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Safe Active Learning for Time-Series Modeling with Gaussian Processes

Christoph Zimmer, Mona Meister, Duy Nguyen-Tuong

TL;DR

This work addresses safe active learning for time-series models expressed as Gaussian processes with nonlinear exogenous inputs (NX). It jointly learns a dynamic model $f$ and a safety predictor $g$ using GP priors, planning trajectory-wise input excitations that maximize information gain while enforcing a probabilistic safety constraint $\xi(\boldsymbol{\tau})>1-\alpha$. The authors prove a bound on unsafe exploration and establish a decay of predictive uncertainty under a determinant-based exploration criterion, with a connection to maximum information gain $\gamma_n$. Empirical results on synthetic toy tasks and a high-pressure fluid system show faster convergence and safer region coverage than random exploration, highlighting practical applicability in safety-critical settings. The approach enables efficient, safety-aware data collection for time-series modeling in industrial contexts.

Abstract

Learning time-series models is useful for many applications, such as simulation and forecasting. In this study, we consider the problem of actively learning time-series models while taking given safety constraints into account. For time-series modeling we employ a Gaussian process with a nonlinear exogenous input structure. The proposed approach generates data appropriate for time series model learning, i.e. input and output trajectories, by dynamically exploring the input space. The approach parametrizes the input trajectory as consecutive trajectory sections, which are determined stepwise given safety requirements and past observations. We analyze the proposed algorithm and evaluate it empirically on a technical application. The results show the effectiveness of our approach in a realistic technical use case.

Safe Active Learning for Time-Series Modeling with Gaussian Processes

TL;DR

This work addresses safe active learning for time-series models expressed as Gaussian processes with nonlinear exogenous inputs (NX). It jointly learns a dynamic model and a safety predictor using GP priors, planning trajectory-wise input excitations that maximize information gain while enforcing a probabilistic safety constraint . The authors prove a bound on unsafe exploration and establish a decay of predictive uncertainty under a determinant-based exploration criterion, with a connection to maximum information gain . Empirical results on synthetic toy tasks and a high-pressure fluid system show faster convergence and safer region coverage than random exploration, highlighting practical applicability in safety-critical settings. The approach enables efficient, safety-aware data collection for time-series modeling in industrial contexts.

Abstract

Learning time-series models is useful for many applications, such as simulation and forecasting. In this study, we consider the problem of actively learning time-series models while taking given safety constraints into account. For time-series modeling we employ a Gaussian process with a nonlinear exogenous input structure. The proposed approach generates data appropriate for time series model learning, i.e. input and output trajectories, by dynamically exploring the input space. The approach parametrizes the input trajectory as consecutive trajectory sections, which are determined stepwise given safety requirements and past observations. We analyze the proposed algorithm and evaluate it empirically on a technical application. The results show the effectiveness of our approach in a realistic technical use case.
Paper Structure (24 sections, 7 theorems, 37 equations, 5 figures, 1 algorithm)

This paper contains 24 sections, 7 theorems, 37 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Safe Active Learning for Time-Series Modeling
  4. Modeling Trajectories with Gaussian Processes
  5. Modeling the Safety Condition
  6. The Algorithm
  7. Theoretical Results
  8. Safe Exploration
  9. Decay of Predictive Variance
  10. Evaluations
  11. Simulated Experiments
  12. Experiment 1
  13. Experiment 2
  14. Learning a Surrogate Model of the High-Pressure Fluid System
  15. The Use Case
  16. ...and 9 more sections

Key Result

Theorem 1

Let us assume that we have recorded $n_0$ initial safe trajectories, and that their observations are enough to model $g$ well, in the sense that our GP quantifies the uncertainty of predictions for $g$ correctly, i.e. $P(\mu_g - \nu \sigma_g \leq z \leq \mu_g + \nu \sigma_g) \!=\! \hbox{Erf}(\nu / \

Figures (5)

  • Figure 1: The columns show the progress of the approximation of $f$ (inlay) and the identified safety region (main figure) at different iterations. Each iteration corresponds to a consecutive planning of a new piecewise trajectory (here: 2D ramp). As shown by the results, the current estimation of the safe region (green area) gradually covers the actual safe area (red line), and the approximation error gradually decreases (as shown in the subfigures). An illustrative video showing all iterations can be found in the Appendix.
  • Figure 2: The first two pictures from the left show the comparison of the SAL-NX (red line) with random selection (blue line). SAL-NX yields faster convergence in model approximation (left picture) and coverage of safe regions (right picture), while having less variance and outliers (indicated as small circles). The last two pictures show the impact of the safety threshold $\alpha$. The left picture shows the RMSE of SAL-NX for 4 different values of $\alpha$. The right picture shows the model approximation error as RMSE (red line) and percentage of unsafe trajectories (blue line) as a function of $\alpha$. All pictures show boxplots over $5$ repetitions. The plot contained inconsistencies which are now corrected. The old plot is left above for comparison.
  • Figure 3: High-pressure fluid injection system with controllable inputs $v_k\,,n_k$ and measured output $\psi_k$ (picture taken from Tietze2014).
  • Figure 4: The first two pictures from the left show the comparison of the SAL-NX (red line) with random selection with safe constraints (blue line), with respect to model approximation and coverage of safe regions. Here, $\alpha\!=\!$$0.5$$0.8$ and $250$ trajectories are planned. The last two pictures show the impact of the safety threshold $\alpha$ on the approximation error, and failures during exploration. The results are displayed as a boxplot over $5$ repetitions. The evaluation and plot contained inconsistencies which are now corrected. The old plot is left above for comparison.
  • Figure 5: Right panel: As figure \ref{['fig:toy1']} but without safety consideration. Blue color representing the approach based on the Fisher information matrix, red color still representing our SAL-NX approach (now without safety consideration).

Theorems & Definitions (16)

  • Example 1: A safety indicator for a high-pressure fluid system
  • Example 2: Consecutive ramps as piecewise trajectory
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm:SafeBound']}
  • proof : Proof of Lemma \ref{['lem:MuteSigma']}
  • Lemma 3
  • ...and 6 more