1 Introduction↩︎

Causal representation learning (CRL) seeks to recover latent causal variables and their structural relations from entangled observations [1]. This problem is especially compelling for time series, where causal structure underlies explanation, forecasting, and decision-making in domains such as climate science, biology, healthcare, and economics [2], [3]. In many temporal settings, however, the measured variables are not the causal variables of interest: video frames, sensor streams, and multivariate readouts are often nonlinear mixtures of latent dynamical factors. Without additional structure, those factors are not identifiable from observations alone [4], [5]. Temporal CRL therefore asks how temporal dependence, environmental variation, or interventions can break nonlinear mixing ambiguities and make latent causal variables identifiable.

1.0.0.1 Motivation.

Many scientific and engineered systems evolve in continuous time, are sampled irregularly, and are intrinsically stochastic. Stochastic differential equations (SDEs) provide a standard language for such dynamics [6], [7]. Continuous-time latent and controlled dynamics are also widely used in machine learning through neural ODEs, latent ODEs, neural controlled differential equations, and neural SDEs [8]–[11]. Continuous time also exposes objects that are blurred by a discrete-time view, including drift, diffusion, quadratic variation, and instantaneous dependence. This distinction is important for representation learning: the drift describes the deterministic law of motion, whereas the diffusion describes the scale and orientation of unresolved stochastic fluctuations.

In many latent dynamical systems, it is natural for the deterministic law of motion to remain stable across environments while the stochastic forcing changes. A bridge may have the same structural dynamics while wind and traffic loads change the amplitude and direction of stochastic excitation; a cardiovascular system may retain the same regulatory feedback loops while exercise or stress changes physiological variability; and a power grid may retain the same network dynamics while stochastic load fluctuations vary across demand or weather regimes. Such regime variation is often easier to observe than direct interventions on latent causal mechanisms. Such variation also provides a form of side information specific to continuous time: diffusion leaves a local quadratic-variation signature in the observed path, even though this signature is distorted by the unknown nonlinear mixing. This leads to the central identifiability question of this paper: when can diffusion shifts alone recover the latent coordinate system of a continuous-time SDE model from nonlinear observations?

1.0.0.2 Challenges.

Existing identifiable nonlinear ICA and temporal CRL results show that side information can break nonlinear mixing ambiguities, but their identifiability theory is largely built for discrete-time settings, using finite-lag transitions, autoregressive mechanisms, or temporal context on a grid [12]–[16]. Conversely, work on continuous-time SDE identification and causal discovery typically assumes that the stochastic state itself is observed, rather than seen only through an unknown nonlinear observation map [17]–[21]. The latent continuous-time setting is harder: the state is observed only after an unknown nonlinear mixing, and the target is not merely a predictive state but the latent coordinate system and its instantaneous drift-Jacobian graph. What is missing is a theorem-level CRL account for continuous-time latent SDE state coordinates observed only through such a mixing. The central difficulty is that diffusion covariance is not invariant under nonlinear coordinate changes: in observed coordinates, local quadratic variation is the latent covariance transformed by the Jacobian of the unknown mixing. A single diffusion regime therefore cannot determine which latent axes are the causal coordinates. Our key observation is that two diagonal diffusion regimes with distinct coordinate-wise variance ratios impose simultaneous constraints on the same unknown Jacobian. These constraints reduce the remaining ambiguity to permutation and scaling, which in turn makes instantaneous drift-graph recovery well defined.

1.0.0.3 Contributions.

We study additive-noise latent SDEs observed through an unknown nonlinear diffeomorphism across two environments that share the latent drift but differ in diffusion covariance. Our contributions are: We formulate this continuous-time CRL problem and identify diffusion shifts as the source of side information: two diagonal regimes with coordinate-wise distinct variance ratios are sufficient to anchor the latent coordinate system. We prove identifiability first for linear Ornstein–Uhlenbeck systems and then for general additive-noise latent SDEs: the latent coordinate system is recovered up to permutation and scaling without any sparsity assumption on the drift, and under mild smoothness the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation. We turn the theory into a practical two-stage estimator: diffusion-based disentanglement followed, when graph recovery is desired, by sparse drift regression. Synthetic experiments validate the predicted identifiability boundary, and a Hardanger Bridge monitoring study illustrates the estimator on real sensor trajectories.

2 Problem setup and target↩︎

We consider two environments, indexed by \(e\in\{1,2\}\). In environment \(e\), the latent state \(z_t^{(e)}\in\mathbb{R}^d\) evolves according to the additive-noise SDE \[\label{eq:generic95latent95sde95setup} dz_t^{(e)} = f(z_t^{(e)})\,dt + G_e\,dW_t^{(e)}, \qquad x_t^{(e)} = g(z_t^{(e)}),\tag{1}\] where \(x_t^{(e)}\in\mathbb{R}^d\) is the observed process, \(W_t^{(e)}\) is a standard \(d\)-dimensional Brownian motion, \(f:\mathbb{R}^d\to\mathbb{R}^d\) is the latent drift, \(G_e\in\mathbb{R}^{d\times d}\) is the environment-specific diffusion matrix, and \(g:\mathbb{R}^d\to\mathbb{R}^d\) is an unknown nonlinear observation map. The two environments share the same latent drift \(f\) and differ only through their diffusion covariances. In the linear Ornstein–Uhlenbeck special case studied first in Section 3.1, the drift takes the form \(f(z)=Az\) for some matrix \(A\in\mathbb{R}^{d\times d}\). Throughout, we write \(\Sigma_e := G_e G_e^\top\) for the latent diffusion covariance in environment \(e\).

The latent dynamics are autonomous: \(f\) has no explicit time dependence, so the causal mechanism is not explicitly time varying. In continuous-time causal models for SDEs, a parent relation is typically induced when one variable enters another variable’s drift or diffusion coefficient [18], [20], [22]. In our additive-noise setting, however, the diffusion matrices \(G_e\) are constant in \(z\), so no variable enters another variable’s diffusion coefficient. We therefore represent the latent causal structure by the drift alone. In the linear OU case, this reduces to the support pattern of \(A\), which is constant over time. In the general nonlinear case, we work with the instantaneous drift-Jacobian graph at state \(z\), namely the support pattern of \(Df(z)\); this graph may depend on the current state, but not explicitly on time.

Our goal is to characterize conditions under which the pair of observed path laws from the two environments is sufficient to recover the latent coordinate system up to permutation and coordinate-wise scaling, and to identify the corresponding latent causal structure up to the same permutation. The precise identifiability statements are given in Section 3.

3 Identifiability theory↩︎

3.1 Linear OU systems↩︎

We begin with the linear-drift case, where the identification mechanism can be seen most transparently. This case is a conceptual starting point rather than a restrictive assumption. In the OU setting, the diffusion comparison already contains the core disentanglement mechanism, and the state-space coverage step is automatic from the Gaussian transition law. This lets us isolate the role of the two regimes with minimal additional technical overhead.

3.1.0.1 Assumptions.

We state the shared assumptions for two observationally equivalent latent representations: the reference parametrization and an alternative tilded parametrization. Both the OU theorem below and the general nonlinear theorem in Section 3.2 rely on these four assumptions.

Assumptions (S1) and (S2) are standard in identifiable nonlinear ICA and causal representation learning: one posits an invertible nonlinear mixing and asks whether two latent parametrizations inducing the same observed law must be equivalent, as in [12]–[15], [23], [24]. Assumption (S3) is a nondegeneracy condition on the additive noise. It fixes coordinatewise diffusion geometry and rules out singular directions. Related diagonal or block-diagonal noise restrictions are also common in continuous-time structure learning and path-space causal discovery [18], [20], [22].

