Learning to Test: Latent Representation Learning for Dynamic Instability Detection

The gist: This paper proposes a test-oriented learning framework that integrates physics-informed representation learning with distributional hypothesis testing to detect instability risk in stochastic constrained dynamical systems without repeated simulation.

Problem Setup

The system under consideration is parameterized by deterministic but unknown physical parameters θ, evolving according to differential–algebraic equations (DAEs) (Page 4). Stability is defined as a chance constraint on the random trajectory sθ(·; X) over the context distribution P, formulated as PX∼P η sθ(·; X) ≤ τ ≥ 1 − α (Definition 1 ((τ, α)-Stability)). The safe regime P0 is the subset of context distributions for which this stability requirement holds (Definition 2 (Safe Regime)) (Page 4). The central question addressed is how to assess stability under stochastically varying inputs without incurring the cost of repeated full-scale analysis, while ensuring reliable generalization beyond observed conditions (Page 2).

Proposed Framework: Learning to Test

The proposed framework, termed “learning to test” (L2T), directly targets the hypothesis testing problem in Eq. (4) rather than first estimating the physical parameters and then simulating state trajectories under the new domain (Page 3). The core idea is to learn a latent representation of contextual variables that encodes stability-relevant features of the constrained dynamics, while being regularized toward a simple and tractable reference distribution (Page 3). This representation is learned end-to-end with respect to this testing objective, bypassing repeated large-scale DAE simulation (Page 3).

Physics-Informed Representation Learning

The framework utilizes neural ordinary differential equations (ODE) as a flexible surrogate to learn a reduced ODE whose dynamics approximate the evolution of the DAE on the constraint manifold (Page 3.1). A compact history embedding h(t) is defined, and its evolution is governed by dh(t)/dt = φ(s(t), h(t); θ, x) (Equation 5) (Page 3.1). The latent representation u = ϕ(x) maps each context x to a latent variable (Page 3.1). The training objective L(ϕ, ψ, φ) combines three components:

1. Reconstruction loss: To ensure that the latent representation captures the DAE-induced trajectory behavior observed in the baseline regime (Equation 8) (Page 3.1).

2. Latent uniformity regularization: To facilitate distributional testing in the latent space, it minimizes KL pU Unif(0, 1)d = -H(pU) + const (Equation 9) (Page 3.1).

3. Stability-aware reweighting: This augments the reconstruction loss with an asymmetric penalty based on the stability metric η(s(·)) = maxt∈[0,T] s(t), using a sample-level weight ωstb(s, sˆ) (Page 3.1).

Latent Goodness-of-Fit Test for Dynamic Instability Detection

After training on the baseline domain, the encoder ϕ is frozen to map contexts into a latent space where distributional testing is convenient (Page 3.2). The key theoretical result is Theorem 1 (Reformulation), which shows that stability verification can be reformulated as a distributional test in the latent space: H0: P1 ∈ P0 vs. H1: P1 ∈ P/ 0 is equivalent to the hypothesis test on the distribution Q1 of U: TV(Q1, Q∗) ≤ α/2 vs. TV(Q1, Q∗) > α/2 (Equation 11) (Page 3.2). This equivalence is established under ideal latent representations and is shown to hold up to an explicit error tolerance in Theorem 3 (Approximate reformulation), which introduces an ambiguity band of width 2¯ε around the threshold α/2 (Page 3.2). The practical testing procedure involves a two-sample test in the latent space using a discrepancy measure D(·, ·) between held-out baseline contexts U0 and new contexts U1, rejecting H0 if Tb > tα (Algorithm 1) (Page 3.2).

Experimental Validation

The approach was validated on synthetic experiments involving a mass–spring–damper system and a Cartesian pendulum system (Page 4). Empirical results on the IEEE 39-bus TSA benchmark demonstrated that L2T consistently achieves reliable Type I error control under the null, with rejection rates close to the nominal significance level 0.1, while maintaining low Type II error and hence high power under the alternative, particularly in the boundary-transition regime where detection is most challenging (Table 1) (Page 9). For synthetic experiments, L2T consistently achieves reliable Type I error control under the null, with rejection rates close to the nominal significance level 0.1, while maintaining low Type II error and hence high power under the alternative, particularly in the boundary-transition regime where detection is most challenging (Figure 3a) (Page 9). On the IEEE 39-bus TSA benchmark, L2T achieves a favorable balance between Type I error control and detection power (Table 1) (Page 10).

Conclusion

The framework establishes a scalable and statistically principled method for safety assessment in stochastic constrained dynamical systems by integrating physics-informed representation learning and distributional hypothesis testing (Page 5). It successfully reduces trajectory-level stability verification to a simple distributional test in the latent space, supported by theoretical guarantees via a reformulation and large deviation analysis (Page 3.2). Empirically, the method achieved reliable type-I error control and strong detection power across synthetic systems and a real-world power system benchmark, particularly near critical stability boundaries where existing approaches struggle (Page 5).

