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In this study, a density-on-density regression model is introduced, where the association between densities is elucidated via a warping function. The proposed model has the advantage of a being straightforward demonstration of how one density transforms into another. Using the Riemannian representation of density functions, which is the square-root function (or half density), the model is defined in the correspondingly constructed Riemannian manifold. To estimate the warping function, it is proposed to minimize the average Hellinger distance, which is equivalent to minimizing the average Fisher-Rao distance between densities. An optimization algorithm is introduced by estimating the smooth monotone transformation of the warping function. Asymptotic properties of the proposed estimator are discussed. Simulation studies demonstrate the superior performance of the proposed approach over competing approaches in predicting outcome density functions. Applying to a proteomic-imaging study from the Alzheimer's Disease Neuroimaging Initiative, the proposed approach illustrates the connection between the distribution of protein abundance in the cerebrospinal fluid and the distribution of brain regional volume. Discrepancies among cognitive normal subjects, patients with mild cognitive impairment, and Alzheimer's disease (AD) are identified and the findings are in line with existing knowledge about AD.

The development of new manufacturing techniques such as 3D printing have enabled the creation of previously infeasible chemical reactor designs. Systematically optimizing the highly parameterized geometries involved in these new classes of reactor is vital to ensure enhanced mixing characteristics and feasible manufacturability. Here we present a framework to rapidly solve this nonlinear, computationally expensive, and derivative-free problem, enabling the fast prototype of novel reactor parameterizations. We take advantage of Gaussian processes to adaptively learn a multi-fidelity model of reactor simulations across a number of different continuous mesh fidelities. The search space of reactor geometries is explored through an amalgam of different, potentially lower, fidelity simulations which are chosen for evaluation based on weighted acquisition function, trading off information gain with cost of simulation. Within our framework we derive a novel criteria for monitoring the progress and dictating the termination of multi-fidelity Bayesian optimization, ensuring a high fidelity solution is returned before experimental budget is exhausted. The class of reactor we investigate are helical-tube reactors under pulsed-flow conditions, which have demonstrated outstanding mixing characteristics, have the potential to be highly parameterized, and are easily manufactured using 3D printing. To validate our results, we 3D print and experimentally validate the optimal reactor geometry, confirming its mixing performance. In doing so we demonstrate our design framework to be extensible to a broad variety of expensive simulation-based optimization problems, supporting the design of the next generation of highly parameterized chemical reactors.

Batch active learning is a popular approach for efficiently training machine learning models on large, initially unlabelled datasets by repeatedly acquiring labels for batches of data points. However, many recent batch active learning methods are white-box approaches and are often limited to differentiable parametric models: they score unlabeled points using acquisition functions based on model embeddings or first- and second-order derivatives. In this paper, we propose black-box batch active learning for regression tasks as an extension of white-box approaches. Crucially, our method only relies on model predictions. This approach is compatible with a wide range of machine learning models, including regular and Bayesian deep learning models and non-differentiable models such as random forests. It is rooted in Bayesian principles and utilizes recent kernel-based approaches. This allows us to extend a wide range of existing state-of-the-art white-box batch active learning methods (BADGE, BAIT, LCMD) to black-box models. We demonstrate the effectiveness of our approach through extensive experimental evaluations on regression datasets, achieving surprisingly strong performance compared to white-box approaches for deep learning models.

In this paper, we aim at establishing an approximation theory and a learning theory of distribution regression via a fully connected neural network (FNN). In contrast to the classical regression methods, the input variables of distribution regression are probability measures. Then we often need to perform a second-stage sampling process to approximate the actual information of the distribution. On the other hand, the classical neural network structure requires the input variable to be a vector. When the input samples are probability distributions, the traditional deep neural network method cannot be directly used and the difficulty arises for distribution regression. A well-defined neural network structure for distribution inputs is intensively desirable. There is no mathematical model and theoretical analysis on neural network realization of distribution regression. To overcome technical difficulties and address this issue, we establish a novel fully connected neural network framework to realize an approximation theory of functionals defined on the space of Borel probability measures. Furthermore, based on the established functional approximation results, in the hypothesis space induced by the novel FNN structure with distribution inputs, almost optimal learning rates for the proposed distribution regression model up to logarithmic terms are derived via a novel two-stage error decomposition technique.

Car-following behavior modeling is critical for understanding traffic flow dynamics and developing high-fidelity microscopic simulation models. Most existing impulse-response car-following models prioritize computational efficiency and interpretability by using a parsimonious nonlinear function based on immediate preceding state observations. However, this approach disregards historical information, limiting its ability to explain real-world driving data. Consequently, serially correlated residuals are commonly observed when calibrating these models with actual trajectory data, hindering their ability to capture complex and stochastic phenomena. To address this limitation, we propose a dynamic regression framework incorporating time series models, such as autoregressive processes, to capture error dynamics. This statistically rigorous calibration outperforms the simple assumption of independent errors and enables more accurate simulation and prediction by leveraging higher-order historical information. We validate the effectiveness of our framework using HighD and OpenACC data, demonstrating improved probabilistic simulations. In summary, our framework preserves the parsimonious nature of traditional car-following models while offering enhanced probabilistic simulations.