Assumption (S4) is the separating condition: each latent coordinate carries a distinct diffusion-ratio fingerprint across environments. This is generically mild: failure occurs only when two ratios coincide exactly, a lower-dimensional subset of the diagonal covariance parameter space. It is also the continuous-time analogue of the variability conditions that drive identifiability in temporal and multi-environment CRL [15], [23], [25]–[27]. In the OU case, no additional support assumption is needed, because every nondegenerate Gaussian transition has full support on \(\mathbb{R}^d\).

Throughout this section, \(\varphi = \tilde{g}^{-1}\circ g\) denotes the change of coordinates between two observationally equivalent latent parametrizations: \(\tilde{z}=\varphi(z)\) describes the same underlying state in the alternative coordinate system. We write \(D\varphi(z)\) for its Jacobian. The main identifiability question is whether \(\varphi\) must reduce to a monomial affine transformation.

3.1.0.2 Proof sketch.

The full proof is deferred to Appendix 10. Equality of observed path laws allows a coupling with \(\tilde{z}_t^{(e)}=\varphi(z_t^{(e)})\) almost surely in each regime. Itô’s formula and quadratic variation give \(D\varphi(z_t^{(e)})\Sigma_e D\varphi(z_t^{(e)})^\top=\tilde{\Sigma}_e\) along trajectories. For OU dynamics, the latent state at any positive time has a strictly positive Gaussian density on \(\mathbb{R}^d\), so this identity extends to every state \(z\in\mathbb{R}^d\). Whitening with respect to the first regime reduces the problem to an orthogonal matrix field. The second regime and the simple-spectrum condition in (S4) force that field to be a signed permutation at every \(z\). Continuity then makes the signed permutation constant, so \(D\varphi\) is a constant monomial matrix and \(\varphi\) is affine; substituting into the drift relation gives \(\tilde{A} L=LA\), yielding the coordinate-system and graph statements.

3.2 General latent SDEs↩︎

We now turn to nonlinear drifts. As in the OU case, \(\varphi := \tilde{g}^{-1}\circ g\) denotes the coordinate change between two observationally equivalent latent parametrizations, with Jacobian \(D\varphi(z)\). The diffusion comparison still provides the core algebraic step. Only the support argument changes: an identity first obtained along sample paths must be extended to \(\mathbb{R}^d\) without explicit Gaussian transition densities.

The shared assumptions (S1)–(S4) remain in force, and (S5) ensures well-defined continuous semimartingales and the open-set positivity step. This is a standard additive-noise SDE regularity condition: it requires continuous drifts and weak well-posedness with continuous sample paths, but not strong solutions or closed-form transition densities. Appendix 10.5 gives further discussion.

3.2.0.1 Proof sketch.

The proof, deferred to Appendix 10, reuses the OU argument. The coupling, quadratic-variation comparison, and whitening/simple-spectrum steps are unchanged. The new ingredient is the open-set positivity lemma, which upgrades the along-trajectory covariance identity to a pointwise identity on \(\mathbb{R}^d\) without Gaussian transition densities. The rest follows as in the OU case: \(D\varphi\) is a constant monomial matrix, hence \(\varphi(z)=Lz+c\), and the same semimartingale comparison gives \(\tilde{f}(Lz+c)=Lf(z)\).

Under (S1)–(S5), Theorem 2 shows that any observationally equivalent latent coordinate change must be a monomial affine map. When \(f\) and \(\tilde{f}\) are \(C^1\), differentiating \(\tilde{f}(Lz+c)=Lf(z)\) yields the drift-Jacobian graph statement.

4 Two-stage estimation↩︎

We translate the identifiability results into a two-stage estimator. Stage 1 learns the latent coordinate system from short-time transition statistics across the two regimes. Stage 2 is optional and estimates a sparse drift graph when explicit graph recovery is desired.

4.1 Model parameterization↩︎

We observe trajectories \(\{x_{t_0}^{(e)},x_{t_1}^{(e)},\dots,x_{t_N}^{(e)}\}\) from two regimes \(e\in\{1,2\}\) at uniform intervals \(\Delta t\). The encoder \(h_\theta\) maps observations into a learned latent coordinate system, playing the estimator-side role of \(\tilde{g}^{-1}\) in Sections 3.1–3.2. We keep the tilde notation \((\tilde{z},\tilde{f},\tilde{\Sigma}_e)\) for learned coordinates, drift, and diffusion parameters, with \(\theta,\psi\) denoting parametric components. Our learnable model consists of three components:

• an encoder \(h_\theta:\mathbb{R}^d\to\mathbb{R}^d\), implemented as an MLP in our experiments, with \(\tilde{z}_t=h_\theta(x_t)\);

• a differentiable drift model \(\tilde{f}_\psi:\mathbb{R}^d\to\mathbb{R}^d\), shared across regimes and implemented as either a linear map or an MLP;

• diagonal per-regime diffusion parameters \(\tilde{\Sigma}_e=\operatorname{diag}(\tilde{\sigma}_{e,1}^2,\ldots,\tilde{\sigma}_{e,d}^2)\), parameterized by \(\log \tilde{\sigma}_{e,i}^2\) to ensure positivity.

We do not enforce \(h_\theta\) to be globally invertible during training, so the Stage 1 likelihood below is interpreted as a pseudo-likelihood in general.

4.2 Stage 1: short-time transition fitting↩︎

Stage 1 uses the short-time behavior of additive-noise SDEs. For small \(\Delta t\), the Euler–Maruyama discretization gives a Gaussian approximation to one-step transitions [6], [7]. In regime \(e\), the encoded-space working model is \[\label{eq:EM95transition} \tilde{z}_{t+\Delta t}\mid \tilde{z}_t,e \sim \mathcal{N}\!\Big(\tilde{z}_t + \tilde{f}_\psi(\tilde{z}_t)\,\Delta t, \tilde{\Sigma}_e\,\Delta t\Big).\tag{6}\] Equivalently, \[\tilde{z}_{t+\Delta t} = \tilde{z}_t + \tilde{f}_\psi(\tilde{z}_t)\,\Delta t + \varepsilon_t^{(e)}, \qquad \varepsilon_t^{(e)}\sim \mathcal{N}(0,\tilde{\Sigma}_e\,\Delta t).\] The diagonal \(\tilde{\Sigma}_e\) makes the one-step residual coordinates conditionally independent under this model.

If the encoded states \(\tilde{z}_t\) were observed, the per-transition negative log-likelihood, up to the additive constant \(\tfrac{d}{2}\log(2\pi\Delta t)\), is \[\label{eq:nll95ztilde} \ell_e(\tilde{z}_t,\tilde{z}_{t+\Delta t}) = \frac{1}{2}\sum_{i=1}^d \log \tilde{\sigma}_{e,i}^2 + \frac{1}{2}\sum_{i=1}^d \frac{\big(\tilde{z}_{t+\Delta t,i}-\tilde{z}_{t,i}-\tilde{f}_{\psi,i}(\tilde{z}_t)\,\Delta t\big)^2}{\tilde{\sigma}_{e,i}^2\,\Delta t}.\tag{7}\] This is the Gaussian transition score for 6 .

