Gaussian processes (GPs) are non-parametric probabilistic regression models that are popular due to their flexibility, data efficiency, and well-calibrated uncertainty estimates. However, standard GP models assume homoskedastic Gaussian noise, while many real-world applications are subject to non-Gaussian corruptions. Variants of GPs that are more robust to alternative noise models have been proposed, and entail significant trade-offs between accuracy and robustness, and between computational requirements and theoretical guarantees. In this work, we propose and study a GP model that achieves robustness against sparse outliers by inferring data-point-specific noise levels with a sequential selection procedure maximizing the log marginal likelihood that we refer to as relevance pursuit. We show, surprisingly, that the model can be parameterized such that the associated log marginal likelihood is strongly concave in the data-point-specific noise variances, a property rarely found in either robust regression objectives or GP marginal likelihoods. This in turn implies the weak submodularity of the corresponding subset selection problem, and thereby proves approximation guarantees for the proposed algorithm. We compare the model's performance relative to other approaches on diverse regression and Bayesian optimization tasks, including the challenging but common setting of sparse corruptions of the labels within or close to the function range.

Inverse problems can be described as limited-data problems in which the signal of interest cannot be observed directly. A physics-based forward model that relates the signal with the observations is typically needed. Unfortunately, unknown model parameters and imperfect forward models can undermine the signal recovery. Even though supervised machine learning offers promising avenues to improve the robustness of the solutions, we have to rely on model-based learning when there is no access to ground truth for the training. Here, we studied a linear inverse problem that included an unknown non-linear model parameter and utilized a Bayesian model-based learning approach that allowed signal recovery and subsequently estimation of the model parameter. This approach, called Bayesian Approximation Error approach, employed a simplified model of the physics of the problem augmented with an approximation error term that compensated for the simplification. An error subspace was spanned with the help of the eigenvectors of the approximation error covariance matrix which allowed, alongside the primary signal, simultaneous estimation of the induced error. The estimated error and signal were then used to determine the unknown model parameter. For the model parameter estimation, we tested different approaches: a conditional Gaussian regression, an iterative (model-based) optimization, and a Gaussian process that was modeled with the help of physics-informed learning. In addition, alternating optimization was used as a reference method. As an example application, we focused on the problem of reconstructing brain activity from EEG recordings under the condition that the electrical conductivity of the patient's skull was unknown in the model. Our results demonstrated clear improvements in EEG source localization accuracy and provided feasible estimates for the unknown model parameter, skull conductivity.

This paper is concerned with the approximation of probability distributions known up to normalization constants, with a focus on Bayesian inference for large-scale inverse problems in scientific computing. In this context, key challenges include costly repeated evaluations of forward models, multimodality, and inaccessible gradients for the forward model. To address them, we develop a variational inference framework that combines Fisher-Rao natural gradient with specialized quadrature rules to enable derivative free updates of Gaussian mixture variational families. The resulting method, termed Derivative Free Gaussian Mixture Variational Inference (DF-GMVI), guarantees covariance positivity and affine invariance, offering a stable and efficient framework for approximating complex posterior distributions. The effectiveness of DF-GMVI is demonstrated through numerical experiments on challenging scenarios, including distributions with multiple modes, infinitely many modes, and curved modes in spaces with up to hundreds of dimensions. The method's practicality is further demonstrated in a large-scale application, where it successfully recovers the initial conditions of the Navier-Stokes equations from solution data at positive times.

A Multi-robot system (MRS) provides significant advantages for intricate tasks such as environmental monitoring, underwater inspections, and space missions. However, addressing potential communication failures or the lack of communication infrastructure in these fields remains a challenge. A significant portion of MRS research presumes that the system can maintain communication with proximity constraints, but this approach does not solve situations where communication is either non-existent, unreliable, or poses a security risk. Some approaches tackle this issue using predictions about other robots while not communicating, but these methods generally only permit agents to utilize first-order reasoning, which involves reasoning based purely on their own observations. In contrast, to deal with this problem, our proposed framework utilizes Theory of Mind (ToM), employing higher-order reasoning by shifting a robot's perspective to reason about a belief of others observations. Our approach has two main phases: i) an efficient runtime plan adaptation using active inference to signal intentions and reason about a robot's own belief and the beliefs of others in the system, and ii) a hierarchical epistemic planning framework to iteratively reason about the current MRS mission state. The proposed framework outperforms greedy and first-order reasoning approaches and is validated using simulations and experiments with heterogeneous robotic systems.

However, the use of neural networks presents some disadvantages, the most significant of which is their lack of explainability. This means that most neural networks In many astrophysical applications, statistical models are treated as a'black box', where the decisions taken can be simulated forward, but their likelihood functions by the artificial intelligence in arriving at the optimized are too complex to calculate directly. Simulation-based solution are not known to researchers, which can hinder inference (SBI) [1] provides an alternative way to perform intellectual oversight [18]. This problem affects the Bayesian analysis on these models, relying solely on forward algorithms discussed above, as NRE constitutes an unsupervised simulations rather than likelihood estimates. However, learning task, where the artificial intelligence is modern cosmological models are typically expensive given unlabeled input data and allowed to discover patterns to simulate and datasets are often high-dimensional, in its distribution without guidance. This combines so traditional methods like the Approximate Bayesian with the problem of over-fitting, where the neural network Computation (ABC) [2], which scale poorly with dimensionality, may attempt to maximize the likelihood without are no longer suitable for parameter estimation.

