Gaussian Process

Batch processes are inherently transient and typically nonlinear, motivating nonlinear model predictive control (NMPC). However, adopting NMPC is hindered by the cost and unavailability of dynamic models. Thus, we propose to use Gaussian Processes (GP) in a model-learning NMPC scheme (GP-MLMPC) for batch processes. We initialize the GP-MLMPC using data from a single initial trajectory, e.g., from a PI controller. We iteratively apply the NMPC embedded with GPs to run batches and update the GP with new observations from each iteration, thereby achieving batch-wise improvements. Using uncertainty quantification from the GPs, we formulate chance constraints to enforce safe operation to the required confidence levels. We demonstrate our approach in \textit{silico} on a semi-batch polymerization reactor for tracking and economic objectives over durations of two hours, and the reactor temperature is constrained in a range of $\pm2^\circ C$ around its setpoint. After only four batch iterations, tracking error from the GP-MLMPC scheme converged to a reduction of $83\%$, compared to the initial trajectory. Furthermore, under an economic objective, the GP-MLMPC resulted in a 17-fold increase in final product mass by iteration 8, compared to the initial trajectory. In both cases, the resulting GP-MLMPC performance is on par with the full-model NMPC, which shows that the optimal controller can be learned by the approach. By collecting samples around the optimal trajectory, the GP-MLMPC remains sample-efficient across iterations and achieves quick convergence. Thus, the proposed GP-MLMPC scheme presents a promising data-efficient approach for the control of nonlinear batch processes without mechanistic knowledge.

This paper presents a framework for mapping unknown scalar fields using a sensor-equipped autonomous robot operating in unsafe environments. The unsafe regions are defined as regions of high-intensity, where the field value exceeds a predefined safety threshold. For safe and efficient mapping of the scalar field, the sensor-equipped robot must avoid high-intensity regions during the measurement process. In this paper, the scalar field is modeled as a sample from a Gaussian process (GP), which enables Bayesian inference and provides closed-form expressions for both the predictive mean and the uncertainty. Concurrently, the spatial structure of the high-intensity regions is estimated in real-time using the Hough transform (HT), leveraging the evolving GP posterior. A safe sampling strategy is then employed to guide the robot towards safe measurement locations, using probabilistic safety guarantees on the evolving GP posterior. The estimated high-intensity regions also facilitate the design of safe motion plans for the robot. The effectiveness of the approach is verified through two numerical simulation studies and an indoor experiment for mapping a light-intensity field using a wheeled mobile robot.

Accurate forecasting of electric vehicle (EV) charging demand is critical for grid management and infrastructure planning. Yet the field continues to rely on legacy benchmarks; such as the Palo Alto (2020) dataset; that fail to reflect the scale and behavioral diversity of modern charging networks. To address this, we introduce a novel large-scale longitudinal dataset collected across Scotland (2022 2025), which release it as an open benchmark for the community. Building on this dataset, we formulate EV charging demand as a spatio-temporal latent Gaussian field and perform approximate Bayesian inference via Integrated Nested Laplace Approximation (INLA). The resulting model jointly captures spatial dependence, temporal dynamics, and covariate effects within a unified proba bilistic framework. On station-level forecasting tasks, our approach achieves competitive predictive accuracy against machine learning baselines, while additionally providing principled uncertainty quan tification and interpretable spatial and temporal decompositions properties that are essential for risk-aware infrastructure planning.

Efficient and robust path planning hinges on combining all accessible information sources. In particular, the task of path planning for robotic environmental exploration and monitoring depends highly on the current belief of the world. To capture the uncertainty in the belief, we present a Gaussian process based path planning method that adapts to multi-modal environmental sensing data and incorporates state and input constraints. To solve the path planning problem, we optimize over future waypoints in a receding horizon fashion, and our cost is thus a function of the Gaussian process posterior over all these waypoints. We demonstrate this method, dubbed OLAhGP, on an autonomous surface vessel using oceanic algal bloom data from both a high-fidelity model and in-situ sensing data in a monitoring scenario. Our simulated and experimental results demonstrate significant improvement over existing methods. With the same number of samples, our method generates more informative paths and achieves greater accuracy in identifying algal blooms in chlorophyll a rich waters, measured with respect to total misclassification probability and binary misclassification rate over the domain of interest.