4.2.0.1 Observation-space objective.

We observe only \(x_t\) and set \(\tilde{z}_t=h_\theta(x_t)\). If \(h_\theta\) is a \(C^1\) diffeomorphism, the change-of-variables formula gives \[p_\theta(x_{t+\Delta t}\mid x_t,e) = p_{\tilde{z},e}\big(h_\theta(x_{t+\Delta t})\mid h_\theta(x_t)\big) \,\big|\det Dh_\theta(x_{t+\Delta t})\big|,\] where \(p_{\tilde{z},e}\) is the Gaussian density in 6 and \(Dh_\theta\) is the encoder Jacobian. The observed-data per-transition negative log-likelihood, up to the same additive constant, is \[\label{eq:nll95obs} s_e(x_t,x_{t+\Delta t}) = \ell_e\big(h_\theta(x_t),h_\theta(x_{t+\Delta t})\big) - \log\big|\det Dh_\theta(x_{t+\Delta t})\big|.\tag{8}\]

In implementation, \(h_\theta\) is differentiable but need not be globally invertible, so 8 is generally a transition pseudo-likelihood. The first term fits encoded increments to the diagonal Gaussian transition model, while the log-determinant term provides a local volume correction. If \(h_\theta\) is a \(C^1\) diffeomorphism, the same score is the exact Euler–Maruyama one-step negative log-likelihood.

4.2.0.2 Stage 1 objective.

We average the per-transition score over regimes and sampled consecutive pairs: \[\label{eq:stage195loss} \mathcal{L}_{\mathrm{S1}}(\theta,\psi,\{\tilde{\sigma}_{e,i}^2\}) = \frac{1}{2}\sum_{e=1}^{2} \mathbb{E}_{(x_t,x_{t+\Delta t})\sim\mathcal{D}_e} \Big[ s_e(x_t,x_{t+\Delta t}) \Big],\tag{9}\] where \(\mathcal{D}_e\) is the empirical distribution of consecutive pairs in regime \(e\). Stage 1 jointly learns \((\theta,\psi,\{\tilde{\sigma}_{e,i}^2\})\).

4.2.0.3 Why this objective matches the theory.

Over a short interval \(\Delta t\), the drift contributes a mean shift of order \(O(\Delta t)\), whereas the stochastic increment has standard deviation \(O(\sqrt{\Delta t})\) and covariance \(\tilde{\Sigma}_e\,\Delta t\). Diffusion therefore dominates the short-time second-order behavior. Our theory shows that this regime-dependent diagonal diffusion structure anchors the latent coordinate system. Thus 7 –9 trains the encoder to find coordinates in which short-time residual increments \[\tilde{z}_{t+\Delta t}-\tilde{z}_t-\tilde{f}_\psi(\tilde{z}_t)\,\Delta t\] fit a shared-drift, regime-specific diagonal Gaussian model. This is the finite-step analogue of the two-regime identifiability mechanism in Section 3.

4.3 Stage 2: sparse drift regression↩︎

Stage 2 is optional and used only to estimate a sparse causal graph. For latent disentanglement alone, Stage 1 is the relevant part of the pipeline.

4.3.0.1 Velocity targets.

With the encoder \(h_\theta\) frozen, we form velocity targets from encoded transitions: \[\label{eq:velocity95target} \hat{v}_t = \frac{h_\theta(x_{t+\Delta t'})-h_\theta(x_t)}{\Delta t'},\tag{10}\] where \(\Delta t'=s\Delta t\) for an integer stride \(s\ge 1\). Larger strides reduce target variance, \(\operatorname{Var}(\hat{v}_{t,i})\approx \tilde{\sigma}_{e,i}^2/\Delta t'\), at the cost of \(O(\Delta t')\) bias from time-averaging the drift.

4.3.0.2 Sparse drift estimation.

We reinitialize \(\tilde{f}_\psi\) and fit it to the velocity targets with a Jacobian sparsity penalty: \[\label{eq:stage295loss} \mathcal{L}_{\mathrm{S2}}(\psi) = \underbrace{ \frac{1}{2}\sum_{e=1}^{2} \mathbb{E}_{(\tilde{z}_t,\hat{v}_t)\sim\mathcal{D}_e} \Big\|\tilde{f}_\psi(\tilde{z}_t)-\hat{v}_t\Big\|^2 }_{\text{velocity MSE}} + \lambda_{\mathrm{sparse}} \underbrace{ \mathbb{E}_{\tilde{z}\sim\mathcal{D}_1\cup\mathcal{D}_2} \sum_{i=1}^d\sum_{j=1}^d \bigg|\frac{\partial \tilde{f}_{\psi,i}}{\partial \tilde{z}_j}(\tilde{z})\bigg| }_{\ell_1\text{ drift-Jacobian penalty}}.\tag{11}\] This stage is estimator-side only and is used when a stable sparse graph estimate is needed.

4.3.0.3 Graph extraction.

By Corollary 1, the learned drift-Jacobian zero pattern is meaningful only up to the latent permutation induced by \(\Pi\). In practice, the binary adjacency matrix is obtained by thresholding the mean absolute Jacobian at a fixed threshold \(\tau>0\): \[\overline{|D\tilde{f}_\psi|} := \mathbb{E}_{\tilde{z}\sim\mathcal{D}_1\cup\mathcal{D}_2} \big[|D\tilde{f}_\psi(\tilde{z})|\big].\]

4.3.0.4 Summary.

Algorithm 3 in Appendix 11 summarizes the training and graph-extraction procedure.

5 Experiments↩︎

5.1 Synthetic experiments↩︎

We test whether the estimator matches the identifiability boundary predicted by the theory. We use three latent drift families: dense linear OU, sparse linear OU, and nonlinear sparse drift. For each family, we compare three diffusion-regime conditions: Distinct ratios, where two environments have coordinate-wise distinct diffusion-variance ratios and hence satisfy the separation requirement in Assumption (S4); One regime, which removes environment variation; and Proportional diffusions, which keeps two environments but makes all variance ratios equal. The theory predicts coordinate recovery only under Distinct ratios.

Each drift family is tested at \(d\in\{5,7\}\) under two nonlinear mixing functions. The main-text setting uses a three-layer MLP with leaky-\(\tanh\) activation \(\phi(z)=\tanh(z)+0.1z\), yielding a \(C^\infty\) diffeomorphism that satisfies the theory’s smoothness assumptions. We also test LeakyReLU mixing as a robustness check beyond the \(C^2\) setting. To keep the main text focused, we report the \(d=5\) leaky-\(\tanh\) results below; LeakyReLU and \(d=7\) results are in Appendix 13, with the data-generating process, training pipeline, hyperparameters, and metric definitions in Appendix 12 (Table 4). We report mean correlation coefficient (MCC) for one-to-one absolute-correlation alignment between learned and true coordinates; monomial score (Mon.) for how close the mean absolute Jacobian of \(h\circ g\) is to a permutation-scaling matrix; and, for sparse linear and nonlinear drifts, graph match rate (GMR) for entrywise agreement between the thresholded learned drift-Jacobian graph and the ground-truth graph. Higher is better for all metrics.

Table 1 summarizes the quantitative results. Across all three drift families, Distinct ratios gives near-perfect MCC and high monomial scores, whereas both controls show substantially lower coordinate recovery and less monomial encoder Jacobians. In the sparse linear and nonlinear families, Distinct ratios also recovers the drift graph almost exactly. GMR is omitted for the dense linear family because sparse graph recovery is not the target there. The same qualitative pattern holds at \(d=7\) and under LeakyReLU mixing (Appendix 13).

