Mixture-of-Experts (MoE) is a flexible framework that combines multiple specialized submodels (``experts''), by assigning covariate-dependent weights (``gating functions'') to each expert, and have been commonly used for analyzing heterogeneous data. Existing statistical MoE formulations typically assume constant coefficients, for covariate effects within the expert or gating models, which can be inadequate for longitudinal, spatial, or other dynamic settings where covariate influences and latent subpopulation structure evolve across a known dimension. We propose a Varying-Coefficient Mixture of Experts (VCMoE) model that allows all coefficient effects in both the gating functions and expert models to vary along an indexing variable. We establish identifiability and consistency of the proposed model, and develop an estimation procedure, label-consistent EM algorithm, for both fully functional and hybrid specifications, along with the corresponding asymptotic distributions of the resulting estimators. For inference, simultaneous confidence bands are constructed using both asymptotic theory for the maximum discrepancy between the estimated functional coefficients and their true counterparts, and with bootstrap methods. In addition, a generalized likelihood ratio test is developed to examine whether a coefficient function is genuinely varying across the index variable. Simulation studies demonstrate good finite-sample performance, with acceptable bias and satisfactory coverage rates. We illustrate the proposed VCMoE model using a dataset of single nucleus gene expression in embryonic mice to characterize the temporal dynamics of the associations between the expression levels of genes Satb2 and Bcl11b across two latent cell subpopulations of neurons, yielding results that are consistent with prior findings.

Physics-informed machine learning (PIML) integrates partial differential equations (PDEs) into machine learning models to solve inverse problems, such as estimating coefficient functions (e.g., the Hamiltonian function) that characterize physical systems. This framework enables data-driven understanding and prediction of complex physical phenomena. While coefficient functions in PIML are typically estimated on the basis of predictive performance, physics as a discipline does not rely solely on prediction accuracy to evaluate models. For example, Kepler's heliocentric model was favored owing to small discrepancies in planetary motion, despite its similar predictive accuracy to the geocentric model. This highlights the inherent uncertainties in data-driven model inference and the scientific importance of selecting physically meaningful solutions. In this paper, we propose a framework to quantify and analyze such uncertainties in the estimation of coefficient functions in PIML. We apply our framework to reduced model of magnetohydrodynamics and our framework shows that there are uncertainties, and unique identification is possible with geometric constraints. Finally, we confirm that we can estimate the reduced model uniquely by incorporating these constraints.

Inverse problems governed by partial differential equations (PDEs) play a crucial role in various fields, including computational science, image processing, and engineering. Particularly, Darcy flow equation is a fundamental equation in fluid mechanics, which plays a crucial role in understanding fluid flow through porous media. Bayesian methods provide an effective approach for solving PDEs inverse problems, while their numerical implementation requires numerous evaluations of computationally expensive forward solvers. Therefore, the adoption of surrogate models with lower computational costs is essential. However, constructing a globally accurate surrogate model for high-dimensional complex problems demands high model capacity and large amounts of data. To address this challenge, this study proposes an efficient locally accurate surrogate that focuses on the high-probability regions of the true likelihood in inverse problems, with relatively low model complexity and few training data requirements. Additionally, we introduce a sequential Bayesian design strategy to acquire the proposed surrogate since the high-probability region of the likelihood is unknown. The strategy treats the posterior evolution process of sequential Bayesian design as a Gaussian process, enabling algorithmic acceleration through one-step ahead prior. The complete algorithmic framework is referred to as Sequential Bayesian design for locally accurate surrogate (SBD-LAS). Finally, three experiments based the Darcy flow equation demonstrate the advantages of the proposed method in terms of both inversion accuracy and computational speed.

In this paper, we develop a novel transfer learning methodology to tackle the challenge of data scarcity in functional linear models. The methodology incorporates samples from the target model (target domain) alongside those from auxiliary models (source domains), transferring knowledge of coefficient shape from the source domains to the target domain. This shape-based knowledge transfer offers two key advantages. First, it is robust to covariate scaling, ensuring effectiveness despite variations in data distributions across different source domains. Second, the notion of coefficient shape homogeneity represents a meaningful advance beyond traditional coefficient homogeneity, allowing the method to exploit a wider range of source domains and achieve significantly improved model estimation. We rigorously analyze the convergence rates of the proposed estimator and examine the minimax optimality. Our findings show that the degree of improvement depends not only on the similarity of coefficient shapes between the target and source domains, but also on coefficient magnitudes and the spectral decay rates of the functional covariates covariance operators. To address situations where only a subset of auxiliary models is informative for the target model, we further develop a data-driven procedure for identifying such informative sources. The effectiveness of the proposed methodology is demonstrated through comprehensive simulation studies and an application to occupation time analysis using physical activity data from the U.S. National Health and Nutrition Examination Survey.