How it works

1. The framework learns a latent representation of contextual variables that captures stability-relevant features of the constrained dynamics while being regularized toward a uniform reference distribution (Page 3).

2. Training involves minimizing an objective L(ϕ, ψ, φ) that combines trajectory reconstruction loss (Lrec), latent uniformity regularization (Lreg), and stability-aware reweighting (Page 3.1).

3. Testing involves mapping new contexts Xj ∼ P1 into the latent space to obtain Uj ∼ Q1, and then performing a distributional test between Q1 and the reference distribution Q0 (or the uniform distribution) in latent space using a two-sample test statistic Tb = D(U0, U1) (Page 3.2).

4. Theoretical analysis shows that stability verification is equivalent to testing uniformity in the latent space, and this equivalence holds up to an explicit error tolerance defined by Assumption 2 (Approximate latent sufficiency and calibration) (Page 3.2).

5. The two-sample test in the latent space provides a robust empirical implementation, where the test statistic Tb = D(U0, U1) is used to reject H0 if Tb exceeds a calibrated critical value tα (Algorithm 1) (Page 3.2).

Implementation Details on Physics-Informed Representation Learning

The representation learning uses neural ordinary differential equations (ODE) as a surrogate to learn a reduced ODE whose dynamics approximate the evolution of the DAE on the constraint manifold (Page 3.1). The history embedding h(t) evolves according to dh(t)/dt = φ(s(t), h(t); θ, x) (Equation 5) (Page 3.1). The training objective L is defined as L = 1/n Σ [ωstb · si,k − sˆi,k2 + λ · Lreg] (Equation 10) (Page 3.1). Uniformity regularization can be implemented via Maximum Mean Discrepancy (MMD) or a flow matching loss to encourage the latent representations u = ϕ(x) to follow Unif(0, 1)d (Page 3.1).

Details of Two-Sample Test Procedure

The practical testing procedure involves mapping baseline contexts X0 and new contexts X1 into latent spaces U0 and U1, respectively (Algorithm 1) (Page 3.2). A discrepancy measure D(·, ·), such as MMD or energy distance, is used to compute the test statistic Tb = D(U0, U1), and H0 is rejected if Tb exceeds a calibrated critical value tα (Page 3.2). The choice of tα depends on the discrepancy D (Remark 2) (Page 3.2).

Conclusion

The proposed L2T framework establishes a scalable and statistically principled method for safety assessment in stochastic constrained dynamical systems by integrating physics-informed representation learning and distributional hypothesis testing (Page 5). It successfully reduces trajectory-level stability verification to a simple distributional test in the latent space, supported by theoretical guarantees via a reformulation and large deviation analysis (Page 3.2). Empirically, the method achieved reliable type-I error control and strong detection power across synthetic systems and a real-world power system benchmark, particularly near critical stability boundaries where existing approaches struggle (Page 5).

Acknowledgments and Disclosure of Funding

The authors gratefully acknowledge Prof. Bai Cui (Iowa State University) for insightful discussions that helped shape the problem formulation and identify relevant real-world applications (Page 11).

References

[1] Zohreh Abbasi and Xinzhi Liu. Switched epidemic models: A contact rate dependent analysis. Applied Mathematical Modelling, 128:168–194, 2024 (Page 13).

[2] Paul M Anderson and Aziz A Fouad. Power system control and stability.

1. No repeated large-scale DAE simulation is required for stability assessment, enabling deployment-time safety monitoring as a distributional hypothesis test in latent space. This allows for real-time risk detection in stochastic constrained dynamical systems without incurring prohibitive computational costs.

2. The system can perform context-only monitoring by learning a physics informed latent representation of contextual variables that captures stability-relevant structure, which is then regularized toward a tractable reference distribution (uniform). This enables the method to detect instability by checking if contexts under a new regime remain statistically consistent with the reference distribution.

3. The framework can integrate uncertainty quantification into safety monitoring by reformulating stability verification as a distributional test in latent space, which allows for uncertainty-aware calibration and provides explicit error bounds via large deviation theory to characterize Type I and Type II errors.

4. The AI system can be trained on limited baseline data from a single safe domain to capture stability-relevant dynamics, allowing it to generalize across domains where no trajectory observations are available, overcoming the limitation that conventional end-to-end learning approaches that map inputs directly to stability labels can be systematically biased.

5. The model can detect instability by comparing latent representations of new contexts against a reference distribution, with the test statistic being based on a discrepancy measure like Maximum Mean Discrepancy (MMD) or energy distance between latent samples.