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Figures 1 and 2 give qualitative diagnostics for the nonlinear \(d=5\) setting, covering both latent disentanglement and graph recovery. Each column shows one regime condition, using the same randomly selected training seed across conditions. Figure 1 shows that Distinct ratios gives a near-permutation alignment between true and learned coordinates, while the controls do not. Figure 2 shows that the corresponding \(\overline{|D\varphi|}\) matrix is near-monomial and the drift graph is recovered exactly under Distinct ratios; the controls produce denser encoder Jacobians and spurious or missed graph edges. Analogous figures for other drift families, dimensions, and mixing functions are in Appendix 13.

5.2 Real-world case study↩︎

We further test the method on operational accelerometer recordings from the Hardanger Bridge, a long-span suspension bridge instrumented for long-term wind and vibration monitoring [28], [29]. The model is trained only on bridge-deck acceleration channels; wind measurements define regimes but are not used as model inputs. This also probes robustness beyond the square diffeomorphic setting of Assumption (S1): the input has \(16\) sensor channels and the learned latent space has dimension \(d=5\). To obtain a real-world analogue of the diffusion-separation condition in Assumption (S4), we split segments within a matched wind-speed bin by turbulence intensity (TI). This keeps mean wind speed fixed while changing stochastic excitation, so low- and high-TI segments are intended to share structural dynamics while carrying different diffusion fingerprints.

We compare three regime constructions: TI-split regimes (ours), using the low- and high-TI groups as two regimes; One regime, merging the two TI groups and removing regime-specific diffusion variation; and Shuffled TI labels, preserving the two-regime training format but destroying the turbulence-based contrast. Since ground-truth latent coordinates and graphs are unavailable, we evaluate cross-seed reproducibility rather than accuracy, using pairwise mean matched absolute correlation coefficient (MCC) and graph match rate (GMR). Table 2 reports results over \(K=10\) seeds at \(d=5\). The TI-split construction gives the highest MCC and GMR, consistent with the synthetic gap between identifiable diffusion shifts and controls. Appendix 14 gives the dataset construction, training protocol, aggregate diagnostics, Assumption (S4)-style diffusion-fingerprint checks, a representative drift-Jacobian interpretation, and limitations of the real-world diagnostic.

6 Related work↩︎

Identifiable nonlinear ICA and temporal CRL show that auxiliary structure, such as nonstationarity, temporal dependence, interventions, grouping, or environment variation, can break nonlinear mixing ambiguities [12]–[16], [30], [31]. Their theorem-level temporal results, however, are largely discrete-time, based on finite-lag transitions, autoregressive mechanisms, or temporal context on a grid. Continuous-time SDE identification and causal discovery usually assume that the stochastic state is directly observed [17]–[21]; the closest latent continuous-time work treats stochastic point processes or system-level parameters rather than latent SDE state coordinates [32]–[34]. We address this gap by identifying the latent coordinate system and drift-Jacobian graph of continuous-time SDE states observed through an unknown nonlinear map, using two environments with shared drift and different diffusion covariances. Appendix 9 gives a fuller comparison.

7 Discussion and limitations↩︎

This paper gives a theorem-level account of identifiable CRL for continuous-time latent SDEs. The main message is that multi-regime diffusion structure can anchor the latent coordinate system: with shared drift and diagonal diffusion covariances whose coordinate-wise variance ratios are distinct, nonlinear observations identify the coordinates up to permutation and scaling, and the instantaneous drift-Jacobian graph follows up to the same permutation, without drift sparsity. This is a step toward CRL for continuous latent dynamics, where most temporal identifiability results are discrete-time and most continuous-time SDE identification assumes observed states. The assumptions are intentionally simple: an invertible observation map, additive state-independent diffusion, a shared drift, and separated diffusion regimes. The diffeomorphism assumption is the strongest idealization, excluding non-invertible or nonsmooth observation maps. Our LeakyReLU experiments and Hardanger study probe robustness beyond the smooth square diffeomorphic setting. Extending the theory to weaker observation maps, state-dependent noise, and imperfect regime labels remains important future work.

8 Notation↩︎

Table 3 summarizes the principal notation used throughout the paper.

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9 Extended related work↩︎

9.0.0.1 Identifiable nonlinear ICA.

General nonlinear mixing is not identifiable without extra structure [4], [5]. Identifiable nonlinear ICA overcomes this via nonstationarity, temporal dependence, or auxiliary variables [12], [13], [35], [36], with structured extensions to hidden Markov and structured dependent sources [37], [38]. This line provides the main conceptual backdrop for CRL: the nonlinear mixing ambiguity can be broken once suitable side information is available.

9.0.0.2 Temporal CRL in discrete time.

A closely related CRL line studies nonlinear mixtures of latent temporal processes in discrete time. LEAP identifies latent temporal causal processes under nonstationary noise [14]; TDRL studies nonparametric time-delayed causal processes [15]; NCTRL addresses unknown nonstationarity [25]; and CtrlNS uses sparse-transition structure to recover nonstationary latent dynamics without known domain variables [16]. A parallel branch uses stronger side information: CITRIS and iCITRIS leverage interventions [30], [39], G-CaRL uses grouping information [40], and CaRiNG allows non-invertible temporal generation processes by using temporal context to recover lost information [31]. More recent extensions study instantaneous dependence via sparse latent dynamics [23], hierarchical temporal latents [41], and continuously evolving mechanisms rather than discrete switches [42]. These papers are close in spirit because they identify latent causal processes from sequential observations, but their identifying arguments are built around discrete observations, transition factorizations, or auxiliary mechanism structure rather than continuous-time SDE diffusion structure. Our object of interest is a continuous-time latent SDE path, and our proof leverages diffusion covariance changes rather than lagged transition sparsity or intervention metadata.

9.0.0.3 Multi-environment identification and partial observability in CRL.

Beyond temporal structure, environment heterogeneity and observation structure have emerged as important sources of identifiability in CRL. Multi-environment results address unknown interventions [26], cross-domain invariance conditions [43], general environment families [27], [44], [45], and the limits of purely observational identification [46]. Multi-view and partial observability settings offer a complementary route by exploiting additional observation structure [24], [47]. Recent work has also sought unifying perspectives connecting invariance-based CRL formulations with exchangeability and causal discovery [48], [49]. Our work shares the multi-environment motivation, but the identifying structure is specifically continuous-time: the drift is invariant across environments while the diffusion covariance changes, and pairwise distinct diffusion-variance ratios pin down the latent coordinates up to permutation and scaling.

9.0.0.4 Continuous-time identification and causal discovery for stochastic processes.

A separate literature studies what can be identified when the stochastic system itself is observed. Wang et al. analyze generator identifiability for linear SDEs [17]. SCOTCH studies observed-space structure learning with neural SDEs under irregular sampling [18]. Guan et al. identify linear drift, diffusion, and causal structure from temporal marginals [19]. Manten et al. derive path-space conditional independence constraints and corresponding tests for SDE models [20], and extend the graphical treatment to asymmetric independence on path spaces [22]. Zweig et al. analyze identifiability of interventional SDEs from stationary distributions [21]. Among continuous-time latent-process CRL papers, Ren et al. [32] study identifiable CRL for continuous-time latent stochastic point processes, not latent SDE state coordinates. Closely related dynamical-systems papers connect CRL with trajectory-specific or system-parameter identification [33], [34]. Our paper differs from all of these in the recovery target: we identify continuously evolving latent state coordinates and their instantaneous Jacobian graph from unknown nonlinear mixtures, using two environments with shared drift and changed diffusion.

