Learning to Test: Latent Representation Learning for Dynamic Instability Detection

Minxing Zheng; Zewei Deng; Liyan Xie; Shixiang Zhu

cs.LGApr 13, 2026

Learning to Test: Latent Representation Learning for Dynamic Instability Detection

Authors: Minxing Zheng, Zewei Deng, Liyan Xie, Shixiang Zhu

Organizations: Carnegie Mellon University · University of Minnesota

Abstract

Stability assessment is central to safety-critical systems governed by differential-algebraic equations (DAEs), whose dynamics are constrained by physical laws and admissibility conditions. These systems operate under stochastically varying environmental inputs, so stability must be reassessed as the context distribution shifts, yet repeated large-scale DAE simulation is prohibitive in high-dimensional or real-time settings. We propose a test-oriented learning framework that tests a trajectory-level chance constraint from deployment contexts alone. Rather than re-estimating physical parameters or re-solving the DAE, we learn a latent representation from baseline context-trajectory data that retains stability-relevant information and is regularized toward a tractable reference distribution, and calibrate a latent discrepancy test over the composite safety null. We establish exact validity for oracle calibration and asymptotic validity for its empirical implementation under class-conditional latent invariance, and show that the guarantee degrades gracefully when this invariance holds only approximately. On simulated dynamical systems and a power-system real-world example, our approach strikes a balance between validity and power, whereas other baseline methods each fail in at least one regime. An ablation that injects controlled violations of the invariance assumption shows that decisions stay correct while the violation is below the safety margin, as the theory predicts.

Figures & tables

Figure 1: Illustration of Cartesian pendulum systems and the one-dimensional latent representations induced by our method. The system maps a context variable ( e.g. , initial conditions or external disturbances) to a pendulum trajectory. A system is considered stable when the context follows the “safe” distribution. The top row shows representative trajectories under three exogenous context distributions: green segments satisfy the angular stability criterion, while red segments indicate violations. Under stable regimes, the learned latent representation is approximately uniform; under unstable regimes, it deviates noticeably from uniformity.

Figure 2: Overview of the “Learning-to-Test” paradigm.

Figure 3: Why Assumption 1 is needed and what violating it costs. Panels (a, b) show two systems that agree on the baseline support $\mathcal{S}_{0}$ and produce identical observed data, yet have violation probabilities on opposite sides of $\alpha=0.1$ (Proposition 1 ). Panel (c) shows the probability that the calibrated test decides correctly on the spring L2T latent under class-conditional drift toward the other class; the dash-dot boundary is predicted by Proposition 2 , not fitted. Hatched cells mark infeasible drift.

Panel of Figure 4 — Figure 4: Comparison between L2T and other baselines under various distribution shifts. Green and red regions denote stable and unstable contexts, respectively, while dashed gray and solid black contours represent the baseline and shifted distributions. The top row of each panel shows baseline ( $P_{0}$ ) and shifted ( $P_{1}$ ) context distributions in the $2$ D space across representative regimes: null ( $H_{0}\to H_{0}$ ), boundary transition ( $H_{0}\to H_{1}$ ), and alternative ( $H_{1}\to H_{1}$ ). The bottom row shows empirical rejection rates over $100$ runs against the shift magnitude, with $90\%$ Wilson intervals shaded. The dotted line marks the test level $\beta=0.1$ .

Dataset	Metric	Context Test	Binary-clf.	Recon-only	Uniform-only	L2T
Mass–spring–damper	Type-I $\downarrow$	$1.00\pm 0.00$	$0.17\pm 0.04$	$1.00\pm 0.00$	$1.00\pm 0.00$	$0.14\pm 0.03$
Mass–spring–damper	Type-II $\downarrow$	$0.47\pm 0.05$	$0.00\pm 0.00$	$0.54\pm 0.05$	$0.00\pm 0.00$	$0.00\pm 0.00$
Cartesian pendulum	Type-I $\downarrow$	$1.00\pm 0.00$	$0.83\pm 0.04$	$0.00\pm 0.00$	$0.13\pm 0.03$	$0.13\pm 0.03$
Cartesian pendulum	Type-II $\downarrow$	$0.00\pm 0.00$	$0.56\pm 0.05$	$1.00\pm 0.00$	$0.90\pm 0.03$	$0.02\pm 0.01$
IEEE 39-bus TSA	Type-I $\downarrow$	$0.01\pm 0.01$	$1.00\pm 0.00$	$0.11\pm 0.02$	$0.12\pm 0.02$	$0.10\pm 0.02$
IEEE 39-bus TSA	Type-II $\downarrow$	$0.18\pm 0.03$	$0.00\pm 0.00$	$0.00\pm 0.00$	$0.71\pm 0.03$	$0.01\pm 0.01$

Table 1: Comparison of empirical Type-I and Type-II errors at test level

\beta=0.1

(mean

\pm

standard error).