This paper presents a novel machine-learning framework for reconstructing low-order gust-encounter flow field and lift coefficients from sparse, noisy surface pressure measurements. Our study thoroughly investigates the time-varying response of sensors to gust-airfoil interactions, uncovering valuable insights into optimal sensor placement. To address uncertainties in deep learning predictions, we implement probabilistic regression strategies to model both epistemic and aleatoric uncertainties. Epistemic uncertainty, reflecting the model's confidence in its predictions, is modeled using Monte Carlo dropout, as an approximation to the variational inference in the Bayesian framework, treating the neural network as a stochastic entity. On the other hand, aleatoric uncertainty, arising from noisy input measurements, is captured via learned statistical parameters, which propagates measurement noise through the network into the final predictions. Our results showcase the efficacy of this dual uncertainty quantification strategy in accurately predicting aerodynamic behavior under extreme conditions while maintaining computational efficiency, underscoring its potential to improve online sensor-based flow estimation in real-world applications.

Time-delayed differential equations (TDDEs) are widely used to model complex dynamic systems where future states depend on past states with a delay. However, inferring the underlying TDDEs from observed data remains a challenging problem due to the inherent nonlinearity, uncertainty, and noise in real-world systems. Conventional equation discovery methods often exhibit limitations when dealing with large time delays, relying on deterministic techniques or optimization-based approaches that may struggle with scalability and robustness. In this paper, we present BayTiDe - Bayesian Approach for Discovering Time-Delayed Differential Equations from Data, that is capable of identifying arbitrarily large values of time delay to an accuracy that is directly proportional to the resolution of the data input to it. BayTiDe leverages Bayesian inference combined with a sparsity-promoting discontinuous spike-and-slab prior to accurately identify time-delayed differential equations. The approach accommodates arbitrarily large time delays with accuracy proportional to the input data resolution, while efficiently narrowing the search space to achieve significant computational savings. We demonstrate the efficiency and robustness of BayTiDe through a range of numerical examples, validating its ability to recover delayed differential equations from noisy data.

Designing effective prompts is essential to guiding large language models (LLMs) toward desired responses. Automated prompt engineering aims to reduce reliance on manual effort by streamlining the design, refinement, and optimization of natural language prompts. This paper proposes an optimal learning framework for automated prompt engineering, designed to sequentially identify effective prompt features while efficiently allocating a limited evaluation budget. We introduce a feature-based method to express prompts, which significantly broadens the search space. Bayesian regression is employed to utilize correlations among similar prompts, accelerating the learning process. To efficiently explore the large space of prompt features for a high quality prompt, we adopt the forward-looking Knowledge-Gradient (KG) policy for sequential optimal learning. The KG policy is computed efficiently by solving mixed-integer second-order cone optimization problems, making it scalable and capable of accommodating prompts characterized only through constraints. We demonstrate that our method significantly outperforms a set of benchmark strategies assessed on instruction induction tasks. The results highlight the advantages of using the KG policy for prompt learning given a limited evaluation budget. Our framework provides a solution to deploying automated prompt engineering in a wider range applications where prompt evaluation is costly.

Importance sampling is a rare event simulation technique used in Monte Carlo simulations to bias the sampling distribution towards the rare event of interest. By assigning appropriate weights to sampled points, importance sampling allows for more efficient estimation of rare events or tails of distributions. However, importance sampling can fail when the proposal distribution does not effectively cover the target distribution. In this work, we propose a method for more efficient sampling by updating the proposal distribution in the latent space of a normalizing flow. Normalizing flows learn an invertible mapping from a target distribution to a simpler latent distribution. The latent space can be more easily explored during the search for a proposal distribution, and samples from the proposal distribution are recovered in the space of the target distribution via the invertible mapping. We empirically validate our methodology on simulated robotics applications such as autonomous racing and aircraft ground collision avoidance.

Outlier detection refers to the identification of anomalous samples that deviate significantly from the distribution of normal data and has been extensively studied and used in a variety of practical tasks. However, most unsupervised outlier detection methods are carefully designed to detect specified outliers, while real-world data may be entangled with different types of outliers. In this study, we propose a fuzzy rough sets-based multi-scale outlier detection method to identify various types of outliers. Specifically, a novel fuzzy rough sets-based method that integrates relative fuzzy granule density is first introduced to improve the capability of detecting local outliers. Then, a multi-scale view generation method based on granular-ball computing is proposed to collaboratively identify group outliers at different levels of granularity. Moreover, reliable outliers and inliers determined by the three-way decision are used to train a weighted support vector machine to further improve the performance of outlier detection. The proposed method innovatively transforms unsupervised outlier detection into a semi-supervised classification problem and for the first time explores the fuzzy rough sets-based outlier detection from the perspective of multi-scale granular balls, allowing for high adaptability to different types of outliers. Extensive experiments carried out on both artificial and UCI datasets demonstrate that the proposed outlier detection method significantly outperforms the state-of-the-art methods, improving the results by at least 8.48% in terms of the Area Under the ROC Curve (AUROC) index. { The source codes are released at \url{https://github.com/Xiaofeng-Tan/MGBOD}. }