10 Deferred technical material↩︎

10.1 Auxiliary lemmas↩︎

10.2 Proof of Theorem 1↩︎

10.3 Open-set positivity for additive-noise SDEs↩︎

10.4 Proof of Theorem 2 and Corollary 1↩︎

10.5 Why weak well-posedness suffices↩︎

11 Estimation algorithm↩︎

Algorithm 3 summarizes the two-stage estimator used in the experiments. Stage 1 fits the shared-drift, regime-specific diagonal-diffusion transition model in the learned coordinate system. Stage 2 is optional and is used only when a sparse drift-Jacobian graph estimate is needed.

12 Simulation setup↩︎

This section specifies the data-generating process, observation model, training procedure, evaluation metrics, and hyperparameters for all simulation experiments.

12.1 Latent process↩︎

Each experiment generates \(N\) independent trajectories from a \(d\)-dimensional latent stochastic differential equation \[\mathrm{d}z_t^{(e)} = f(z_t^{(e)})\,\mathrm{d}t + \Sigma_e^{1/2}\,\mathrm{d}W_t^{(e)},\] where \(f\colon \mathbb{R}^d \to \mathbb{R}^d\) is the latent drift, \(\Sigma_e = \operatorname{diag}(\sigma_{e,1}^2,\dots,\sigma_{e,d}^2)\) is the environment-specific diagonal diffusion covariance, and \(W_t^{(e)}\) is a standard \(d\)-dimensional Brownian motion. Trajectories are discretized via the Euler–Maruyama scheme with step size \(\Delta t\). In the implementation, we set \(n_{\mathrm{steps}}=\lfloor T/\Delta t\rfloor=20\) and store \(n_{\mathrm{steps}}\) points including the initial condition, yielding \(n_{\mathrm{steps}}-1=19\) adjacent one-step transition pairs. Initial conditions are drawn i.i.d.from \(\mathcal{N}(0,\, 0.25\,I_d)\).

We consider three drift families: dense linear OU, sparse linear OU, and nonlinear sparse drift, whose constructions are detailed in Sections 13.1 to 13.3. Within each drift family and dimension, the drift \(f\) is fixed and shared across all regime conditions and all training seeds.

12.2 Regime conditions and diffusion covariances↩︎

Each drift family is evaluated under three regime conditions. These conditions vary only the number and choice of diagonal diffusion covariances; all other components are held fixed. In all conditions, the first environment uses \[\sigma_{1,j}^2 = 0.5\,j, \qquad j = 1, \dots, d,\] i.e.\(\Sigma_1 = \operatorname{diag}(0.5,\,1.0,\,1.5,\dots,0.5d)\).

12.2.0.1 Distinct ratios (identifiable).

Two environments (\(e \in \{1,2\}\)) with \(\Sigma_2\) chosen so that the variance ratios \(\sigma_{2,j}^2 / \sigma_{1,j}^2\) are pairwise distinct across all coordinates \(j = 1, \dots, d\). The diagonal entries of \(\Sigma_2\) are \[\begin{align} d=5\colon \quad \sigma_{2,j}^2 &= (2.0,\; 1.5,\; 0.8,\; 0.5,\; 0.4), \\ d=7\colon \quad \sigma_{2,j}^2 &= (2.5,\; 0.3,\; 2.1,\; 0.35,\; 1.8,\; 0.43,\; 1.6). \end{align}\]

12.2.0.2 One regime (control).

A single environment with diffusion covariance \(\Sigma_1\) only. Since there is only one environment, the distinct-ratios condition is not applicable.

12.2.0.3 Proportional diffusions (control).

Two environments with \(\sigma_{2,j}^2 = 2\,\sigma_{1,j}^2\) for all \(j\), so that all variance ratios \(\sigma_{2,j}^2/\sigma_{1,j}^2\) equal \(2\). This violates the two-regime separation requirement in Assumption (S4), since all ratios are identical.

For a fixed dimension \(d\), the same diffusion-covariance choices are used across all three drift families.

12.3 Observation model↩︎

The learner observes \(x_t^{(e)} = g(z_t^{(e)})\). We implement \(g\colon \mathbb{R}^d \to \mathbb{R}^d\) as an invertible MLP mixing map with orthogonal linear layers and componentwise activations. We test two choices of activation within the same \(m\)-layer orthogonal MLP architecture: \[y_1=\phi(R_1z+b_1), \qquad y_k=\phi(R_k y_{k-1}+b_k)\;\;(2\le k\le m-1), \qquad g(z)=R_m y_{m-1},\] where each \(R_k \in \mathrm{O}(d)\) is a random orthogonal matrix (drawn via the Haar measure), \(b_k \sim \mathcal{N}(0, I_d)\) for \(k=1,\ldots,m-1\) are bias vectors, and \(\phi\) is a componentwise nonlinearity. The final layer is a pure rotation with no bias or nonlinearity. In all experiments \(m = 3\).

12.3.0.1 Leaky-\(\tanh\) (main text).

\(\phi(z) = \tanh(z) + 0.1\,z\). Since \(\phi'(z) = \operatorname{sech}^2(z) + 0.1 > 0\) everywhere and \(\phi \in C^\infty\), the composition \(g\) is a \(C^\infty\) diffeomorphism, consistent with the smoothness assumptions of our identifiability results.

12.3.0.2 LeakyReLU (appendix).

\(\phi(z) = \max(z,\, 0.2\,z)\). This map is invertible and piecewise linear, but it is not \(C^1\) at zero and hence not \(C^2\), providing a robustness check beyond the smoothness assumptions.

The orthogonal matrices and biases are drawn with a fixed seed, so the same \(g\) is used across all regime conditions and training seeds within each fixed drift family, dimension, and activation choice.

12.4 Training procedure↩︎

We follow the two-stage procedure described in Section 4 and summarized in Algorithm 3.

12.4.0.1 Stage 1: Short-time transition fitting.

The encoder \(h_\theta\colon \mathbb{R}^d \to \mathbb{R}^d\) is a fully connected MLP with hidden dimension 256, four hidden layers, and ELU activations. We minimize the Stage 1 objective \(\mathcal{L}_{\mathrm{S1}}\) (Eq. 9 ), including the change-of-variables log-determinant term from Eq. 8 : \[-\log\big|\det Dh_\theta(x_{t+\Delta t})\big|.\] The log-determinant term is the local volume correction in the observation-space pseudo-likelihood and penalizes collapse of the encoder Jacobian. All parameters \((\theta, \psi, \{\tilde{\sigma}_{e,i}^2\})\) are optimized jointly with Adam (learning rate \(10^{-3}\), batch size \(4{,}096\)) for \(20{,}000\) epochs.

12.4.0.2 Stage 2: Sparse drift-Jacobian graph recovery (sparse linear and nonlinear only).

After Stage 1, the encoder is frozen and we minimize \(\mathcal{L}_{\mathrm{S2}}\) (Eq. 11 ) to fit a drift model to the encoded velocity targets. For the sparse linear family, the drift model is a linear map \(\tilde{f}_\psi(\tilde{z}) = \hat{A}\tilde{z}\), with sparsity penalty \(\sum_{i,j}|\hat{A}_{ij}|\). For the nonlinear family, \(\tilde{f}_\psi\) is a three-layer MLP with ELU activations, with the Jacobian \(L_1\) penalty in Eq. 11 evaluated on encoded samples. We optimize with Adam (learning rate \(5 \times 10^{-4}\)). Dense linear experiments skip Stage 2 because the drift is dense by design, making sparse drift-Jacobian graph recovery inapplicable.