We study the deterministic global optimization of trained Gaussian process posterior mean functions over hyperrectangular domains. Although the posterior mean function has a compact closed-form representation, its global optimization is challenging because it remains nonlinear and nonconvex. Existing exact deterministic approaches become increasingly difficult to scale as the number of training data points grows, leading to approximation-based methods that improve tractability by optimizing a modified (inexact) objective. In this work, we propose PALM-Mean, a piecewise-analytic lower-bounding framework embedded in reduced-space spatial branch-and-bound. At each node, kernel terms that are locally important are replaced by a sign-aware piecewise-linear relaxation in an appropriate scalar distance variable, while the remaining terms are bounded analytically in closed form. We show this hybrid approach yields a valid lower bound for the posterior mean, while limiting the size of the branch-and-bound subproblems. We establish validity of the node lower bounds and $\varepsilon$-global convergence of the resulting algorithm. Computational results on synthetic benchmarks and real-world application problems show that PALM-Mean improves scalability relative to representative general-purpose deterministic global solvers, particularly as the number of training data points increases.

Generalized Additive Models (GAMs) can be used to create non-linear glass-box (i.e. explicitly interpretable) models, where the predictive function is fully observable over the complete input space. However, glass-box interpretability itself does not allow for the incorporation of expert knowledge from the modeller. In this paper, we present ParamBoost, a novel GAM whose shape functions (i.e. mappings from individual input features to the output) are learnt using a Gradient Boosting algorithm that fits cubic polynomial functions at leaf nodes. ParamBoost incorporates several constraints commonly used in parametric analysis to ensure well-refined shape functions. These constraints include: (i) continuity of the shape functions and their derivatives (up to C2); (ii) monotonicity; (iii) convexity; (iv) feature interaction constraints; and (v) model specification constraints. Empirical results show that the unconstrained ParamBoost model consistently outperforms state-of-the-art GAMs across several real-world datasets. We further demonstrate that modellers can selectively impose required constraints at a modest trade-off in predictive performance, allowing the model to be fully tailored to application-specific interpretability and parametric-analysis requirements.

While Large Language Models (LLMs) exhibit exceptional zero-shot relevance modeling, their high computational cost necessitates framing passage retrieval as a budget-constrained global optimization problem. Existing approaches passively rely on first-stage dense retrievers, which leads to two limitations: (1) failing to retrieve relevant passages in semantically distinct clusters, and (2) failing to propagate relevance signals to the broader corpus. To address these limitations, we propose Bayesian Active Learning with Gaussian Processes guided by LLM relevance scoring (BAGEL), a novel framework that propagates sparse LLM relevance signals across the embedding space to guide global exploration. BAGEL models the multimodal relevance distribution across the entire embedding space with a query-specific Gaussian Process (GP) based on LLM relevance scores. Subsequently, it iteratively selects passages for scoring by strategically balancing the exploitation of high-confidence regions with the exploration of uncertain areas. Extensive experiments across four benchmark datasets and two LLM backbones demonstrate that BAGEL effectively explores and captures complex relevance distributions and outperforms LLM reranking methods under the same LLM budget on all four datasets.

We present GP-4DGS, a novel framework that integrates Gaussian Processes (GPs) into 4D Gaussian Splatting (4DGS) for principled probabilistic modeling of dynamic scenes. While existing 4DGS methods focus on deterministic reconstruction, they are inherently limited in capturing motion ambiguity and lack mechanisms to assess prediction reliability. By leveraging the kernel-based probabilistic nature of GPs, our approach introduces three key capabilities: (i) uncertainty quantification for motion predictions, (ii) motion estimation for unobserved or sparsely sampled regions, and (iii) temporal extrapolation beyond observed training frames. To scale GPs to the large number of Gaussian primitives in 4DGS, we design spatio-temporal kernels that capture the correlation structure of deformation fields and adopt variational Gaussian Processes with inducing points for tractable inference. Our experiments show that GP-4DGS enhances reconstruction quality while providing reliable uncertainty estimates that effectively identify regions of high motion ambiguity. By addressing these challenges, our work takes a meaningful step toward bridging probabilistic modeling and neural graphics.

