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Conformal inference is a fundamental and versatile tool that provides distribution-free guarantees for many machine learning tasks. We consider the transductive setting, where decisions are made on a test sample of $m$ new points, giving rise to $m$ conformal $p$-values. {While classical results only concern their marginal distribution, we show that their joint distribution follows a P\'olya urn model, and establish a concentration inequality for their empirical distribution function.} The results hold for arbitrary exchangeable scores, including {\it adaptive} ones that can use the covariates of the test+calibration samples at training stage for increased accuracy. We demonstrate the usefulness of these theoretical results through uniform, in-probability guarantees for two machine learning tasks of current interest: interval prediction for transductive transfer learning and novelty detection based on two-class classification.

Ensemble forecasts and their combination are explored from the perspective of a probability space. Manipulating ensemble forecasts as discrete probability distributions, multi-model ensembles (MMEs) are reformulated as barycenters of these distributions. Barycenters are defined with respect to a given distance. The barycenter with respect to the L2-distance is shown to be equivalent to the pooling method. Then, the barycenter-based approach is extended to a different distance with interesting properties in the distribution space: the Wasserstein distance. Another interesting feature of the barycenter approach is the possibility to give different weights to the ensembles and so to naturally build weighted MME. As a proof of concept, the L2- and the Wasserstein-barycenters are applied to combine two models from the S2S database, namely the European Centre Medium-Range Weather Forecasts (ECMWF) and the National Centers for Environmental Prediction (NCEP) models. The performance of the two (weighted-) MMEs are evaluated for the prediction of weekly 2m-temperature over Europe for seven winters. The weights given to the models in the barycenters are optimized with respect to two metrics, the CRPS and the proportion of skilful forecasts. These weights have an important impact on the skill of the two barycenter-based MMEs. Although the ECMWF model has an overall better performance than NCEP, the barycenter-ensembles are generally able to outperform both. However, the best MME method, but also the weights, are dependent on the metric. These results constitute a promising first implementation of this methodology before moving to combination of more models.

The fundamental computational issues in Bayesian inverse problems (BIPs) governed by partial differential equations (PDEs) stem from the requirement of repeated forward model evaluations. A popular strategy to reduce such cost is to replace expensive model simulations by computationally efficient approximations using operator learning, motivated by recent progresses in deep learning. However, using the approximated model directly may introduce a modeling error, exacerbating the already ill-posedness of inverse problems. Thus, balancing between accuracy and efficiency is essential for the effective implementation of such approaches. To this end, we develop an adaptive operator learning framework that can reduce modeling error gradually by forcing the surrogate to be accurate in local areas. This is accomplished by fine-tuning the pre-trained approximate model during the inversion process with adaptive points selected by a greedy algorithm, which requires only a few forward model evaluations. To validate our approach, we adopt DeepOnet to construct the surrogate and use unscented Kalman inversion (UKI) to approximate the solution of BIPs, respectively. Furthermore, we present rigorous convergence guarantee in the linear case using the framework of UKI. We test the approach on several benchmarks, including the Darcy flow, the heat source inversion problem, and the reaction diffusion problems. Numerical results demonstrate that our method can significantly reduce computational costs while maintaining inversion accuracy.

A novel spatial autoregressive model for panel data is introduced, which incorporates multilayer networks and accounts for time-varying relationships. Moreover, the proposed approach allows the structural variance to evolve smoothly over time and enables the analysis of shock propagation in terms of time-varying spillover effects. The framework is applied to analyse the dynamics of international relationships among the G7 economies and their impact on stock market returns and volatilities. The findings underscore the substantial impact of cooperative interactions and highlight discernible disparities in network exposure across G7 nations, along with nuanced patterns in direct and indirect spillover effects.

Generative diffusion models have achieved spectacular performance in many areas of generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We argue that this lead to a form of instability that lies at the heart of their generative capabilities and that can be described by a set of mean field critical exponents. We conclude by analyzing recent work connecting diffusion models and associative memory networks in view of the thermodynamic formulations.

We address speech enhancement based on variational autoencoders, which involves learning a speech prior distribution in the time-frequency (TF) domain. A zero-mean complex-valued Gaussian distribution is usually assumed for the generative model, where the speech information is encoded in the variance as a function of a latent variable. In contrast to this commonly used approach, we propose a weighted variance generative model, where the contribution of each spectrogram time-frame in parameter learning is weighted. We impose a Gamma prior distribution on the weights, which would effectively lead to a Student's t-distribution instead of Gaussian for speech generative modeling. We develop efficient training and speech enhancement algorithms based on the proposed generative model. Our experimental results on spectrogram auto-encoding and speech enhancement demonstrate the effectiveness and robustness of the proposed approach compared to the standard unweighted variance model.

