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The swift progression of machine learning (ML) have not gone unnoticed in the realm of statistical mechanics. ML techniques have attracted attention by the classical density-functional theory (DFT) community, as they enable discovery of free-energy functionals to determine the equilibrium-density profile of a many-particle system. Within DFT, the external potential accounts for the interaction of the many-particle system with an external field, thus, affecting the density distribution. In this context, we introduce a statistical-learning framework to infer the external potential exerted on a many-particle system. We combine a Bayesian inference approach with the classical DFT apparatus to reconstruct the external potential, yielding a probabilistic description of the external potential functional form with inherent uncertainty quantification. Our framework is exemplified with a grand-canonical one-dimensional particle ensemble with excluded volume interactions in a confined geometry. The required training dataset is generated using a Monte Carlo (MC) simulation where the external potential is applied to the grand-canonical ensemble. The resulting particle coordinates from the MC simulation are fed into the learning framework to uncover the external potential. This eventually allows us to compute the equilibrium density profile of the system by using the tools of DFT. Our approach benchmarks the inferred density against the exact one calculated through the DFT formulation with the true external potential. The proposed Bayesian procedure accurately infers the external potential and the density profile. We also highlight the external-potential uncertainty quantification conditioned on the amount of available simulated data. The seemingly simple case study introduced in this work might serve as a prototype for studying a wide variety of applications, including adsorption and capillarity.

This study focuses on the use of model and data fusion for improving the Spalart-Allmaras (SA) closure model for Reynolds-averaged Navier-Stokes solutions of separated flows. In particular, our goal is to develop of models that not-only assimilate sparse experimental data to improve performance in computational models, but also generalize to unseen cases by recovering classical SA behavior. We achieve our goals using data assimilation, namely the Ensemble Kalman Filtering approach (EnKF), to calibrate the coefficients of the SA model for separated flows. A holistic calibration strategy is implemented via a parameterization of the production, diffusion, and destruction terms. This calibration relies on the assimilation of experimental data collected velocity profiles, skin friction, and pressure coefficients for separated flows. Despite using of observational data from a single flow condition around a backward-facing step (BFS), the recalibrated SA model demonstrates generalization to other separated flows, including cases such as the 2D-bump and modified BFS. Significant improvement is observed in the quantities of interest, i.e., skin friction coefficient ($C_f$) and pressure coefficient ($C_p$) for each flow tested. Finally, it is also demonstrated that the newly proposed model recovers SA proficiency for external, unseparated flows, such as flow around a NACA-0012 airfoil without any danger of extrapolation, and that the individually calibrated terms in the SA model are targeted towards specific flow-physics wherein the calibrated production term improves the re-circulation zone while destruction improves the recovery zone.

Modern high-throughput sequencing assays efficiently capture not only gene expression and different levels of gene regulation but also a multitude of genome variants. Focused analysis of alternative alleles of variable sites at homologous chromosomes of the human genome reveals allele-specific gene expression and allele-specific gene regulation by assessing allelic imbalance of read counts at individual sites. Here we formally describe an advanced statistical framework for detecting the allelic imbalance in allelic read counts at single-nucleotide variants detected in diverse omics studies (ChIP-Seq, ATAC-Seq, DNase-Seq, CAGE-Seq, and others). MIXALIME accounts for copy-number variants and aneuploidy, reference read mapping bias, and provides several scoring models to balance between sensitivity and specificity when scoring data with varying levels of experimental noise-caused overdispersion.

The design dataset is the backbone of data-driven design. Ideally, the dataset should be fairly distributed in both shape and property spaces to efficiently explore the underlying relationship. However, the classical experimental design focuses on shape diversity and thus yields biased exploration in the property space. Recently developed methods either conduct subset selection from a large dataset or employ assumptions with severe limitations. In this paper, fairness- and uncertainty-aware data generation (FairGen) is proposed to actively detect and generate missing properties starting from a small dataset. At each iteration, its coverage module computes the data coverage to guide the selection of the target properties. The uncertainty module ensures that the generative model can make certain and thus accurate shape predictions. Integrating the two modules, Bayesian optimization determines the target properties, which are thereafter fed into the generative model to predict the associated shapes. The new designs, whose properties are analyzed by simulation, are added to the design dataset. An S-slot design dataset case study was implemented to demonstrate the efficiency of FairGen in auxetic structural design. Compared with grid and randomized sampling, FairGen increased the coverage score at twice the speed and significantly expanded the sampled region in the property space. As a result, the generative models trained with FairGen-generated datasets showed consistent and significant reductions in mean absolute errors.

