We propose a family of Gaussian processes (GP) for dynamical systems with linear time-invariant responses, which are nonlinear only in initial conditions. This linearity allows us to tractably quantify forecasting and representational uncertainty, simultaneously alleviating the challenge of computing the distribution of trajectories from a GP-based dynamical system and enabling a new probabilistic treatment of learning Koopman operator representations. Using a trajectory-based equivariance – which we refer to as Koopman equivariance – we obtain a GP model with enhanced generalization capabilities. To allow for large-scale regression, we equip our framework with variational inference based on suitable inducing points. Experiments demonstrate on-par and often better forecasting performance compared to kernel-based methods for learning dynamical systems.

Control tasks with safety requirements under high levels of model uncertainty are increasingly common. Machine learning techniques are frequently used to address such tasks, typically by leveraging model error bounds to specify robust constraint-based safety filters. However, if the learned model uncertainty is very high, the corresponding filters are potentially invalid, meaning no control input satisfies the constraints imposed by the safety filter. While most works address this issue by assuming some form of safe backup controller, ours tackles it by collecting additional data on the fly using a Gaussian process bandit-type algorithm. We combine a control barrier function with a learned model to specify a robust certificate that ensures safety if feasible. Whenever infeasibility occurs, we leverage the control barrier function to guide exploration, ensuring the collected data contributes toward the closed-loop system safety. By combining a safety filter with exploration in this manner, our method provably achieves safety in a general setting that does not require any prior model or backup controller, provided that the true system lies in a reproducing kernel Hilbert space. To the best of our knowledge, it is the first safe learning-based control method that achieves this.

Machine learning algorithms often struggle to control complex real-world systems. In the case of nuclear fusion, these challenges are exacerbated, as the dynamics are notoriously complex, data is poor, hardware is subject to failures, and experiments often affect dynamics beyond the experiment’s duration. Existing tools like reinforcement learning, supervised learning, and Bayesian optimization address some of these challenges but fail to provide a comprehensive solution. To overcome these limitations, we present a multi-scale Bayesian optimization approach that integrates a high-frequency data-driven dynamics model with a low-frequency Gaussian process. By updating the Gaussian process between experiments, the method rapidly adapts to new data, refining the predictions of the less reliable dynamical model. We validate our approach by controlling tearing instabilities in the DIII-D nuclear fusion plant. Offline testing on historical data shows that our method significantly outperforms several baselines. Results on live experiments on the DIII-D tokamak, conducted under high-performance plasma scenarios prone to instabilities, shows a 50% success rate — marking a 117% improvement over historical outcomes.

Model-based reinforcement learning (MBRL) hinges on a learned dynamics model whose errors can compound along imagined rollouts. We study how encouraging \emph{flatness} in the model’s training loss affects downstream control, and show that steering optimization toward flatter minima yields a better policy. Concretely, we integrate \emph{Sharpness-Aware Minimization} (SAM) into world-model training as a drop-in objective, leaving the planner and policy components unchanged. On the theory side, we derive PAC-Bayesian bounds that link first-order sharpness to the value-estimation gap and the performance gap between model-optimal and true-optimal policies, implying that flatter minima tighten both. Empirically, SAM reduces measured sharpness and value-prediction error and improves returns across HumanoidBench, Atari-100k, and high-DoF DeepMind Control tasks. Augmenting existing MBRL algorithms with SAM increases mean return, with especially large gains in settings with high dimensional state–action space. We further observe positive transfer across algorithms and input modalities, including a transformer-based world-model. These results position flat-minima training as a simple, general mechanism for more robust MBRL without architectural changes.

While Gaussian processes are a mainstay for various engineering and scientific applications, the uncertainty estimates don't satisfy frequentist guarantees and can be miscalibrated in practice. State-of-the-art approaches for designing calibrated models rely on inflating the Gaussian process posterior variance, which yields confidence intervals that are potentially too coarse. To remedy this, we present a calibration approach that generates predictive quantiles using a computation inspired by the vanilla Gaussian process posterior variance but using a different set of hyperparameters chosen to satisfy an empirical calibration constraint. This results in a calibration approach that is considerably more flexible than existing approaches, which we optimize to yield tight predictive quantiles. Our approach is shown to yield a calibrated model under reasonable assumptions. Furthermore, it outperforms existing approaches in sharpness when employed for calibrated regression.

Gaussian processes have become a promising tool for various safety-critical settings, since the posterior variance can be used to directly estimate the model error and quantify risk. However, state-of-the-art techniques for safety-critical settings hinge on the assumption that the kernel hyperparameters are known, which does not apply in general. To mitigate this, we introduce robust Gaussian process uniform error bounds in settings with unknown hyperparameters. Our approach computes a confidence region in the space of hyperparameters, which enables us to obtain a probabilistic upper bound for the model error of a Gaussian process with arbitrary hyperparameters. We do not require to know any bounds for the hyperparameters a priori, which is an assumption commonly found in related work. Instead, we are able to derive bounds from data in an intuitive fashion. We additionally employ the proposed technique to derive performance guarantees for a class of learning-based control problems. Experiments show that the bound performs significantly better than vanilla and fully Bayesian Gaussian processes.