Numerically solving partial differential equations (PDEs) can be challenging and computationally expensive. This has led to the development of reduced-order models (ROMs) that are accurate but faster than full order models (FOMs). Recently, machine learning advances have enabled the creation of non-linear projection methods, such as Latent Space Dynamics Identification (LaSDI). LaSDI maps full-order PDE solutions to a latent space using autoencoders and learns the system of ODEs governing the latent space dynamics. By interpolating and solving the ODE system in the reduced latent space, fast and accurate ROM predictions can be made by feeding the predicted latent space dynamics into the decoder. In this paper, we introduce GPLaSDI, a novel LaSDI-based framework that relies on Gaussian process (GP) for latent space ODE interpolations. Using GPs offers two significant advantages. First, it enables the quantification of uncertainty over the ROM predictions. Second, leveraging this prediction uncertainty allows for efficient adaptive training through a greedy selection of additional training data points. This approach does not require prior knowledge of the underlying PDEs. Consequently, GPLaSDI is inherently non-intrusive and can be applied to problems without a known PDE or its residual. We demonstrate the effectiveness of our approach on the Burgers equation, Vlasov equation for plasma physics, and a rising thermal bubble problem. Our proposed method achieves between 200 and 100,000 times speed-up, with up to 7% relative error.

We investigate the problem of compound estimation of normal means while accounting for the presence of side information. Leveraging the empirical Bayes framework, we develop a nonparametric integrative Tweedie (NIT) approach that incorporates structural knowledge encoded in multivariate auxiliary data to enhance the precision of compound estimation. Our approach employs convex optimization tools to estimate the gradient of the log-density directly, enabling the incorporation of structural constraints. We conduct theoretical analyses of the asymptotic risk of NIT and establish the rate at which NIT converges to the oracle estimator. As the dimension of the auxiliary data increases, we accurately quantify the improvements in estimation risk and the associated deterioration in convergence rate. The numerical performance of NIT is illustrated through the analysis of both simulated and real data, demonstrating its superiority over existing methods.

Selective inference methods are developed for group lasso estimators for use with a wide class of distributions and loss functions. The method includes the use of exponential family distributions, as well as quasi-likelihood modeling for overdispersed count data, for example, and allows for categorical or grouped covariates as well as continuous covariates. A randomized group-regularized optimization problem is studied. The added randomization allows us to construct a post-selection likelihood which we show to be adequate for selective inference when conditioning on the event of the selection of the grouped covariates. This likelihood also provides a selective point estimator, accounting for the selection by the group lasso. Confidence regions for the regression parameters in the selected model take the form of Wald-type regions and are shown to have bounded volume. The selective inference method for grouped lasso is illustrated on data from the national health and nutrition examination survey while simulations showcase its behaviour and favorable comparison with other methods.

Neural network approaches to approximate the ground state of quantum hamiltonians require the numerical solution of a highly nonlinear optimization problem. We introduce a statistical learning approach that makes the optimization trivial by using kernel methods. Our scheme is an approximate realization of the power method, where supervised learning is used to learn the next step of the power iteration. We show that the ground state properties of arbitrary gapped quantum hamiltonians can be reached with polynomial resources under the assumption that the supervised learning is efficient. Using kernel ridge regression, we provide numerical evidence that the learning assumption is verified by applying our scheme to find the ground states of several prototypical interacting many-body quantum systems, both in one and two dimensions, showing the flexibility of our approach.

Neural networks (NN) have become almost ubiquitous with image classification, but in their standard form produce point estimates, with no measure of confidence. Bayesian neural networks (BNN) provide uncertainty quantification (UQ) for NN predictions and estimates through the posterior distribution. As NN are applied in more high-consequence applications, UQ is becoming a requirement. BNN provide a solution to this problem by not only giving accurate predictions and estimates, but also an interval that includes reasonable values within a desired probability. Despite their positive attributes, BNN are notoriously difficult and time consuming to train. Traditional Bayesian methods use Markov Chain Monte Carlo (MCMC), but this is often brushed aside as being too slow. The most common method is variational inference (VI) due to its fast computation, but there are multiple concerns with its efficacy. We apply and compare MCMC- and VI-trained BNN in the context of target detection in hyperspectral imagery (HSI), where materials of interest can be identified by their unique spectral signature. This is a challenging field, due to the numerous permuting effects practical collection of HSI has on measured spectra. Both models are trained using out-of-the-box tools on a high fidelity HSI target detection scene. Both MCMC- and VI-trained BNN perform well overall at target detection on a simulated HSI scene. This paper provides an example of how to utilize the benefits of UQ, but also to increase awareness that different training methods can give different results for the same model. If sufficient computational resources are available, the best approach rather than the fastest or most efficient should be used, especially for high consequence problems.

