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Causal representation learning algorithms discover lower-dimensional representations of data that admit a decipherable interpretation of cause and effect; as achieving such interpretable representations is challenging, many causal learning algorithms utilize elements indicating prior information, such as (linear) structural causal models, interventional data, or weak supervision. Unfortunately, in exploratory causal representation learning, such elements and prior information may not be available or warranted. Alternatively, scientific datasets often have multiple modalities or physics-based constraints, and the use of such scientific, multimodal data has been shown to improve disentanglement in fully unsupervised settings. Consequently, we introduce a causal representation learning algorithm (causalPIMA) that can use multimodal data and known physics to discover important features with causal relationships. Our innovative algorithm utilizes a new differentiable parametrization to learn a directed acyclic graph (DAG) together with a latent space of a variational autoencoder in an end-to-end differentiable framework via a single, tractable evidence lower bound loss function. We place a Gaussian mixture prior on the latent space and identify each of the mixtures with an outcome of the DAG nodes; this novel identification enables feature discovery with causal relationships. Tested against a synthetic and a scientific dataset, our results demonstrate the capability of learning an interpretable causal structure while simultaneously discovering key features in a fully unsupervised setting.

Causal investigations in observational studies pose a great challenge in scientific research where randomized trials or intervention-based studies are not feasible. Leveraging Shannon's seminal work on information theory, we consider a framework of asymmetry where any causal link between putative cause and effect must be explained through a mechanism governing the cause as well as a generative process yielding an effect of the cause. Under weak assumptions, this framework enables the assessment of whether X is a stronger predictor of Y or vice-versa. Under stronger identifiability assumptions our framework is able to distinguish between cause and effect using observational data. We establish key statistical properties of this framework. Our proposed methodology relies on scalable non-parametric density estimation using fast Fourier transformation. The resulting estimation method is manyfold faster than the classical bandwidth-based density estimation while maintaining comparable mean integrated squared error rates. We investigate key asymptotic properties of our methodology and introduce a data-splitting technique to facilitate inference. The key attraction of our framework is its inference toolkit, which allows researchers to quantify uncertainty in causal discovery findings. We illustrate the performance of our methodology through simulation studies as well as multiple real data examples.

The optimization of open-loop shallow geothermal systems, which includes both design and operational aspects, is an important research area aimed at improving their efficiency and sustainability and the effective management of groundwater as a shallow geothermal resource. This paper investigates various approaches to address optimization problems arising from these research and implementation questions about GWHP systems. The identified optimization approaches are thoroughly analyzed based on criteria such as computational cost and applicability. Moreover, a novel classification scheme is introduced that categorizes the approaches according to the types of groundwater simulation model and the optimization algorithm used. Simulation models are divided into two types: numerical and simplified (analytical or data-driven) models, while optimization algorithms are divided into gradient-based and derivative-free algorithms. Finally, a comprehensive review of existing approaches in the literature is provided, highlighting their strengths and limitations and offering recommendations for both the use of existing approaches and the development of new, improved ones in this field.

Happ and Greven (2018) developed a methodology for principal components analysis of multivariate functional data for data observed on different dimensional domains. Their approach relies on an estimation of univariate functional principal components for each univariate functional feature. In this paper, we present extensive simulations to investigate choosing the number of principal components to retain. We show empirically that the conventional approach of using a percentage of variance explained threshold for each univariate functional feature may be unreliable when aiming to explain an overall percentage of variance in the multivariate functional data, and thus we advise practitioners to be careful when using it.

We propose three test criteria each of which is appropriate for testing, respectively, the equivalence hypotheses of symmetry, of homogeneity, and of independence, with multivariate data. All quantities have the common feature of involving weighted--type distances between characteristic functions and are convenient from the computational point of view if the weight function is properly chosen. The asymptotic behavior of the tests under the null hypothesis is investigated, and numerical studies are conducted in order to examine the performance of the criteria in finite samples.

We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes.

We study optimal data pooling for shared learning in two common maintenance operations: condition-based maintenance and spare parts management. We consider a set of systems subject to Poisson input -- the degradation or demand process -- that are coupled through an a-priori unknown rate. Decision problems involving these systems are high-dimensional Markov decision processes (MDPs) and hence notoriously difficult to solve. We present a decomposition result that reduces such an MDP to two-dimensional MDPs, enabling structural analyses and computations. Leveraging this decomposition, we (i) demonstrate that pooling data can lead to significant cost reductions compared to not pooling, and (ii) show that the optimal policy for the condition-based maintenance problem is a control limit policy, while for the spare parts management problem, it is an order-up-to level policy, both dependent on the pooled data.

Thanks to the singularity of the solution of linear subdiffusion problems, most time-stepping methods on uniform meshes can result in $O(\tau)$ accuracy where $\tau$ denotes the time step. The present work aims to discover the reason why some type of Crank-Nicolson schemes (the averaging Crank-Nicolson scheme) for the subdiffusion can only yield $O(\tau^\alpha)$$(\alpha<1)$ accuracy, which is much lower than the desired. The existing well developed error analysis for the subdiffusion, which has been successfully applied to many time-stepping methods such as the fractional BDF-$p (1\leq p\leq 6)$, all requires singular points be out of the path of contour integrals involved. The averaging Crank-Nicolson scheme in this work is quite natural but fails to meet this requirement. By resorting to the residue theorem, some novel sharp error analysis is developed in this study, upon which correction methods are further designed to obtain the optimal $O(\tau^2)$ accuracy. All results are verified by numerical tests.

In prediction settings where data are collected over time, it is often of interest to understand both the importance of variables for predicting the response at each time point and the importance summarized over the time series. Building on recent advances in estimation and inference for variable importance measures, we define summaries of variable importance trajectories. These measures can be estimated and the same approaches for inference can be applied regardless of the choice of the algorithm(s) used to estimate the prediction function. We propose a nonparametric efficient estimation and inference procedure as well as a null hypothesis testing procedure that are valid even when complex machine learning tools are used for prediction. Through simulations, we demonstrate that our proposed procedures have good operating characteristics, and we illustrate their use by investigating the longitudinal importance of risk factors for suicide attempt.

Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.