-- Certifying safety in dynamical systems is crucial, but barrier certificates -- widely used to verify that system trajectories remain within a safe region -- typically require explicit system models. When dynamics are unknown, data-driven methods can be used instead, yet obtaining a valid certificate requires rigorous uncertainty quantification. For this purpose, existing methods usually rely on full-state measurements, limiting their applicability. This paper proposes a novel approach for synthesizing barrier certificates for unknown systems with latent states and polynomial dynamics. A Bayesian framework is employed, where a prior in state-space representation is updated using input-output data via a targeted marginal Metropolis-Hastings sampler . The resulting samples are used to construct a candidate barrier certificate through a sum-of-squares program. It is shown that if the candidate satisfies the required conditions on a test set of additional samples, it is also valid for the true, unknown system with high probability. The approach and its probabilistic guarantees are illustrated through a numerical simulation. Ensuring the safety of dynamical systems is a critical concern in applications such as human-robot interaction, autonomous driving, and medical devices, where failures can lead to severe consequences. In such scenarios, safety constraints typically mandate that the system state remains within a predefined allowable region. Barrier certificates [1] provide a systematic framework for verifying safety by establishing mathematical conditions that guarantee that system trajectories remain within these regions.

Credible forecasting and representation learning of dynamical systems are of ever-increasing importance for reliable decision-making. To that end, 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 \textit{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.

Additionally, the investigation into the nested pointwise aggregation of In the realm of real-time online Gaussian Process (GP) experts has been undertaken [20], [21]. Nevertheless, the regression, continuously collecting the training data becomes application of pointwise aggregation across the entirety of impractical for dynamic systems due to the constraints in the training dataset proves unattainable within distributed physical storage space and the escalating computational burden systems. Instead of employing the entire dataset for prediction, poses substantial practical challenges, particularly in real-time several approximation techniques prove instrumental. Moreover, local approximation B. Agent-based Gaussian Process methods, such as the naive local experts, the mixture of Distributed learning finds prominent application in multiagent experts, and the product of experts, present viable alternatives. Consequently, joint predictions are aggregated [8]. Several efforts have been dedicated to implementing distributed Prominently, cooperative learning within distributed systems Gaussian Process (DGP) methodologies within MASs.

This paper presents a novel approach for optimal control of nonlinear stochastic systems using infinitesimal generator learning within infinite-dimensional reproducing kernel Hilbert spaces. Our learning framework leverages data samples of system dynamics and stage cost functions, with only control penalties and constraints provided. The proposed method directly learns the diffusion operator of a controlled Fokker-Planck-Kolmogorov equation in an infinite-dimensional hypothesis space. This operator models the continuous-time evolution of the probability measure of the control system's state. We demonstrate that this approach seamlessly integrates with modern convex operator-theoretic Hamilton-Jacobi-Bellman recursions, enabling a data-driven solution to the optimal control problem. Furthermore, our statistical learning framework includes nonparametric estimators for uncontrolled forward infinitesimal generators as a special case. Numerical experiments, ranging from synthetic differential equations to simulated robotic systems, showcase the advantages of our approach compared to both modern data-driven and classical nonlinear programming methods for optimal control.

In real-world scenarios, the impacts of decisions may not manifest immediately. Taking these delays into account facilitates accurate assessment and management of risk in real-world environments, thereby ensuring the efficacy of strategies. In this paper, we investigate risk-averse learning using Conditional Value at Risk (CVaR) as risk measure, while incorporating delayed feedback with unknown but bounded delays. We develop two risk-averse learning algorithms that rely on one-point and two-point zeroth-order optimization approaches, respectively. The regret achieved by the algorithms is analyzed in terms of the cumulative delay and the number of total samplings. The results suggest that the two-point risk-averse learning achieves a smaller regret bound than the one-point algorithm. Furthermore, the one-point risk-averse learning algorithm attains sublinear regret under certain delay conditions, and the two-point risk-averse learning algorithm achieves sublinear regret with minimal restrictions on the delay. We provide numerical experiments on a dynamic pricing problem to demonstrate the performance of the proposed algorithms.

Ensuring safety and adapting to the user's behavior are of paramount importance in physical human-robot interaction. Thus, incorporating elastic actuators in the robot's mechanical design has become popular, since it offers intrinsic compliance and additionally provide a coarse estimate for the interaction force by measuring the deformation of the elastic components. While observer-based methods have been shown to improve these estimates, they rely on accurate models of the system, which are challenging to obtain in complex operating environments. In this work, we overcome this issue by learning the unknown dynamics components using Gaussian process (GP) regression. By employing the learned model in a Bayesian filtering framework, we improve the estimation accuracy and additionally obtain an observer that explicitly considers local model uncertainty in the confidence measure of the state estimate. Furthermore, we derive guaranteed estimation error bounds, thus, facilitating the use in safety-critical applications. We demonstrate the effectiveness of the proposed approach experimentally in a human-exoskeleton interaction scenario.