Autonomous experimental systems are increasingly used in materials research to accelerate scientific discovery, but their performance is often limited by low-quality, noisy data. This issue is especially problematic in data-intensive structure-property learning tasks such as Image-to-Spectrum (Im2Spec) and Spectrum-to-Image (Spec2Im) translations, where standard active learning strategies can mistakenly prioritize poor-quality measurements. We introduce a gated active learning framework that combines curiosity-driven sampling with a physics-informed quality control filter based on the Simple Harmonic Oscillator model fits, allowing the system to automatically exclude low-fidelity data during acquisition. Evaluations on a pre-acquired dataset of band-excitation piezoresponse spectroscopy (BEPS) data from PbTiO3 thin films with spatially localized noise show that the proposed method outperforms random sampling, standard active learning, and multitask learning strategies. The gated approach enhances both Im2Spec and Spec2Im by handling noise during training and acquisition, leading to more reliable forward and inverse predictions. In contrast, standard active learners often misinterpret noise as uncertainty and end up acquiring bad samples that hurt performance. Given its promising applicability, we further deployed the framework in real-time experiments on BiFeO3 thin films, demonstrating its effectiveness in real autonomous microscopy experiments. Overall, this work supports a shift toward hybrid autonomy in self-driving labs, where physics-informed quality assessment and active decision-making work hand-in-hand for more reliable discovery.

Supervised machine learning describes the practice of fitting a parameterized model to labeled input-output data. Supervised machine learning methods have demonstrated promise in learning efficient surrogate models that can (partially) replace expensive high-fidelity models, making many-query analyses, such as optimization, uncertainty quantification, and inference, tractable. However, when training data must be obtained through the evaluation of an expensive model or experiment, the amount of training data that can be obtained is often limited, which can make learned surrogate models unreliable. In many engineering and scientific settings, cheaper low-fidelity models may be available, for example arising from simplified physics modeling or coarse grids. These models may be used to generate additional low-fidelity training data. The goal of multifidelity machine learning is to use both high- and low-fidelity training data to learn a surrogate model which is cheaper to evaluate than the high-fidelity model, but more accurate than any available low-fidelity model. This work proposes a new multifidelity training approach for Gaussian process regression which uses low-fidelity data to define additional features that augment the input space of the learned model. Similarly to cokriging estimators, the proposed approach conditions the high-fidelity surrogate model on the predictions of all available low-fidelity surrogate models, while benefiting from the computational efficiency of autoregressive estimators. Numerical experiments on several test problems demonstrate both increased predictive accuracy and reduced computational cost relative to the state of the art.

Non-conservative uncertainty bounds are essential for making reliable predictions about latent functions from noisy data, and thus, a key enabler for safe learning-based control. In this domain, kernel methods such as Gaussian process regression are established techniques, thanks to their inherent uncertainty quantification mechanism. Still, existing bounds either pose strong assumptions on the underlying noise distribution, are conservative, do not directly apply in the multi-output case, or are difficult to integrate into downstream tasks. This paper addresses these limitations by presenting a tight, deterministic bound for multi-output functions in Reproducing Kernel Hilbert Spaces (RKHSs) subject to bounded noise. It is obtained through an unconstrained, duality-based formulation, which shares the same structure as classic Gaussian process confidence bounds, and can thus be straightforwardly integrated into downstream optimization pipelines. We show that the proposed bound generalizes existing results and illustrate its application using an example inspired by quadrotor dynamics learning.

We analyze the Bayesian regret of the Gaussian process posterior sampling reinforcement learning (GP-PSRL) algorithm. Posterior sampling is a heuristic for decision-making under uncertainty that has been used to develop successful algorithms for a variety of continuous control problems. However, theoretical work on GP-PSRL is limited. All known regret bounds either have a sub-optimal growth rate, require strong smoothness assumptions, or fail to properly account for the fact that the set of possible system states is unbounded. Through a recursive application of the Borell-Tsirelson-Ibragimov-Sudakov inequality, we show that, with high probability, the states actually visited by the algorithm are contained within a ball of near-constant radius. We then use the chaining method to control the regret suffered by GP-PSRL under weak smoothness conditions. Our main result is a Bayesian regret bound of the order $\widetilde{\mathcal{O}}(H\sqrt{γ_TT})$, where $H$ is the horizon, $T$ is the number of time steps and $γ_T$ is the expected information gain. With this result, we resolve the limitations with prior theoretical work on PSRL, and provide the theoretical foundation and tools for analyzing PSRL in complex settings.