The evolutionary distance between two species is proportional to some (unknown) function of the time of divergence from their common ancestor. One way to estimate this time is by comparing the underlying macromolecular sequences that cascade the information of accumulated evolutionary changes across DNA RNA Proteins (sequence structure function). Since the introduction of the molecular evolutionary clock by Zuckerkandl and Pauling (1965) to perform phylogenetic studies, several statistical models have been proposed to estimate the divergence of extant sequences from common ancestors, and to correlate the estimates of time from other sources of information (e.g., fossil records) when they exist (Sarich and Wilson, 1967). Such divergence time estimates require reliable statistical models of DNA/RNA/Proteins macromolecules (Bromham and Penny, 2003). For protein amino acid sequences, several statistical models have been proposed to explain sequence variation as a function of time. The point accepted mutation (PAM) matrix of Dayhoff et al. (1978) was the first successful model to explain the mutability of amino acid sequences. PAM is a stochastic (Markov) matrix defined in PAM (time) units where PAM-1 is a Markov matrix that embodies a 1% expected change to the amino acids. Subsequent studies highlighted the importance of incorporating evolutionary time-dependent substitution and gap models as an elegant way to model the divergent relationships of proteins (Holmes, 1998; Gonnet et al., 1992). The recent approach of Sumanaweera et al. (2022) derives a unified statistical model for quantifying the evolution of pairs of protein sequences

Traditional deep learning (DL) models are powerful classifiers, but many approaches do not provide uncertainties for their estimates. Uncertainty quantification (UQ) methods for DL models have received increased attention in the literature due to their usefulness in decision making, particularly for high-consequence decisions. However, there has been little research done on how to evaluate the quality of such methods. We use statistical methods of frequentist interval coverage and interval width to evaluate the quality of credible intervals, and expected calibration error to evaluate classification predicted confidence. These metrics are evaluated on Bayesian neural networks (BNN) fit using Markov Chain Monte Carlo (MCMC) and variational inference (VI), bootstrapped neural networks (NN), Deep Ensembles (DE), and Monte Carlo (MC) dropout. We apply these different UQ for DL methods to a hyperspectral image target detection problem and show the inconsistency of the different methods' results and the necessity of a UQ quality metric. To reconcile these differences and choose a UQ method that appropriately quantifies the uncertainty, we create a simulated data set with fully parameterized probability distribution for a two-class classification problem. The gold standard MCMC performs the best overall, and the bootstrapped NN is a close second, requiring the same computational expense as DE. Through this comparison, we demonstrate that, for a given data set, different models can produce uncertainty estimates of markedly different quality. This in turn points to a great need for principled assessment methods of UQ quality in DL applications.

Human-Robot Interaction (HRI) becomes more and more important in a world where robots integrate fast in all aspects of our lives but HRI applications depend massively on the utilized robotic system as well as the deployment environment and cultural differences. Because of these variable dependencies it is often not feasible to use a data-driven approach to train a model for human intent recognition. Expert systems have been proven to close this gap very efficiently. Furthermore, it is important to support understandability in HRI systems to establish trust in the system. To address the above-mentioned challenges in HRI we present an adaptable python library in which current state-of-the-art Models for context recognition can be integrated. For Context-Based Intention Recognition a two-layer Bayesian Network (BN) is used. The bayesian approach offers explainability and clarity in the creation of scenarios and is easily extendable with more modalities. Additionally, it can be used as an expert system if no data is available but can as well be fine-tuned when data becomes available.

Large skew-t factor copula models are attractive for the modeling of financial data because they allow for asymmetric and extreme tail dependence. We show that the copula implicit in the skew-t distribution of Azzalini and Capitanio (2003) allows for a higher level of pairwise asymmetric dependence than two popular alternative skew-t copulas. Estimation of this copula in high dimensions is challenging, and we propose a fast and accurate Bayesian variational inference (VI) approach to do so. The method uses a conditionally Gaussian generative representation of the skew-t distribution to define an augmented posterior that can be approximated accurately. A fast stochastic gradient ascent algorithm is used to solve the variational optimization. The new methodology is used to estimate copula models for intraday returns from 2017 to 2021 on 93 U.S. equities. The copula captures substantial heterogeneity in asymmetric dependence over equity pairs, in addition to the variability in pairwise correlations. We show that intraday predictive densities from the skew-t copula are more accurate than from some other copula models, while portfolio selection strategies based on the estimated pairwise tail dependencies improve performance relative to the benchmark index.

Spike trains data find a growing list of applications in computational neuroscience, imaging, streaming data and finance. Machine learning strategies for spike trains are based on various neural network and probabilistic models. The probabilistic approach is relying on parametric or nonparametric specifications of the underlying spike generation model. In this paper we consider the two-class statistical classification problem for a class of spike train data characterized by nonparametrically specified intensity functions. We derive the optimal Bayes rule and next form the plug-in nonparametric kernel classifier. Asymptotical properties of the rules are established including the limit with respect to the increasing recording time interval and the size of a training set. In particular the convergence of the kernel classifier to the Bayes rule is proved. The obtained results are supported by a finite sample simulation studies.