐜\\mathbf\{c\}is unknown, setting the estimated dimensiond^≥d\\hat\{d\}\\geq densures that the original latent variable𝐜\\mathbf\{c\}is recovered up to an affine transformation and permutation within a subset of the estimated latent space, while the remaining dimensions encode only noise\.

To validate this corollary, we conducted experiments on the CartPole dataset where the true environment variable𝐜\\mathbf\{c\}is one\-dimensional \(d=1d=1\)\. We trained Meta\-iLaD with varying estimated dimensionsd^∈\{2,3,4\}\\hat\{d\}\\in\\\{2,3,4\\\}and evaluated the learned latent environment dimensions by computing their Mean Correlation Coefficient \(MCC\) with the true environment parameter\. For visualization, we sorted the estimated dimensions\{c^1,c^2,…,c^d^\}\\\{\\hat\{c\}\_\{1\},\\hat\{c\}\_\{2\},\\ldots,\\hat\{c\}\_\{\\hat\{d\}\}\\\}in descending order of their MCC values\.

Results:As shown in Figure[8](https://arxiv.org/html/2610.27180#A4.F8), across all configurations, only the top\-ranked dimensionc^1\\hat\{c\}\_\{1\}\(with highest MCC\) exhibits a clear monotonic relationship with the true parameter, successfully achieving identifiability\. All remaining dimensions\{c^2,c^3,…,c^d^\}\\\{\\hat\{c\}\_\{2\},\\hat\{c\}\_\{3\},\\ldots,\\hat\{c\}\_\{\\hat\{d\}\}\\\}display random noise\-like patterns with no systematic structure:

- •d^=2\\hat\{d\}=2:c^1\\hat\{c\}\_\{1\}shows strong negative linear correlation with true𝐜\\mathbf\{c\};c^2\\hat\{c\}\_\{2\}contains only noise\.
- •d^=3\\hat\{d\}=3:c^1\\hat\{c\}\_\{1\}captures the true parameter;c^2,c^3\\hat\{c\}\_\{2\},\\hat\{c\}\_\{3\}are noisy\.
- •d^=4\\hat\{d\}=4:c^1\\hat\{c\}\_\{1\}remains informative;c^2,c^3,c^4\\hat\{c\}\_\{2\},\\hat\{c\}\_\{3\},\\hat\{c\}\_\{4\}encode noise\.

![Refer to caption](https://arxiv.org/html/2610.27180v1/Fig/Figure_D.png)Figure 8:Scatter plots of estimated latent environment dimensions\{c^1,c^2,c^3,c^4\}\\\{\\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 dimensionsd^∈\{2,3,4\}\\hat\{d\}\\in\\\{2,3,4\\\}with true dimensiond=1d=1\. Onlyc^1\\hat\{c\}\_\{1\}exhibits identifiable relationship with the true parameter \(clear monotonic trend\), while all remaining dimensions contain only noise, confirming Corollary 1\.Conclusion:These results empirically confirm Corollary 1: overestimating the dimensionality of𝐜\\mathbf\{c\}does not compromise identifiability\. The model successfully recovers the true one\-dimensional structure in exactly one dimension \(the highest\-MCC dimension\), while all excess dimensions naturally collapse to noise, demonstrating that Meta\-iLaD’s identifiability guarantees remain robust under dimension misspecification\.

## Appendix EBeyond Known Dynamics Environments

Finally, we ventured into settings where there are no prior knowledge about the underlying dynamics environments\. Based on the assumption of local stationarity \(dynamics environments do not change rapidly over time\), Meta\-iLaD usedkk\-shot preceding time windows as context samples for the current query window to predict\.

Baselines:Because identifiable latent dynamics utilizing predefined environment labelsuudoes not apply here, we considered NCTRL\([Song et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib24)\)as representatives of identifiable latent dynamicsℱ⁡\(𝐳<t,𝐜t\)\\mathcal\{F\}\(\\mathbf\{z\}\_\{<t\};\\mathbf\{c\}\_\{t\}\)where𝐜t\\mathbf\{c\}\_\{t\}is assumed to represent the latent states of a Markov process\.

Real data & results:We considered a real double\-pendulum dataset\([Wehenkel et al\., 2023](https://arxiv.org/html/2610.27180#bib.bib28)\)with unknown dynamics environments\. NCTRL was initialized with an arbitrary number of hidden states=10=10\. Since ground truth for𝐳t\\mathbf\{z\}\_\{t\}’s was unknown, we followed\([Willetts & Paige, 2021](https://arxiv.org/html/2610.27180#bib.bib29)\)to calculate MCC between𝐳t\\mathbf\{z\}\_\{t\}’s obtained between multiple runs\. As shown in Fig\.[9](https://arxiv.org/html/2610.27180#A5.F9)B, NCTRL demonstrated a strong MSE performance for reconstructing𝐱t\\mathbf\{x\}\_\{t\}’s, although less successful with the identifiability for the latent𝐳t\\mathbf\{z\}\_\{t\}’s\. In comparison, even tested in the more challenging setting of forecasting, Meta\-iLaD was able to achieve stronger results in both metrics\.

![Refer to caption](https://arxiv.org/html/2610.27180v1/Fig/Fig_real.png)

Figure 9:Results on real double pendulum data \(left\), including MCC on latent𝐳t\\mathbf\{z\}\_\{t\}’s \(middle\) and examples on reconstructing \(NCTRL\) or forecasting \(Meta\-iLaD\)𝐱t\\mathbf\{x\}\_\{t\}’s \(right\)\.This set of exploratory experiments demonstrated the real\-world versatility of Meta\-iLaD in problems where the underlying latent dynamics environments are not expected to rapidly switch\. Expectedly, incorrect context\-query pairing near the near the unknown boundary of environment switch will likely impact the performance of Meta\-iLaD\.

## Appendix FModel Architecture and Implementation Details

### F\.1E\.1\. Model Architecture

Table[6](https://arxiv.org/html/2610.27180#A6.T6)summarizes the neural network architectures for the main components of Meta\-iLaD\.

Table 6:Architecture details of Meta\-iLaD components\.
### F\.2E\.2\. Implementation Details

Training configuration\.All models were trained using Adam optimizer with learning rate1×10−31\\times 10^\{\-3\}for the main network and5×10−45\\times 10^\{\-4\}for the discriminator\.

Loss hyperparameters\.The total loss isℒTotal=−ℒELBO\+λTC​ℒTC\+λFlow​ℒFlow\\mathcal\{L\}\_\{\\text\{Total\}\}=\-\\mathcal\{L\}\_\{\\text\{ELBO\}\}\+\\lambda\_\{\\text\{TC\}\}\\mathcal\{L\}\_\{\\text\{TC\}\}\+\\lambda\_\{\\text\{Flow\}\}\\mathcal\{L\}\_\{\\text\{Flow\}\}\. We setλTC=5\.0\\lambda\_\{\\text\{TC\}\}=5\.0following FactorVAE\. ForλFlow\\lambda\_\{\\text\{Flow\}\}, we used a warm\-up annealing schedule: starting at 0, linearly increasing to 0\.1 over 50 epochs\.

Hardware\.All experiments were conducted on a single NVIDIA RTX 3090 GPU with 24GB memory\. Training time ranged from 1\-2 hours for synthetic physics system datasets\.