12.4.0.3 Computational resources.

The reported experiments were run as standard Slurm jobs requesting one NVIDIA L40S GPU and 32 GB memory per job. For a fixed setting, the five training seeds used in the synthetic tables typically completed in a few minutes. The implementation does not rely on specialized hardware and can also run on CPU or on other CUDA GPUs, with CPU runs expected to be slower.

12.4.0.4 Graph thresholding.

Following Algorithm 3, the learned drift-Jacobian graph is obtained by thresholding the mean absolute Jacobian \(\overline{|D\tilde{f}_\psi|}\) at a fixed threshold \(\tau = 0.05\) in the encoded space, before any scaling correction. The resulting binary adjacency is then permuted back to the original coordinate ordering for comparison with the ground truth.

12.4.0.5 Evaluation.

Each experiment is repeated over five independent training seeds. We report mean \(\pm\) standard deviation of three metrics defined below. For diagnostic plots (scatter matrices, Jacobian heatmaps, graph comparisons), we use one representative seed, held fixed across the regime conditions shown in each figure.

12.4.0.5.1 Mean correlation coefficient (MCC).

Given \(n\) samples of the true latent coordinates \(z \in \mathbb{R}^{n \times d}\) and the learned coordinates \(\tilde{z} \in \mathbb{R}^{n \times d}\), we compute the \(d \times d\) absolute Pearson correlation matrix \(C\) with entries \(C_{ij} = \bigl|\operatorname{corr}(z_{\cdot,i},\, \tilde{z}_{\cdot,j})\bigr|\). The optimal one-to-one assignment \(\pi^*\) is found by the Hungarian algorithm: \[\pi^* = \arg\max_{\pi \in S_d} \sum_{i=1}^{d} C_{i,\pi(i)}, \qquad \mathrm{MCC} = \frac{1}{d}\sum_{i=1}^{d} C_{i,\pi^*(i)}.\] A value near 1 indicates that each learned coordinate aligns with exactly one true latent.

12.4.0.5.2 Monomial score (Mon.).

Let \(\bar{J} = \overline{|D(h_\theta \circ g)(z)|}\) denote the mean absolute Jacobian of the composition of the encoder and observation map, a \(d \times d\) non-negative matrix. We normalize \(\bar{J}\) along rows and columns separately: \[R_{ij} = \frac{\bar{J}_{ij}}{\sum_{k} \bar{J}_{ik}}, \qquad Q_{ij} = \frac{\bar{J}_{ij}}{\sum_{k} \bar{J}_{kj}},\] and define the row and column concentration as the mean of the row-wise and column-wise maxima: \[\rho_{\mathrm{row}} = \frac{1}{d}\sum_{i=1}^{d} \max_{j}\, R_{ij}, \qquad \rho_{\mathrm{col}} = \frac{1}{d}\sum_{j=1}^{d} \max_{i}\, Q_{ij}.\] The monomial score is \(\mathrm{Mon.} = \tfrac{1}{2}(\rho_{\mathrm{row}} + \rho_{\mathrm{col}})\). For a perfect monomial matrix every row and column has exactly one nonzero entry, giving \(\mathrm{Mon.} = 1\); a uniform matrix yields \(\mathrm{Mon.} = 1/d\).

12.4.0.5.3 Graph match rate (GMR).

The mean absolute Jacobian of the learned drift, \(\overline{|D\tilde{f}_\psi|}\), is binarized at threshold \(\tau\): \(\hat{G}^{\mathrm{enc}}_{ij} = \mathbf{1}\!\bigl[\,\overline{|D\tilde{f}_\psi|}_{ij} > \tau\,\bigr]\). Let \(p\) denote the encoder permutation extracted from the encoder Jacobian, where \(p(k)\) is the true coordinate matched to encoded coordinate \(k\). This learned drift-Jacobian adjacency is then permuted back to the original coordinate ordering: \(\hat{G}_{ij} = \hat{G}^{\mathrm{enc}}_{p^{-1}(i),\,p^{-1}(j)}\). In practice, \(p\) is obtained by applying the Hungarian algorithm to the mean absolute encoder Jacobian \(\bar{J}\). The ground-truth adjacency \(G^*\) includes self-loops (\(G^*_{ii} = 1\) for all \(i\)) because every coordinate has a self-regulating drift term. The graph match rate is the fraction of all \(d^2\) entries that agree: \[\mathrm{GMR} = \frac{1}{d^2}\sum_{i,j=1}^{d} \mathbf{1}\!\bigl[\hat{G}_{ij} = G^*_{ij}\bigr].\]

Table 4 summarizes the key simulation and training hyperparameters. Values that differ across drift families are noted explicitly.

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\(^\dagger\)Nonlinear experiments only; sparse linear uses a linear drift model \(\tilde{f}_\psi(\tilde{z}) = \hat{A}\tilde{z}\).
\(^\ddagger\)Leaky-\(\tanh\) mixing; the LeakyReLU experiments use \(\lambda_{\mathrm{sparse}} = 0.9\) for both \(d=5\) and \(d=7\).

13 Additional simulation results↩︎

Table 5 reports results under the same three drift families and regime conditions as in the main text (Table 1), but with a three-layer MLP using LeakyReLU activations as the mixing function. The same identifiability gap persists: Distinct ratios yields near-perfect disentanglement across all drift families, whereas both controls substantially reduce MCC and monomial score.

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Tables 6 and 7 extend the analysis to \(d=7\) under leaky-\(\tanh\) and LeakyReLU mixing, respectively. The same qualitative pattern holds: Distinct ratios yields near-perfect disentanglement across all three drift families, whereas both controls substantially reduce coordinate recovery.

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13.1 Dense linear drift↩︎

The dense linear experiments use an Ornstein–Uhlenbeck process \(\mathrm{d}z = Az\,\mathrm{d}t + G\,\mathrm{d}W\) with a symmetric negative-definite drift matrix \(A\). This family evaluates coordinate recovery without using graph sparsity as an evaluation target. We construct \(A\) as \[A = -(MM^\top + \lambda I), \qquad M_{ij} \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(0,1),\] with \(\lambda = 0.5\) to ensure a stability margin. The random matrix \(M\) is drawn with a fixed seed, so the same \(A\) is shared across all training seeds and all three regime conditions. For \(d=5\) the resulting drift matrix, rounded to two decimals, is \[A = \begin{pmatrix} -4.16 & 0.00 & -2.79 & 1.30 & -4.78 \\ 0.00 & -1.74 & -1.56 & 1.26 & 1.57 \\ -2.79 & -1.56 & -5.71 & 2.56 & -0.36 \\ 1.30 & 1.26 & 2.56 & -3.51 & 0.15 \\ -4.78 & 1.57 & -0.36 & 0.15 & -10.27 \end{pmatrix}.\]

Figure 4 shows the pairwise scatter plots of true latent coordinates \(z_j\) versus learned representations \(\tilde{z}_i\) for each regime condition. Under Distinct ratios, each learned coordinate aligns tightly with exactly one true coordinate, producing a near-permutation pattern. Both controls lose this structure.

Figure 5 displays the mean absolute Jacobian \(\overline{|D\varphi|}\) of the composition \(\varphi = h \circ g\). Under Distinct ratios the matrix is near-monomial, confirming that the encoder inverts the mixing up to coordinate permutation and scaling. Both controls yield less monomial, more mixed encoder Jacobians. Graph recovery is not shown because the drift is dense, so sparse graph recovery is not the evaluation target.

