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In applications such as biomedical studies, epidemiology, and social sciences, recurrent events often co-occur with longitudinal measurements and a terminal event, such as death. Therefore, jointly modeling longitudinal measurements, recurrent events, and survival data while accounting for their dependencies is critical. While joint models for the three components exist in statistical literature, many of these approaches are limited by heavy parametric assumptions and scalability issues. Recently, incorporating deep learning techniques into joint modeling has shown promising results. However, current methods only address joint modeling of longitudinal measurements at regularly-spaced observation times and survival events, neglecting recurrent events. In this paper, we develop TransformerLSR, a flexible transformer-based deep modeling and inference framework to jointly model all three components simultaneously. TransformerLSR integrates deep temporal point processes into the joint modeling framework, treating recurrent and terminal events as two competing processes dependent on past longitudinal measurements and recurrent event times. Additionally, TransformerLSR introduces a novel trajectory representation and model architecture to potentially incorporate a priori knowledge of known latent structures among concurrent longitudinal variables. We demonstrate the effectiveness and necessity of TransformerLSR through simulation studies and analyzing a real-world medical dataset on patients after kidney transplantation.

Model-based offline reinforcement learning methods (RL) have achieved state-of-the-art performance in many decision-making problems thanks to their sample efficiency and generalizability. Despite these advancements, existing model-based offline RL approaches either focus on theoretical studies without developing practical algorithms or rely on a restricted parametric policy space, thus not fully leveraging the advantages of an unrestricted policy space inherent to model-based methods. To address this limitation, we develop MoMA, a model-based mirror ascent algorithm with general function approximations under partial coverage of offline data. MoMA distinguishes itself from existing literature by employing an unrestricted policy class. In each iteration, MoMA conservatively estimates the value function by a minimization procedure within a confidence set of transition models in the policy evaluation step, then updates the policy with general function approximations instead of commonly-used parametric policy classes in the policy improvement step. Under some mild assumptions, we establish theoretical guarantees of MoMA by proving an upper bound on the suboptimality of the returned policy. We also provide a practically implementable, approximate version of the algorithm. The effectiveness of MoMA is demonstrated via numerical studies.

We study an optimal control problem governed by elliptic PDEs with interface, which the control acts on the interface. Due to the jump of the coefficient across the interface and the control acting on the interface, the regularity of solution of the control problem is limited on the whole domain, but smoother on subdomains. The control function with pointwise inequality constraints is served as the flux jump condition which we called Neumann interface control. We use a simple uniform mesh that is independent of the interface. The standard linear finite element method can not achieve optimal convergence when the uniform mesh is used. Therefore the state and adjoint state equations are discretized by piecewise linear immersed finite element method (IFEM). While the accuracy of the piecewise constant approximation of the optimal control on the interface is improved by a postprocessing step which possesses superconvergence properties; as well as the variational discretization concept for the optimal control is used to improve the error estimates. Optimal error estimates for the control, suboptimal error estimates for state and adjoint state are derived. Numerical examples with and without constraints are provided to illustrate the effectiveness of the proposed scheme and correctness of the theoretical analysis., Comment: 31pages, 12 figures, 4 tables

Interface problems depict many fundamental physical phenomena and widely apply in the engineering. However, it is challenging to develop efficient fully decoupled numerical methods for solving degenerate interface problems in which the coefficient of a PDE is discontinuous and greater than or equal to zero on the interface. The main motivation in this paper is to construct fully decoupled numerical methods for solving nonlinear degenerate interface problems with ``double singularities". An efficient fully decoupled numerical method is proposed for nonlinear degenerate interface problems. The scheme combines deep neural network on the singular subdomain with finite difference method on the regular subdomain. The key of the new approach is to split nonlinear degenerate partial differential equation with interface into two independent boundary value problems based on deep learning. The outstanding advantages of the proposed schemes are that not only the convergence order of the degenerate interface problems on whole domain is determined by the finite difference scheme on the regular subdomain, but also can calculate $\mathbf{VERY}$ $\mathbf{BIG}$ jump ratio(such as $10^{12}:1$ or $1:10^{12}$) for the interface problems including degenerate and non-degenerate cases. The expansion of the solutions does not contains any undetermined parameters in the numerical method. In this way, two independent nonlinear systems are constructed in other subdomains and can be computed in parallel. The flexibility, accuracy and efficiency of the methods are validated from various experiments in both 1D and 2D. Specially, not only our method is suitable for solving degenerate interface case, but also for non-degenerate interface case. Some application examples with complicated multi-connected and sharp edge interface examples including degenerate and nondegenerate cases are also presented.

The current study explores the generalized embedded wormhole solutions in the background of $f(\mathcal{R},\mathcal{G})$ gravity, where $\mathcal{R}$ represents the Ricci scalar and $\mathcal{G}$ denotes the Gauss-Bonnet invariant. To investigate the necessary structures of the wormhole solutions we thoroughly analyzed the energy conditions under $f(\mathcal{R},\mathcal{G})$ gravity within the anisotropic source of matter. To meet this aim, we consider spherically symmetric geometry with the most generic gravity model of the gravity. A modified version of the field equations is calculated for two different embedded wormhole solutions. All the energy conditions are calculated and shown graphically with the regional ranges of the model parameter. Further, the invalid region of the energy conditions confirms the presence of exotic matter. Finally, we have concluding remarks.

Clustering, a fundamental activity in unsupervised learning, is notoriously difficult when the feature space is high-dimensional. Fortunately, in many realistic scenarios, only a handful of features are relevant in distinguishing clusters. This has motivated the development of sparse clustering techniques that typically rely on k-means within outer algorithms of high computational complexity. Current techniques also require careful tuning of shrinkage parameters, further limiting their scalability. In this paper, we propose a novel framework for sparse k-means clustering that is intuitive, simple to implement, and competitive with state-of-the-art algorithms. We show that our algorithm enjoys consistency and convergence guarantees. Our core method readily generalizes to several task-specific algorithms such as clustering on subsets of attributes and in partially observed data settings. We showcase these contributions thoroughly via simulated experiments and real data benchmarks, including a case study on protein expression in trisomic mice.