We consider the optimization problem of an expensive-to-evaluate black-box function, in which we can obtain noisy function values in parallel. For this problem, parallel Bayesian optimization (PBO) is a promising approach, which aims to optimize with fewer function evaluations by selecting a diverse input set for parallel evaluation. However, existing PBO methods suffer from poor practical performance or lack theoretical guarantees. In this study, we propose a PBO method, called randomized kriging believer (KB), based on a well-known KB heuristic and inheriting the advantages of the original KB: low computational complexity, a simple implementation, versatility across various BO methods, and applicability to asynchronous parallelization. Furthermore, we show that our randomized KB achieves Bayesian expected regret guarantees. We demonstrate the effectiveness of the proposed method through experiments, including those on real-data emulators.

Simulating a Gaussian process requires sampling from a high-dimensional Gaussian distribution, which scales cubically with the number of sample locations. Spectral methods address this challenge by exploiting the Fourier representation and treating the spectral density as a probability distribution suitable for Monte Carlo approximation. Although this probabilistic interpretation is valid for stationary processes, it is overly restrictive for the nonstationary case, where spectral densities are generally not probability measures. To avoid this limitation, we propose regular Fourier features for harmonizable processes with one-dimensional inputs. Our method discretizes the spectral representation directly, preserving the correlation structure among spectral weights without requiring probability assumptions. Assuming finite spectral support, this yields an efficient low-rank approximation that is positive semi-definite by construction and consistent under mild regularity conditions. When the spectral density is unknown, the framework also extends to kernel learning from data, which we explore as a proof of concept. We demonstrate the approximation on locally stationary and harmonizable mixture kernels, the latter with a complex-valued spectral density. As a feasibility study, we then apply the kernel-learning extension to real and synthetic data, where it matches competitive baselines.

Accurate production forecasts are essential for the integration of renewable energy sources into the power grid. This paper illustrates how to obtain probabilistic forecasts of wind power generation using gradient boosting trees and an ensemble of weather forecasts. To this end, we perform a comparative analysis across three state-of-the-art probabilistic prediction methods-conformalized quantile regression, natural gradient boosting and conditional diffusion models-all of which can be combined with tree-based machine learning. The methods are validated using four years of data for all Belgian offshore wind farms. We benchmark the models against the power curve and a calibrated wake model as well as a probabilistic method using stochastic variational Gaussian process regression. The tree-based models significantly reduce the mean absolute error in comparison to the deterministic baselines. Additionally, all three methods outperform the Gaussian process baseline in probabilistic skill, while two out of the three also improve point forecast accuracy. The conditional diffusion model attains the best performance, with improvements of 5% in mean absolute error and 12% in continuous rank probability score compared to the probabilistic baseline. Last, the results indicate an average improvement in point forecast accuracy of 17% by using an ensemble of weather forecasts instead of a single provider.

Mutual information (MI) is a principled and widely used objective for robotic information gathering (RIG), providing strong theoretical guarantees for sensor placement (SP) and informative path planning (IPP). However, its high computational cost - dominated by repeated log-determinant evaluations - has limited its use in real-time planning. This paper presents Schur-MI, a Gaussian process (GP) MI formulation that (i) leverages the iterative structure of RIG to precompute and reuse expensive intermediate quantities across planning steps, and (ii) uses a Schur-complement factorization to avoid large determinant computations. Together, these methods reduce the per-evaluation cost of MI from $\mathcal{O}(|\mathcal{V}|^3)$ to $\mathcal{O}(|\mathcal{A}|^3)$, where $\mathcal{V}$ and $\mathcal{A}$ denote the candidate and selected sensing locations, respectively. Experiments on real-world bathymetry datasets show that Schur-MI achieves up to a $12.7\times$ speedup over the standard MI formulation. Field trials with an autonomous surface vehicle (ASV) performing adaptive IPP further demonstrate the method's practicality. By making MI computation tractable for online planning, Schur-MI helps bridge the gap between information-theoretic objectives and real-time robotic exploration. Our code is available at: www.sgp-tools.com

