Safe Active Learning for Gaussian Differential Equations

TL;DR

This paper introduces Safe Active Learning for Gaussian Process ODEs (SAL GPODE), a framework that combines a mutual-information based acquisition with a safety constraint to guide data collection for learning the unknown dynamics g in a GPODE. The method uses decoupled GP sampling to draw trajectory realizations, computes a Monte Carlo estimate of the information gain from potential measurements, and imposes a probabilistic safety constraint via a separate safety GP. Data collection is planned by solving a constrained optimization that selects initial states to maximize information while keeping the probability of unsafe behavior below a user-defined threshold. Experiments on the Van der Pol oscillator and Lotka-Volterra dynamics show SAL GPODE achieves faster, safer model learning than non-active baselines, highlighting its potential for efficient and safe data-driven dynamical modeling.

Abstract

Gaussian Process differential equations (GPODE) have recently gained momentum due to their ability to capture dynamics behavior of systems and also represent uncertainty in predictions. Prior work has described the process of training the hyperparameters and, thereby, calibrating GPODE to data. How to design efficient algorithms to collect data for training GPODE models is still an open field of research. Nevertheless high-quality training data is key for model performance. Furthermore, data collection leads to time-cost and financial-cost and might in some areas even be safety critical to the system under test. Therefore, algorithms for safe and efficient data collection are central for building high quality GPODE models. Our novel Safe Active Learning (SAL) for GPODE algorithm addresses this challenge by suggesting a mechanism to propose efficient and non-safety-critical data to collect. SAL GPODE does so by sequentially suggesting new data, measuring it and updating the GPODE model with the new data. In this way, subsequent data points are iteratively suggested. The core of our SAL GPODE algorithm is a constrained optimization problem maximizing information of new data for GPODE model training constrained by the safety of the underlying system. We demonstrate our novel SAL GPODE's superiority compared to a standard, non-active way of measuring new data on two relevant examples.

Figures (5)

• Figure 1: The visualization shows the Safe Active Learning loop, from upper right: i) train/update a GPODE model, ii) plan new measurement by solving the constrained optimization problem that maximizes information constraint on safety, iii) conduct new measurement on the test system.
• Figure 2: SAL GPODE outperforming random benchmark: Validation negative log likelihood as a performance indicator in dependence of number of measurements. Lower values mean smaller error. Our SAL GPODE quickly reduces the error and as already learned a good model with as few as 2--3 measurements. The random benchmark needs much longer. Solid line is the mean and shaded area the mean plus/minus two standard deviations of five repetitions with different random seeds.
• Figure 3: Exploration of the safe set
• Figure 4: Validation negative log likelihood on the Lotka-Volterra task
• Figure 5: F1 score on the Lotka-Volterra task