The \(d=7\) case uses the same construction with identical \(\lambda = 0.5\). The resulting drift matrix, rounded to two decimals, is \[A = \begin{pmatrix} -4.89 & 1.11 & -2.12 & -0.13 & 0.98 & -2.94 & 4.09 \\ 1.11 & -5.21 & -2.30 & -0.70 & 4.00 & -1.50 & 0.93 \\ -2.12 & -2.30 & -9.39 & -0.37 & -0.35 & -5.45 & 4.37 \\ -0.13 & -0.70 & -0.37 & -8.22 & 4.36 & -3.82 & -1.67 \\ 0.98 & 4.00 & -0.35 & 4.36 & -8.66 & 3.26 & -1.52 \\ -2.94 & -1.50 & -5.45 & -3.82 & 3.26 & -7.56 & 4.88 \\ 4.09 & 0.93 & 4.37 & -1.67 & -1.52 & 4.88 & -8.83 \end{pmatrix}.\]

Figures 6 and 7 show the \(d=7\) results. The same qualitative pattern holds: Distinct ratios achieves near-perfect disentanglement, while both controls show substantially less coordinate recovery.

13.2 Sparse linear drift↩︎

The sparse linear experiments use an Ornstein–Uhlenbeck process with a sparse drift matrix \(A\) built from a directed cycle support. The nonzero off-diagonal support entries are placed at \((i,i+1)\) for \(i=1,\ldots,d-1\) and at \((d,1)\), and the drift matrix is constructed as \[A_{ii} = -\alpha, \qquad A_{ij} = \beta \, c_{ij} \;\text{if } \mathrm{adj}(i,j)=1, \qquad A_{ij} = 0 \;\text{otherwise},\] where \(\alpha = 1.0\), \(\beta = 1.5\), and \(c_{ij} \stackrel{\mathrm{iid}}{\sim} \mathrm{Uniform}(0.8, 1.2)\) with a fixed seed. The same \(A\) is shared across all training seeds and all three regime conditions.

For \(d=5\) the drift matrix, rounded to two decimals, is \[A = \begin{pmatrix} -1.00 & 1.77 & 0 & 0 & 0 \\ 0 & -1.00 & 1.72 & 0 & 0 \\ 0 & 0 & -1.00 & 1.33 & 0 \\ 0 & 0 & 0 & -1.00 & 1.37 \\ 1.57 & 0 & 0 & 0 & -1.00 \end{pmatrix}.\]

Figure 8 shows the pairwise scatter plots for the sparse linear \(d=5\) setting. Figure 9 displays the encoder Jacobian and graph recovery. Under Distinct ratios, the encoder inverts the mixing up to permutation and scaling, and the thresholded drift-Jacobian graph is recovered exactly (GMR \(=1\) for the representative seed).

The \(d=7\) case uses the same construction. The drift matrix, rounded to two decimals, is \[A = \begin{pmatrix} -1.00 & 1.77 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1.00 & 1.62 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1.00 & 1.51 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1.00 & 1.67 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1.00 & 1.77 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1.00 & 1.50 \\ 1.22 & 0 & 0 & 0 & 0 & 0 & -1.00 \end{pmatrix}.\]

Figures 10 and 11 show the corresponding results for \(d=7\). The same qualitative pattern holds: Distinct ratios gives near-monomial encoder Jacobians and near-perfect graph recovery, whereas the controls show mixed encoder Jacobians and lower GMR.

13.3 Nonlinear sparse drift↩︎

The nonlinear experiments use the same directed cycle support as the sparse linear case (Section 13.2), but replace the linear coupling with a \(\tanh\) nonlinearity. Concretely, the drift function is \[f_i(z) \;=\; -\alpha \, z_i \;+\; \beta \sum_{j:\,\mathrm{adj}(i,j)=1} c_{ij}\,\tanh(z_j),\] where \(\alpha = 1.0\) enforces mean-reversion along the diagonal and \(\beta = 1.5\) controls the coupling strength. The coefficients \(c_{ij}\) are drawn independently from \(\mathrm{Uniform}(0.8, 1.2)\) with a fixed seed and are nonzero only where the adjacency is \(1\). For \(d=5\) the nonzero coupling coefficients, rounded to two decimals, are \[c_{12} = 1.18,\quad c_{23} = 1.15,\quad c_{34} = 0.88,\quad c_{45} = 0.92,\quad c_{51} = 1.04.\] Here \(\mathrm{adj}(i,j)=1\) means that coordinate \(z_j\) enters the \(i\)th drift component \(f_i\).

For \(d=7\) the nonzero off-diagonal support entries are again placed at \((i,i+1)\) for \(i=1,\ldots,6\) and at \((7,1)\), and the nonzero coupling coefficients, rounded to two decimals, are \[c_{12} = 1.09,\quad c_{23} = 0.95,\quad c_{34} = 1.13,\quad c_{45} = 1.06,\quad c_{56} = 1.03,\quad c_{67} = 0.97,\quad c_{71} = 1.08.\] The Jacobian of \(f\) evaluated at the origin recovers the same sparse support pattern as in Section 13.2, but away from the origin the \(\tanh\) saturates, making the dynamics genuinely nonlinear.

For each fixed dimension, the same drift function is shared across the leaky-\(\tanh\) and LeakyReLU mixing experiments. The main text (Figures 1 and 2) reports the \(d=5\) results under leaky-\(\tanh\) mixing. Here we show the corresponding results under LeakyReLU mixing for both \(d=5\) and \(d=7\). All three regime conditions in each figure use the same representative training seed for a fair comparison. The identifiability gap is consistent with the leaky-\(\tanh\) results in the main text. Under Distinct ratios, the displayed run recovers clear coordinate alignment and a substantially more accurate thresholded graph; the controls show mixed coordinates and graph errors.

Figures 14 and 15 show the \(d=7\) results under LeakyReLU mixing. The identifiability gap remains clear: Distinct ratios achieves near-perfect disentanglement and high graph match rate, whereas both controls show weaker coordinate recovery and less reliable graph recovery.

14 Real-world case study: Hardanger Bridge↩︎

This appendix gives the setup and diagnostics behind the Hardanger Bridge experiment reported in Section 5.2. The goal of this experiment is not to evaluate recovery against ground-truth latent coordinates, which are unavailable in the real bridge data. Instead, we test whether the TI-split two-regime construction produces a more reproducible latent representation and drift structure than controls that remove the turbulence-intensity (TI) contrast. The appendix first describes the data and training protocol, then reports cross-seed reproducibility, diffusion-fingerprint diagnostics, and a representative learned drift structure.

14.1 Dataset and regime construction↩︎

We use operational acceleration recordings from the Hardanger Bridge, a long-span suspension bridge monitored with wind and acceleration sensors [28], [29]. The dataset DOI/repository record lists a Creative Commons Attribution 4.0 International (CC BY 4.0) license [28]. The model receives only bridge-deck accelerometer channels as observations. Wind measurements are used only to construct regimes, not as model inputs.

