This paper addresses the challenge of reconstructing full-field structural mode shapes from sparse sensor data. While Gaussian Process Regression (GPR) offers a robust non-parametric framework for spatial interpolation and uncertainty quantification, standard formulations often yield physically inconsistent mode-shape reconstructions under sparse sensing conditions. A Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework is derived that utilizes independent modal kernels while coupling the optimization via a mass-orthogonality penalty. The paper presents derivations for the marginal likelihood, hyperparameter gradients, and penalty coupling. Numerical verification on a multi-degree-of-freedom structure demonstrates that the proposed method overcomes existing limitations in GP-based prediction, providing more accurate and reliable expanded mode shapes.

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.

We provide the supplements of "Contextual Gaussian Process Bandits with Neural Networks" here. Specifically, we discuss alternative acquisition functions that can be incorporated with the neural network-accompanied Gaussian process (NN-AGP) model in Section 6. In Section 7, we discuss the bandit algorithm with NN-AGP, where the neural network approximation error is considered. In Section 8, we provide the detailed proof of theorems. We provide the experimental details and include additional numerical experiments in Section 9. Last we discuss the limitations of NN-AGP and propose the potential approaches to addressing the limitations for future work, including sparse NN-AGP for alleviating computational burdens and transfer learning with NN-AGP to address cold-start issue; see Section 10. In the main text, we employ the upper confidence bound function as the acquisition function in the contextual Bayesian optimization approach. Here, we provide two alternative choices: Thompson sampling (TS) and knowledge gradient (KG). We describe the two procedures of the contextual GP bandit problems with NN-AGP, where the acquisition function is replaced by TS or KG. It chooses the action that maximizes the expected reward with respect to a random belief that is drawn for a posterior distribution. Besides the multi-armed bandit problems, TS has also achieved both theoretical and practical success in BO and Gaussian process regression. For more detailed discussions on TS, we refer to [87, 88]. Specifically, we propose a neural network-accompanied Gaussian process Thompson sampling (NNAGP-TS) approach to address contextual GP bandits. The approach works as follows. In each iteration, NN-AGP-TS first fits an NN-AGP model with the historic data. Then, given the current contextual variable, a realization of the Gaussian process with respect to x X is sampled from the posterior distribution conditional on the historic data1.

Kernel selection plays a central role in determining the performance of Gaussian Process (GP) models, as the chosen kernel determines both the inductive biases and prior support of functions under the GP prior. This work addresses the challenge of constructing custom kernel functions for high-dimensional GP regression models. Drawing inspiration from recent progress in deep learning, we introduce a novel approach named KITT: Kernel Identification Through Transformers. KITT exploits a transformer-based architecture to generate kernel recommendations in under 0.1 seconds, which is several orders of magnitude faster than conventional kernel search algorithms. We train our model using synthetic data generated from priors over a vocabulary of known kernels. By exploiting the nature of the selfattention mechanism, KITT is able to process datasets with inputs of arbitrary dimension. We demonstrate that kernels chosen by KITT yield strong performance over a diverse collection of regression benchmarks.

We say a real-valued random variable X is -sub-Gaussian if it its mean is zero and for all " 2 R we have E[exp("X)] exp Such assumptions on the noise variables are frequently used in bandit optimization. Typically, in kernelized bandits, we assume that unknown f 2F k(D;B)= {f 2H k(D): kfkk B}, where Hk(D) is the reproducing kernel Hilbert space of functions associated with the given positive-definite kernel function. Typically, the learner knows Fk(D;B), meaning that both k(,) and B are considered as input to the learner's algorithm. We outline some commonly used kernel functions k: D D! R, that we also consider: Linear kernel: klin(x,x0)= xTx0, Squared exponential kernel: kSE(x,x0)=exp kx x0k2 2l2, Matérn kernel: kMat(x,x0)= 2 Maximum information gain is a kernel-dependent quantity that measures the complexity of the given function class. It has first been introduced in [40], and since then it has been used in numerous works on Gaussian process bandits.

Given only information in the form of similarity triplets Object A is more similar to object B than to object C about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similarity triplets, we aim at defining kernel functions that correspond to high-dimensional embeddings. These kernel functions can subsequently be used to apply any kernel method to the data set.

Out-of-Distribution (OoD) detection is vital for the reliability of Deep Neural Networks (DNNs).

In this paper we present a novel method for efficient and effective 3D surface reconstruction in open scenes.

Parameter-efficient fine-tuning (PEFT) is an effective method for adapting pre-trained vision models to downstream tasks by tuning a small subset of parameters.