Learning identifiable representations and models from low-level observations is helpful for an intelligent spacecraft to complete downstream tasks reliably. For temporal observations, to ensure that the data generating process is provably inverted, most existing works either assume the noise variables in the dynamic mechanisms are (conditionally) independent or require that the interventions can directly affect each latent variable. However, in practice, the relationship between the exogenous inputs/interventions and the latent variables may follow some complex deterministic mechanisms. In this work, we study the problem of identifiable representation and model learning for latent dynamic systems. The key idea is to use an inductive bias inspired by controllable canonical forms, which are sparse and input-dependent by definition. We prove that, for linear and affine nonlinear latent dynamic systems with sparse input matrices, it is possible to identify the latent variables up to scaling and determine the dynamic models up to some simple transformations. The results have the potential to provide some theoretical guarantees for developing more trustworthy decision-making and control methods for intelligent spacecrafts.

As the frequency of extreme weather events rises, it becomes increasingly crucial to understand and detect them at the earliest opportunity. Statistical models provide a way to enhance their interpretability and offer insights into the connections between extreme events. Since geophysical data is often coupled across both space and time this poses challenges for modeling, often leading to highly complex statistical models. For spatial data, such as precipitation, a common way to describe and analyze extremes are max-stable processes, which arise as the unique limit of pointwise maxima of random fields. These processes are an essential tool in analyzing spatial extremes (Davison et al., 2012), as they allow for flexible modeling of the underlying dependence structure. However, when it comes to modeling these extremes, usually only a few observations are available, even less so as the underlying process is usually changing across time. For that reason traditional statistical methods often fail to identify parameters correctly, particularly as these models are high dimensional and complex. Furthermore, estimating parameters becomes especially challenging when dealing with extreme values. Therefore, specifying a distribution rather than relying on point estimators can be beneficial for quantifying uncertainty.

We develop a learning algorithm for closed signal flow graphs - a graphical model of signal transducers. The algorithm relies on the correspondence between closed signal flow graphs and weighted finite automata on a singleton alphabet. We demonstrate that this procedure results in a genuine reduction of complexity: our algorithm fares better than existing learning algorithms for weighted automata restricted to the case of a singleton alphabet.

Shapley values, originating in game theory and increasingly prominent in explainable AI, have been proposed to assess the contribution of facts in query answering over databases, along with other similar power indices such as Banzhaf values. In this work we adapt these Shapley-like scores to probabilistic settings, the objective being to compute their expected value. We show that the computations of expected Shapley values and of the expected values of Boolean functions are interreducible in polynomial time, thus obtaining the same tractability landscape. We investigate the specific tractable case where Boolean functions are represented as deterministic decomposable circuits, designing a polynomial-time algorithm for this setting. We present applications to probabilistic databases through database provenance, and an effective implementation of this algorithm within the ProvSQL system, which experimentally validates its feasibility over a standard benchmark.

In multivariate functional data analysis, different functional covariates can be homogeneous. The hidden homogeneity structure is informative about the connectivity or association of different covariates. The covariates with pronounced homogeneity can be analyzed jointly within the same group, which gives rise to a way of parsimoniously modeling multivariate functional data. In this paper, a novel grouped multivariate functional regression model with a new regularization approach termed "coefficient shape alignment" is developed to tackle the potential homogeneity of different functional covariates. The modeling procedure includes two main steps: first detect the unknown grouping structure with the new regularization approach to aggregate covariates into disjoint groups; and then the grouped multivariate functional regression model is established based on the detected grouping structure. In this new grouped model, the coefficient functions of covariates in the same homogeneous group share the same shape invariant to scaling. The new regularization approach builds on penalizing the discrepancy of coefficient shape. The consistency property of the detected grouping structure is thoroughly investigated, and the conditions that guarantee uncovering the underlying true grouping structure are developed. The asymptotic properties of the model estimates are also developed. Extensive simulation studies are conducted to investigate the finite-sample properties of the developed methods. The practical utility of the proposed methods is illustrated in the real data analysis on sugar quality evaluation. This work provides a novel means for analyzing the underlying homogeneity of functional covariates and developing parsimonious model structures for multivariate functional data.

We introduce a method for inferring an explicit PDE from a data sample generated by previously unseen dynamics, based on a learned context. The training phase integrates knowledge of the form of the equation with a differential scheme, while the inference phase yields a PDE that fits the data sample and enables both signal prediction and data explanation. We include results of extensive experimentation, comparing our method to SOTA approaches, together with ablation studies that examine different flavors of our solution in terms of prediction error and explainability. Many scientific fields use the language of Partial Differential Equations (PDEs; Evans, 2010) to describe the physical laws governing observed natural phenomena with spatio-temporal dynamics. Typically, a PDE system is derived from first principles and a mechanistic understanding of the problem after experimentation and data collection by domain experts of the field. Well-known examples for such systems include Navier-Stokes and Burgers' equations in fluid dynamics, Maxwell's equations for electromagnetic theory, and Schrödinger's equations for quantum mechanics. Solving a PDE model could provide users with crucial information on how a signal evolves over time and space, and could be used for both prediction and control tasks. While creating PDE-based models holds great value, it is still a difficult task in many cases. For many complex real-world phenomena, we might only know some of the dynamics of the system. For example, an expert might tell us that a heat equation PDE has a specific functional form but we do not know the values of the diffusion and drift coefficient functions.