In many experimental settings, active learning can improve sample efficiency by sequentially selecting where to measure, which is particularly valuable when experiments are expensive. Gaussian processes with variance-based acquisition criteria are widely used for this purpose, but have two limitations. First, they are observation-independent: their posterior variance depends only on where samples are acquired, not on what is measured, impairing their sensitivity to the structure of the acquired data. Second, they inflate the variance near boundaries, leading to excessive sampling at the edges of the space compared to the interior. These limitations undermine the gains in sampling efficiency expected from sequential acquisition. We address both limitations. We derive a reconstruction-driven design density and use the posterior mean to build a training-free warp that places more measurements where the target function varies rapidly. A geometric equalizer separately corrects boundary bias. Across sixteen synthetic and two real-data benchmarks, the geometric equalizer consistently improves function reconstruction by correcting boundary bias, while the reconstruction warp provides further gains by concentrating measurements where the posterior mean varies rapidly.

Large Vision-Language Models (LVLMs) often produce plausible but unreliable outputs, making robust uncertainty estimation essential. Recent work on semantic uncertainty estimates relies on external models to cluster multiple sampled responses and measure their semantic consistency. However, these clustering methods are often fragile, highly sensitive to minor phrasing variations, and can incorrectly group or separate semantically similar answers, leading to unreliable uncertainty estimates. We propose Semantic Gaussian Process Uncertainty (SGPU), a Bayesian framework that quantifies semantic uncertainty by analyzing the geometric structure of answer embeddings, avoiding brittle clustering. SGPU maps generated answers into a dense semantic space, computes the Gram matrix of their embeddings, and summarizes their semantic configuration via the eigenspectrum. This spectral representation is then fed into a Gaussian Process Classifier that learns to map patterns of semantic consistency to predictive uncertainty, and that can be applied in both black-box and white-box settings. Across six LLMs and LVLMs on eight datasets spanning VQA, image classification, and textual QA, SGPU consistently achieves state-of-the-art calibration (ECE) and discriminative (AUROC, AUARC) performance. We further show that SGPU transfers across models and modalities, indicating that its spectral representation captures general patterns of semantic uncertainty.

Monte Carlo Tree Search is a cornerstone algorithm for online planning, and its root-parallel variant is widely used when wall clock time is limited but best performance is desired. In environments with continuous action spaces, how to best aggregate statistics from different threads is an important yet underexplored question. In this work, we introduce a method that uses Gaussian Process Regression to obtain value estimates for promising actions that were not trialed in the environment. We perform a systematic evaluation across 6 different domains, demonstrating that our approach outperforms existing aggregation strategies while requiring a modest increase in inference time.

Sequential optimization of black-box functions from noisy evaluations has been widely studied, with Gaussian Process bandit algorithms such as GP-UCB guaranteeing no-regret in stationary settings. However, for time-varying objectives, no-regret is unattainable under pure bandit feedback unless strong and often unrealistic assumptions are imposed. We propose a novel method for optimizing time-varying rewards in the frequentist setting, where the objective has bounded RKHS norm almost surely. Time variations are captured through uncertainty injection, enabling heteroscedastic Gaussian process regression that adapts past observations to the current time step. As no-regret is unattainable in general in the strict bandit setting, we relax the latter allowing additional queries on previously observed points. Building on sparse inference and the effect of uncertainty injection on regret, we propose W-SparQ-GP-UCB, an online algorithm that achieves no-regret with a vanishing number of additional queries per iteration. To assess the theoretical limits of this approach, we establish a lower bound on the number of additional queries required for no-regret, proving the efficiency of our method. Finally, we provide a comprehensive analysis linking the temporal regime of the function to achievable regret rates, together with upper and lower bounds on the number of additional queries needed in each regime.