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Example	Shift	Null	Boundary-transition	Alternative
Spring	Mean	$\mu_{0}=(0,0)$ , $v=(0,1)$	$\mu_{0}=(0,0)$ , $v=(1,0)$	$\mu_{0}=(-1,0)$ , $v=(0,1)$
Spring	Covariance	$\Sigma_{\delta}=\begin{pmatrix}1&0\\ 0&1+\delta\end{pmatrix}$	$\Sigma_{\delta}=\begin{pmatrix}1+\delta&0\\ 0&1\end{pmatrix}$	N/A
Pendulum	Marginal	$(\alpha_{1},\beta_{1})=\left(1+\tfrac{10}{3}\delta,\ 2+\tfrac{20}{3}\delta\right)$ $(\alpha_{2},\beta_{2})=(1,4)$	$(\alpha_{1},\beta_{1})=(1,2)$ $\alpha_{2}=1+1.5\delta,\ \beta_{2}=4-1.5\delta$	$(\alpha_{1},\beta_{1})=(1.5,1.5)$ $\alpha_{2}=1+1.5\delta,\ \beta_{2}=4-1.5\delta$
Pendulum	Dependence	$\rho=-\delta$	$\rho=\delta$	N/A

Appendix

Table 2: Summary of the synthetic shift settings.

Figure 5: Sensitivity of the learned representation to the loss weights over a reconstruction-dominant simplex parameterized by $(\lambda_{\mathrm{rec}},\lambda_{\mathrm{reg}},\lambda_{\mathrm{stb}})$ , evaluated with the uncalibrated two-sample test between baseline and shifted latent samples. The three panels correspond to the baseline without shift, a null shift, and a boundary-transition shift. Under the null shift a lower rejection rate indicates a latent that stays invariant to the benign shift; under the boundary-transition shift a lower Type-II error indicates a latent that responds to the risk-increasing shift.

Figure 6: Latent-space under increasing shift magnitude $\delta$ . Columns correspond to values of $\delta$ , and rows show the latent density relative to the uniform reference under the null, boundary-transition, and alternative shifts. Under the null shift, the L2T latent stays near uniform across shift levels, whereas the uniform-only latent tilts away from it, which leads to false alarms. Under the boundary-transition and alternative shifts, both latents deviate increasingly from uniformity as $\delta$ grows.

Figure 7: Additional baselines on the mass–spring–damper system; the panels are columns 1, 2, 4 and 5 of Figure 4(a) . Oracle simulation simulates the first

k

of the

2{,}000

deployment contexts and applies an exact binomial test at

\alpha

; trajectory surrogate + label applies the same test to the labels of trajectories predicted by a

2\times 32

MLP. The dashed vertical line marks where the true violation rate crosses

\alpha

, and the dotted line is the test level

0.1

Figure 8: Empirical check of Assumption 1 for the trained L2T encoder on the mass–spring–damper system. Left two panels: class-conditional empirical CDFs of the latent for the baseline, an independent baseline sample, and the two null shifts of Figure 4(a) at

\delta=1

. Right two panels: class-conditional drifts and mixture residual (MMD), divided by the null radius

r_{\alpha}=(\alpha/2)D(R_{0},R_{1})

, along the mean shift and the variance increase of

X_{2}

(columns 1 and 2 of Figure 4(a) ), where the violation rate stays at

8.0

–

8.1\%

and

7.6

–

8.0\%

;

\delta=0

is the independent-baseline sampling floor.

Figure 9: Trajectory visualization for generator G01 across four system variables: (a) active power, (b) excitation current, (c) rotor angle, and (d) terminal voltage. Blue curves denote stable trajectories and red curves denote unstable ones. Thin lines correspond to individual sampled trajectories, while solid lines represent the mean trajectories and shaded regions indicate one standard deviation. Stable trajectories remain tightly concentrated and exhibit smooth, consistent dynamics, whereas unstable trajectories show larger dispersion and systematic deviations, particularly in rotor angle and terminal voltage. These differences highlight the distributional characteristics that enable detection of instability.

Figure 10: TSA rejection curves under three distribution-shift regimes. The null setting is defined by the reference mixture with unstable ratio $0.05$ . Left: null shift from $H_{0}$ to $H_{0}$ , where the unstable ratio decreases from $0.05$ to $0.00$ . Middle: transition shift from $H_{0}$ to $H_{1}$ , where the unstable ratio increases from $0.05$ to $0.20$ . Right: alternative-regime shift from $H_{1}$ to $H_{1}$ , where the unstable ratio varies from $0.20$ to $0.30$ . The horizontal axis shows the normalized shift magnitude $\delta\in[0,1]$ for each regime. The empirical rejection rate of L2T remains close to the nominal test level under the null shift, increases smoothly through the transition regime, and is close to one in the alternative regime, indicating a favorable balance between Type-I error control and detection power.

Method	Type-I @ 0.00	Type-II @ 0.20
L2T (flow, $d_{v}=8$ )	$0.100\pm 0.035$	$0.005\pm 0.008$
L2T-MMD ( $d_{v}=1$ )	$0.145\pm 0.041$	$0.435\pm 0.058$
Reconstruction Only	$0.110\pm 0.036$	$0.000\pm 0.000$
Regularization Only	$0.115\pm 0.037$	$0.710\pm 0.053$
Context Test	$0.010\pm 0.011$	$0.175\pm 0.044$
Binary Classifier	$1.000\pm 0.000$	$0.000\pm 0.000$

Appendix

Table 3: TSA results on Type-I and Type-II errors.

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