This paper is devoted to the statistical and numerical properties of the geometric median, and its applications to the problem of robust mean estimation via the median of means principle. Our main theoretical results include (a) an upper bound for the distance between the mean and the median for general absolutely continuous distributions in R^d, and examples of specific classes of distributions for which these bounds do not depend on the ambient dimension $d$; (b) exponential deviation inequalities for the distance between the sample and the population versions of the geometric median, which again depend only on the trace-type quantities and not on the ambient dimension. As a corollary, we deduce improved bounds for the (geometric) median of means estimator that hold for large classes of heavy-tailed distributions. Finally, we address the error of numerical approximation, which is an important practical aspect of any statistical estimation procedure. We demonstrate that the objective function minimized by the geometric median satisfies a "local quadratic growth" condition that allows one to translate suboptimality bounds for the objective function to the corresponding bounds for the numerical approximation to the median itself. As a corollary, we propose a simple stopping rule (applicable to any optimization method) which yields explicit error guarantees. We conclude with the numerical experiments including the application to estimation of mean values of log-returns for S&P 500 data.

Motivated by control with communication constraints, in this work we develop a time-invariant data compression architecture for linear-quadratic-Gaussian (LQG) control with minimum bitrate prefix-free feedback. For any fixed control performance, the approach we propose nearly achieves known directed information (DI) lower bounds on the time-average expected codeword length. We refine the analysis of a classical achievability approach, which required quantized plant measurements to be encoded via a time-varying lossless source code. We prove that the sequence of random variables describing the quantizations has a limiting distribution and that the quantizations may be encoded with a fixed source code optimized for this distribution without added time-asymptotic redundancy. Our result follows from analyzing the long-term stochastic behavior of the system, and permits us to additionally guarantee that the time-average codeword length (as opposed to expected length) is almost surely within a few bits of the minimum DI. To our knowledge, this time-invariant achievability result is the first in the literature. The originally published version of the supplementary material included a proof that contained an error that turned out to be inconsequential. This updated preprint corrects this error, which originally appeared under Lemma A.7.

Forward simulation-based uncertainty quantification that studies the distribution of quantities of interest (QoI) is a crucial component for computationally robust engineering design and prediction. There is a large body of literature devoted to accurately assessing statistics of QoIs, and in particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy tradeoffs between a given ensemble of models. However, effective algorithms that can estimate the full distribution of QoIs are still under active development. In this paper, we introduce a general multifidelity framework for estimating the cumulative distribution function (CDF) of a vector-valued QoI associated with a high-fidelity model under a budget constraint. Given a family of appropriate control variates obtained from lower-fidelity surrogates, our framework involves identifying the most cost-effective model subset and then using it to build an approximate control variates estimator for the target CDF. We instantiate the framework by constructing a family of control variates using intermediate linear approximators and rigorously analyze the corresponding algorithm. Our analysis reveals that the resulting CDF estimator is uniformly consistent and asymptotically optimal as the budget tends to infinity, with only mild moment and regularity assumptions on the joint distribution of QoIs. The approach provides a robust multifidelity CDF estimator that is adaptive to the available budget, does not require \textit{a priori} knowledge of cross-model statistics or model hierarchy, and applies to multiple dimensions. We demonstrate the efficiency and robustness of the approach using test examples of parametric PDEs and stochastic differential equations including both academic instances and more challenging engineering problems.

The accurate and efficient simulation of Partial Differential Equations (PDEs) in and around arbitrarily defined geometries is critical for many application domains. Immersed boundary methods (IBMs) alleviate the usually laborious and time-consuming process of creating body-fitted meshes around complex geometry models (described by CAD or other representations, e.g., STL, point clouds), especially when high levels of mesh adaptivity are required. In this work, we advance the field of IBM in the context of the recently developed Shifted Boundary Method (SBM). In the SBM, the location where boundary conditions are enforced is shifted from the actual boundary of the immersed object to a nearby surrogate boundary, and boundary conditions are corrected utilizing Taylor expansions. This approach allows choosing surrogate boundaries that conform to a Cartesian mesh without losing accuracy or stability. Our contributions in this work are as follows: (a) we show that the SBM numerical error can be greatly reduced by an optimal choice of the surrogate boundary, (b) we mathematically prove the optimal convergence of the SBM for this optimal choice of the surrogate boundary, (c) we deploy the SBM on massively parallel octree meshes, including algorithmic advances to handle incomplete octrees, and (d) we showcase the applicability of these approaches with a wide variety of simulations involving complex shapes, sharp corners, and different topologies. Specific emphasis is given to Poisson's equation and the linear elasticity equations.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.