Efficient Gaussian process (GP) inference is critical for sequential decision-making tasks such as active learning, online prediction, and Bayesian optimization. Iterative approaches of approximating the GP posterior using solvers like conjugate gradients, stochastic gradient descent, or alternating projections avoid cubic costs, but often require many iterations to converge, limiting their efficacy when the posterior is updated frequently with new data. To address this, we introduce three warm-start strategies that exploit solutions of smaller linear systems to substantially speed-up convergence when updating the posterior with new data. Our methods are supported by theoretical analysis showing reduced initialization error in reproducing kernel Hilbert space (RKHS) distance, and by empirical results on regression benchmarks and Bayesian optimization tasks. Across solvers, warm-starting achieves speed-ups of up to 19x when solving to tolerance, and produces more accurate posterior estimates under fixed compute budgets, directly improving optimization performance. These results establish warm-starting as a simple, effective, and broadly applicable tool for scaling Gaussian processes in sequential settings.

Collaborative partnerships play a crucial role in inquiry-oriented education. However, most learning partners are currently assigned through experience-driven heuristics or rule-based machine assistants, which often result in limited knowledge expansion and low adaptability. To address these challenges, this study introduces InqEduAgent, an LLM-empowered generative agent framework designed to simulate and select adaptive learning partners for inquiry-based learning. InqEduAgent integrates a Gaussian process-augmented matching mechanism to model the cognitive and evaluative characteristics of learners, allowing adaptive partner selection based on prior knowledge patterns. Comprehensive experiments demonstrate that InqEduAgent consistently achieves superior performance across diverse learning scenarios and large language model configurations. This study advances human-AI collaborative learning by enabling intelligent pairing between human- and AI-based learning partners, and contributes to adaptive user modeling and personalized recommendation within Web-based educational environments.

This monograph studies the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis. We study connections and equivalences for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. The monograph serves as a basis to bridge many other methods based on Gaussian processes and reproducing kernels, which are developed in parallel by the two research communities.

In this paper, we propose a Gaussian Process (GP)-based policy iteration framework for addressing both forward and inverse problems in Hamilton--Jacobi--Bellman (HJB) equations and mean field games (MFGs). Policy iteration is formulated as an alternating procedure between evaluating the value function under a fixed control policy and improving the policy. In our approach, we model the unknown fields using GPs within a policy-iteration framework that converts the nonlinear system into a sequence of linear PDE subproblems. Then, leveraging the linear structure, the updates for the value function and, in the MFG setting, the population density admit explicit representer formulas under linear PDE collocation constraints. The policy is subsequently updated pointwise via a Legendre transform step, which involves a low-dimensional maximization over the control variable. This maximization is explicit for standard quadratic costs. For smooth, strictly convex costs, this pointwise maximization is solved through its first-order optimality condition, whereas in constrained or non-smooth cases, it becomes a low-dimensional constrained maximization problem. To improve convergence, we incorporate the additive Schwarz acceleration as a preconditioning step following each policy update. Numerical experiments demonstrate the effectiveness of the Schwarz acceleration in improving computational efficiency.

Novelty search (NS) aims to uncover diverse system behaviors through simulation or experiment without requiring a pre-specified scalar objective. This capability is especially relevant to modern discovery problems in chemistry, materials science, and molecular design, where researchers often seek broad coverage of attainable property space rather than a single optimum and where each evaluation may require a costly computation or experiment. For such expensive black-box settings, we propose BEACON, a sample-efficient NS strategy inspired by Bayesian optimization principles. BEACON models the input-to-outcome mapping using multi-output Gaussian processes and selects new inputs by scoring how far plausible posterior outcomes lie from a denoised archive of previously observed outcomes. This gives a distance-based novelty acquisition that accounts for predictive uncertainty and observational noise while operating directly in continuous outcome space, rather than requiring direct optimization over a discretized partition of behaviors. By leveraging efficient posterior sampling together with scalable high-dimensional Gaussian process models, the proposed framework can be extended to settings with large data sets and high-dimensional design variables. We demonstrate BEACON on established benchmark problems together with real-world case studies in materials and molecular discovery. Across these settings, BEACON consistently discovers broader sets of distinct behaviors than several competing baselines under limited evaluation budgets.