To keep the shared-drift assumption physically meaningful, we restrict the data to a single mean-wind-speed bin, \(4\)–\(6\,\mathrm{m/s}\). Within this bin, each raw recording is divided into non-overlapping \(120\,\mathrm{s}\) sub-windows. For each sub-window we compute the turbulence intensity \[\mathrm{TI} = \frac{\mathrm{std}(\text{wind speed})}{\mathrm{mean}(\text{wind speed})}.\] The two regimes are then defined by a quantile split of TI: the bottom \(20\%\) of sub-windows form the low-TI regime and the top \(20\%\) form the high-TI regime; the middle \(60\%\) is discarded. This construction is the real-world analogue of the distinct-diffusion-regime condition in the synthetic experiments: the bridge dynamics are approximately shared within a matched wind speed bin, while turbulence intensity changes the stochastic excitation.

Table 8 summarizes the resulting dataset. The accelerometer signals are detrended, low-pass filtered at \(2.5\,\mathrm{Hz}\), resampled to \(10\,\mathrm{Hz}\), and standardized with one scaler fitted jointly across both regimes. The selected \(120\,\mathrm{s}\) sub-windows are then converted into overlapping \(12.8\,\mathrm{s}\) trajectory windows with \(6.4\,\mathrm{s}\) stride for training.

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14.2 Training protocol and controls↩︎

We use the same two-stage estimator as in Section 4. The encoder is a three-layer MLP with hidden dimension \(64\), and the drift model is linear in the learned \(d_z=5\) coordinates. Both stages are trained for up to \(10{,}000\) epochs with early stopping patience \(100\). Stage 2 uses stride \(s=6\), sparse-Jacobian penalty \(\lambda_{\mathrm{sparse}}=1.0\), and graph threshold \(\tau=0.05\). All three regime constructions use the same list of ten random training seeds. The Hardanger runs use the same single-GPU compute setup as the simulation experiments in Appendix 12.4.

We compare three regime constructions:

• TI-split regimes (ours): the bottom and top TI quantiles are used as two regimes.

• One regime (control): the two TI groups are merged into a single environment. This preserves the total amount of data but removes regime-specific diffusion parameters.

• Shuffled TI labels (control): the two TI groups are pooled and randomly re-split into two equally sized groups. This preserves the two-regime training format but destroys the turbulence-based contrast.

14.3 Evaluation metrics↩︎

Because ground-truth latent states are not available, we evaluate reproducibility across random seeds. For each seed, we encode the same set of \(5{,}000\) evaluation points. For every pair of seeds, we align learned coordinates by Hungarian matching on absolute correlation and compute the mean matched absolute correlation coefficient (MCC). We then use the same alignment to compare the thresholded drift-Jacobian graphs and report the graph match rate (GMR). With ten seeds, this gives \(\binom{10}{2}=45\) seed pairs. These are reproducibility diagnostics rather than accuracy metrics: high values mean that independent trainings recover compatible coordinate systems and drift graphs. As an auxiliary within-seed diagnostic, we also report the mean absolute off-diagonal correlation among learned latent coordinates; lower values indicate less redundant latent coordinates.

We also report diffusion-fingerprint diagnostics for the two-regime conditions. For parametric fingerprints, we use the learned diagonal variances \(\tilde{\sigma}_{e,k}^2\) from Stage 1 and compute \(r_k^{\mathrm{par}}=\tilde{\sigma}_{2,k}^2/\tilde{\sigma}_{1,k}^2\). For empirical fingerprints, we compute residual covariance matrices \(C_e\) after Stage 2 and use \(r_k^{\mathrm{emp}}=C_{2,kk}/C_{1,kk}\). Since the learned coordinates are identified only up to permutation and scaling, and since the sign of the log-ratio follows the internal regime ordering, we summarize contrast magnitude by \(|\log r_k|\) pooled over seeds and latent dimensions.

14.4 Cross-seed consistency↩︎

Table 9 expands the main-text result with descriptive standard deviations across seed pairs. TI-split regimes yield the highest latent reproducibility and the highest drift-graph reproducibility. The MCC gain is substantial relative to both controls: \(+0.146\) over One regime and \(+0.205\) over Shuffled TI labels. The same construction also gives the lowest mean latent off-diagonal correlation, suggesting that the learned coordinates are less redundant within each learned representation. Because the thresholded drift graphs are mostly diagonal with sparse off-diagonal edges, absolute GMR values can be helped by matching many absent edges; we therefore interpret the relative TI-split gain together with MCC and the diffusion fingerprints below.

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14.5 Diffusion-fingerprint diagnostics↩︎

Table 10 shows that the TI-split regimes produce a strong learned diffusion contrast, whereas Shuffled TI labels collapse the fingerprint ratios to nearly one. This is the intended behavior of the control: it preserves the existence of two batches in the objective but removes the physical TI-based regime information. Thus the reproducibility gap in Table 9 is accompanied by the expected difference in learned diffusion fingerprints. Since Assumption (S4) requires latent coordinates to carry separating diffusion-ratio fingerprints across regimes, the large TI-split contrasts provide empirical support that the TI-based regime construction supplies the kind of diffusion variation required by the theory; the shuffled-label control does not. This is a diagnostic check rather than a formal verification of S4 in the absence of ground-truth latent coordinates.

The One-regime control is omitted from the fingerprint table because it has only one diffusion covariance and therefore no variance ratio.

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14.6 Representative learned structure↩︎

Figure 16 shows the learned drift Jacobians for a randomly selected seed, shared across the three regime constructions and ordered as in the main text. In each heatmap, row \(i\) corresponds to drift component \(\tilde{f}_{\psi,i}\) and column \(j\) corresponds to source coordinate \(\tilde{z}_j\); the plotted value is the evaluation mean of \(|\partial \tilde{f}_{\psi,i}/\partial \tilde{z}_j|\). Each panel is diagonal-dominant, but the off-diagonal structure differs across regime constructions. This is the qualitative pattern one would expect if the learned coordinates approximate a modal-like coordinate system for bridge vibrations: self-dynamics should dominate, while cross-coordinate effects should appear as a small number of structured couplings rather than as dense or scattered dependence. In this representative TI-split-regime run, the largest off-diagonal entries form a reciprocal pair between two learned coordinates: one drift component depends on the other coordinate with magnitude about \(0.48\), and the reverse dependence has magnitude about \(0.40\). A reciprocal pair of this kind is consistent with a learned two-dimensional dynamical subspace, as can arise from nearby vibration modes, non-proportional damping, aeroelastic or environmental coupling, or residual mixing in modal-like coordinates. By contrast, the controls under the same representative seed also contain off-diagonal entries, but these appear as more scattered one-way couplings across different coordinate pairs rather than as a single coherent coupled block. The TI-split structure is therefore more naturally aligned with this modal-like physical interpretation, while still serving only as an illustrative structural diagnostic that complements the aggregate reproducibility and fingerprint results above.

14.7 Limitations of the real-world diagnostic↩︎

This experiment should be read as a real-world diagnostic rather than a ground-truth recovery benchmark. The Hardanger data do not provide true latent coordinates or a true causal graph, so MCC and GMR measure cross-seed reproducibility after alignment, not absolute accuracy; absolute GMR values can also be influenced by matching many absent edges in sparse thresholded graphs. The diffusion-fingerprint ratios support the presence of TI-dependent diffusion variation, but they do not formally verify Assumption (S4) because the true latent coordinates and diffusion covariances are unobserved. Likewise, the modal-like interpretation of the representative drift Jacobian is not a substitute for external validation against finite-element models or modal analysis labels. Finally, the TI quantile split is a proxy for diffusion-regime variation. Its usefulness depends on maintaining enough stochastic-forcing contrast while keeping the deterministic structural dynamics approximately shared, exactly the tradeoff made explicit by the theory.

References↩︎