In recent decades, the use of optical detection systems for meteor studies has increased dramatically, resulting in huge amounts of data being analyzed. Automated meteor detection tools are essential for studying the continuous meteoroid incoming flux, recovering fresh meteorites, and achieving a better understanding of our Solar System. Concerning meteor detection, distinguishing false positives between meteor and non-meteor images has traditionally been performed by hand, which is significantly time-consuming. To address this issue, we developed a fully automated pipeline that uses Convolutional Neural Networks (CNNs) to classify candidate meteor detections. Our new method is able to detect meteors even in images that contain static elements such as clouds, the Moon, and buildings. To accurately locate the meteor within each frame, we employ the Gradient-weighted Class Activation Mapping (Grad-CAM) technique. This method facilitates the identification of the region of interest by multiplying the activations from the last convolutional layer with the average of the gradients across the feature map of that layer. By combining these findings with the activation map derived from the first convolutional layer, we effectively pinpoint the most probable pixel location of the meteor. We trained and evaluated our model on a large dataset collected by the Spanish Meteor Network (SPMN) and achieved a precision of 98\%. Our new methodology presented here has the potential to reduce the workload of meteor scientists and station operators and improve the accuracy of meteor tracking and classification.

This proposed model introduces novel deep learning methodologies. The objective here is to create a reliable intrusion detection mechanism to help identify malicious attacks. Deep learning based solution framework is developed consisting of three approaches. The first approach is Long-Short Term Memory Recurrent Neural Network (LSTM-RNN) with seven optimizer functions such as adamax, SGD, adagrad, adam, RMSprop, nadam and adadelta. The model is evaluated on NSL-KDD dataset and classified multi attack classification. The model has outperformed with adamax optimizer in terms of accuracy, detection rate and low false alarm rate. The results of LSTM-RNN with adamax optimizer is compared with existing shallow machine and deep learning models in terms of accuracy, detection rate and low false alarm rate. The multi model methodology consisting of Recurrent Neural Network (RNN), Long-Short Term Memory Recurrent Neural Network (LSTM-RNN), and Deep Neural Network (DNN). The multi models are evaluated on bench mark datasets such as KDD99, NSL-KDD, and UNSWNB15 datasets. The models self-learnt the features and classifies the attack classes as multi-attack classification. The models RNN, and LSTM-RNN provide considerable performance compared to other existing methods on KDD99 and NSL-KDD dataset

The multispecies Landau collision operator describes the two-particle, small scattering angle or grazing collisions in a plasma made up of different species of particles such as electrons and ions. Recently, a structure preserving deterministic particle method arXiv:1910.03080 has been developed for the single species spatially homogeneous Landau equation. This method relies on a regularization of the Landau collision operator so that an approximate solution, which is a linear combination of Dirac delta distributions, is well-defined. Based on a weak form of the regularized Landau equation, the time dependent locations of the Dirac delta functions satisfy a system of ordinary differential equations. In this work, we extend this particle method to the multispecies case, and examine its conservation of mass, momentum, and energy, and decay of entropy properties. We show that the equilibrium distribution of the regularized multispecies Landau equation is a Maxwellian distribution, and state a critical condition on the regularization parameters that guarantees a species independent equilibrium temperature. A convergence study comparing an exact multispecies BKW solution to the particle solution shows approximately 2nd order accuracy. Important physical properties such as conservation, decay of entropy, and equilibrium distribution of the particle method are demonstrated with several numerical examples.

In variable selection, a selection rule that prescribes the permissible sets of selected variables (called a "selection dictionary") is desirable due to the inherent structural constraints among the candidate variables. Such selection rules can be complex in real-world data analyses, and failing to incorporate such restrictions could not only compromise the interpretability of the model but also lead to decreased prediction accuracy. However, no general framework has been proposed to formalize selection rules and their applications, which poses a significant challenge for practitioners seeking to integrate these rules into their analyses. In this work, we establish a framework for structured variable selection that can incorporate universal structural constraints. We develop a mathematical language for constructing arbitrary selection rules, where the selection dictionary is formally defined. We demonstrate that all selection rules can be expressed as combinations of operations on constructs, facilitating the identification of the corresponding selection dictionary. Once this selection dictionary is derived, practitioners can apply their own user-defined criteria to select the optimal model. Additionally, our framework enhances existing penalized regression methods for variable selection by providing guidance on how to appropriately group variables to achieve the desired selection rule. Furthermore, our innovative framework opens the door to establishing new l0 norm-based penalized regression techniques that can be tailored to respect arbitrary selection rules, thereby expanding the possibilities for more robust and tailored model development.