It is crucial to detect when an instance lies downright too far from the training samples for the machine learning model to be trusted, a challenge known as out-of-distribution (OOD) detection. For neural networks, one approach to this task consists of learning a diversity of predictors that all can explain the training data. This information can be used to estimate the epistemic uncertainty at a given newly observed instance in terms of a measure of the disagreement of the predictions. Evaluation and certification of the ability of a method to detect OOD require specifying instances which are likely to occur in deployment yet on which no prediction is available. Focusing on regression tasks, we choose a simple yet insightful model for this OOD distribution and conduct an empirical evaluation of the ability of various methods to discriminate OOD samples from the data. Moreover, we exhibit evidence that a diversity of parameters may fail to translate to a diversity of predictors. Based on the choice of an OOD distribution, we propose a new way of estimating the entropy of a distribution on predictors based on nearest neighbors in function space. This leads to a variational objective which, combined with the family of distributions given by a generative neural network, systematically produces a diversity of predictors that provides a robust way to detect OOD samples.

A fundamental aspect of statistics is the integration of data from different sources. Classically, Fisher and others were focused on how to integrate homogeneous (or only mildly heterogeneous) sets of data. More recently, as data is becoming more accessible, the question of if data sets from different sources should be integrated is becoming more relevant. The current literature treats this as a question with only two answers: integrate or don't. Here we take a different approach, motivated by information-sharing principles coming from the shrinkage estimation literature. In particular, we deviate from the do/don't perspective and propose a dial parameter that controls the extent to which two data sources are integrated. How far this dial parameter should be turned is shown to depend, for example, on the informativeness of the different data sources as measured by Fisher information. In the context of generalized linear models, this more nuanced data integration framework leads to relatively simple parameter estimates and valid tests/confidence intervals. Moreover, we demonstrate both theoretically and empirically that setting the dial parameter according to our recommendation leads to more efficient estimation compared to other binary data integration schemes.

With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting.

The evaluation of noisy binary classifiers on unlabeled data is treated as a streaming task: given a data sketch of the decisions by an ensemble, estimate the true prevalence of the labels as well as each classifier's accuracy on them. Two fully algebraic evaluators are constructed to do this. Both are based on the assumption that the classifiers make independent errors. The first is based on majority voting. The second, the main contribution of the paper, is guaranteed to be correct. But how do we know the classifiers are independent on any given test? This principal/agent monitoring paradox is ameliorated by exploiting the failures of the independent evaluator to return sensible estimates. A search for nearly error independent trios is empirically carried out on the \texttt{adult}, \texttt{mushroom}, and \texttt{two-norm} datasets by using the algebraic failure modes to reject evaluation ensembles as too correlated. The searches are refined by constructing a surface in evaluation space that contains the true value point. The algebra of arbitrarily correlated classifiers permits the selection of a polynomial subset free of any correlation variables. Candidate evaluation ensembles are rejected if their data sketches produce independent estimates too far from the constructed surface. The results produced by the surviving ensembles can sometimes be as good as 1\%. But handling even small amounts of correlation remains a challenge. A Taylor expansion of the estimates produced when independence is assumed but the classifiers are, in fact, slightly correlated helps clarify how the independent evaluator has algebraic `blind spots'.

We provide a construction of Gabor frames that encode local linearizations of a signal detected on a curved smooth manifold of arbitrary dimension, with Gabor filters that can detect the presence of higher-dimensional boundaries in the manifold signal. We describe an application in configuration spaces in robotics with sharp constrains. The construction is a higher-dimensional generalization of the geometric setting developed for the study of signal analysis in the visual cortex.