Learning from expert demonstrations to flexibly program an autonomous system with complex behaviors or to predict an agent's behavior is a powerful tool, especially in collaborative control settings. A common method to solve this problem is inverse reinforcement learning (IRL), where the observed agent, e.g., a human demonstrator, is assumed to behave according to the optimization of an intrinsic cost function that reflects its intent and informs its control actions. While the framework is expressive, it is also computationally demanding and generally lacks convergence guarantees. We therefore propose a novel, stability-certified IRL approach by reformulating the cost function inference problem to learning control Lyapunov functions (CLF) from demonstrations data. By additionally exploiting closed-form expressions for associated control policies, we are able to efficiently search the space of CLFs by observing the attractor landscape of the induced dynamics. For the construction of the inverse optimal CLFs, we use a Sum of Squares and formulate a convex optimization problem. We present a theoretical analysis of the optimality properties provided by the CLF and evaluate our approach using both simulated and real-world data.

Linearity of Koopman operators and simplicity of their estimators coupled with model-reduction capabilities has lead to their great popularity in applications for learning dynamical systems. While nonparametric Koopman operator learning in infinite-dimensional reproducing kernel Hilbert spaces is well understood for autonomous systems, its control system analogues are largely unexplored. Addressing systems with control inputs in a principled manner is crucial for fully data-driven learning of controllers, especially since existing approaches commonly resort to representational heuristics or parametric models of limited expressiveness and scalability. We address the aforementioned challenge by proposing a universal framework via control-affine reproducing kernels that enables direct estimation of a single operator even for control systems. The proposed approach, called control-Koopman operator regression (cKOR), is thus completely analogous to Koopman operator regression of the autonomous case. First in the literature, we present a nonparametric framework for learning Koopman operator representations of nonlinear control-affine systems that does not suffer from the curse of control input dimensionality. This allows for reformulating the infinite-dimensional learning problem in a finite-dimensional space based solely on data without apriori loss of precision due to a restriction to a finite span of functions or inputs as in other approaches. For enabling applications to large-scale control systems, we also enhance the scalability of control-Koopman operator estimators by leveraging random projections (sketching). The efficacy of our novel cKOR approach is demonstrated on both forecasting and control tasks.

Bone non-union is among the most severe complications associated with trauma surgery, occurring in 10-30% of cases after long bone fractures. Treating non-unions requires a high level of surgical expertise and often involves multiple revision surgeries, sometimes even leading to amputation. Thus, more accurate prognosis is crucial for patient well-being. Recent advances in machine learning (ML) hold promise for developing models to predict non-union healing, even when working with smaller datasets, a commonly encountered challenge in clinical domains. To demonstrate the effectiveness of ML in identifying candidates at risk of failed non-union healing, we applied three ML models (logistic regression, support vector machine, and XGBoost) to the clinical dataset TRUFFLE, which includes 797 patients with long bone non-union. The models provided prediction results with 70% sensitivity, and the specificities of 66% (XGBoost), 49% (support vector machine), and 43% (logistic regression). These findings offer valuable clinical insights because they enable early identification of patients at risk of failed non-union healing after the initial surgical revision treatment protocol.

Considering non-stationary environments in online optimization enables decision-maker to effectively adapt to changes and improve its performance over time. In such cases, it is favorable to adopt a strategy that minimizes the negative impact of change to avoid potentially risky situations. In this paper, we investigate risk-averse online optimization where the distribution of the random cost changes over time. We minimize risk-averse objective function using the Conditional Value at Risk (CVaR) as risk measure. Due to the difficulty in obtaining the exact CVaR gradient, we employ a zeroth-order optimization approach that queries the cost function values multiple times at each iteration and estimates the CVaR gradient using the sampled values. To facilitate the regret analysis, we use a variation metric based on Wasserstein distance to capture time-varying distributions. Given that the distribution variation is sub-linear in the total number of episodes, we show that our designed learning algorithm achieves sub-linear dynamic regret with high probability for both convex and strongly convex functions. Moreover, theoretical results suggest that increasing the number of samples leads to a reduction in the dynamic regret bounds until the sampling number reaches a specific limit. Finally, we provide numerical experiments of dynamic pricing in a parking lot to illustrate the efficacy of the designed algorithm.