In deep learning, dense layer connectivity has become a key design principle in deep neural networks (DNNs), enabling efficient information flow and strong performance across a range of applications. In this work, we model densely connected DNNs mathematically and analyze their learning problems in the deep-layer limit. For a broad applicability, we present our analysis in a framework setting of DNNs with densely connected layers and general non-local feature transformations (with local feature transformations as special cases) within layers, which is called dense non-local (DNL) framework and includes standard DenseNets and variants as special examples. In this formulation, the densely connected networks are modeled as nonlinear integral equations, in contrast to the ordinary differential equation viewpoint commonly adopted in prior works. We study the associated training problems from an optimal control perspective and prove convergence results from the network learning problem to its continuous-time counterpart. In particular, we show the convergence of optimal values and the subsequence convergence of minimizers, using a piecewise linear extension and $Γ$-convergence analysis. Our results provide a mathematical foundation for understanding densely connected DNNs and further suggest that such architectures can offer stability of training deep models.

Understanding the dynamic nature of biological systems is fundamental to deciphering cellular behavior, developmental processes, and disease progression. Single-cell RNA sequencing (scRNA-seq) has provided static snapshots of gene expression, offering valuable insights into cellular states at a single time point. Recent advancements in temporally resolved scRNA-seq, spatial transcriptomics (ST), and time-series spatial transcriptomics (temporal-ST) have further revolutionized our ability to study the spatiotemporal dynamics of individual cells. These technologies, when combined with computational frameworks such as Markov chains, stochastic differential equations (SDEs), and generative models like optimal transport and Schr\"odinger bridges, enable the reconstruction of dynamic cellular trajectories and cell fate decisions. This review discusses how these dynamical system approaches offer new opportunities to model and infer cellular dynamics from a systematic perspective.

The identification of nonlinear dynamics from observations is essential for the alignment of the theoretical ideas and experimental data. The last, in turn, is often corrupted by the side effects and noise of different natures, so probabilistic approaches could give a more general picture of the process. At the same time, high-dimensional probabilities modeling is a challenging and data-intensive task. In this paper, we establish a parallel between the dynamical systems modeling and language modeling tasks. We propose a transformer-based model that incorporates geometrical properties of the data and provide an iterative training algorithm allowing the fine-grid approximation of the conditional probabilities of high-dimensional dynamical systems.

Well-being and mental health are directly associated with relationship status particularly in the context of relatedness and support. A key factor in relationship functioning is emotional arousal. We examine the interplay between emotional arousal manifested through acoustic and physiological cues and its association to relationship satisfaction. We propose a dynamical systems model to infer the within- and across-modality as well as the between-partner relations. Our results suggest that increased emotional regulation is negatively associated with relationship satisfaction and indicate that the proposed system consists a viable framework for analyzing such multimodal interrelations within romantic partners.