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We prove that deep neural networks with ReLU activation function are capable of approximating solutions of semilinear partial integro-differential equations in the case of gradient-independent and Lipschitz-continuous nonlinearities, while the required number of parameters in the neural networks grows at most polynomially in both the dimension $ d\in\mathbb{N} $ and the reciprocal of the prescribed accuracy $ \epsilon $.

Spinal cord segmentation is clinically relevant and is notably used to compute spinal cord cross-sectional area (CSA) for the diagnosis and monitoring of cord compression or neurodegenerative diseases such as multiple sclerosis. While several semi and automatic methods exist, one key limitation remains: the segmentation depends on the MRI contrast, resulting in different CSA across contrasts. This is partly due to the varying appearance of the boundary between the spinal cord and the cerebrospinal fluid that depends on the sequence and acquisition parameters. This contrast-sensitive CSA adds variability in multi-center studies where protocols can vary, reducing the sensitivity to detect subtle atrophies. Moreover, existing methods enhance the CSA variability by training one model per contrast, while also producing binary masks that do not account for partial volume effects. In this work, we present a deep learning-based method that produces soft segmentations of the spinal cord. Using the Spine Generic Public Database of healthy participants ($\text{n}=267$; $\text{contrasts}=6$), we first generated participant-wise soft ground truth (GT) by averaging the binary segmentations across all 6 contrasts. These soft GT, along with a regression-based loss function, were then used to train a UNet model for spinal cord segmentation. We evaluated our model against state-of-the-art methods and performed ablation studies involving different GT mask types, loss functions, and contrast-specific models. Our results show that using the soft average segmentations along with a regression loss function reduces CSA variability ($p < 0.05$, Wilcoxon signed-rank test). The proposed spinal cord segmentation model generalizes better than the state-of-the-art contrast-specific methods amongst unseen datasets, vendors, contrasts, and pathologies (compression, lesions), while accounting for partial volume effects.

Here, we explain and illustrate a geometric perspective on causal inference in cohort studies that can help epidemiologists understand the role of standardization in causal inference as well as the distinctions between confounding, effect modification, and noncollapsibility. For simplicity, we focus on a binary exposure X, a binary outcome D, and a binary confounder C that is not causally affected by X. Rothman diagrams plot risk in the unexposed on the x-axis and risk in the exposed on the y-axis. The crude risks define one point in the unit square, and the stratum-specific risks define two other points in the unit square. These three points can be used to identify confounding and effect modification, and we show briefly how these concepts generalize to confounders with more than two levels. We propose a simplified but equivalent definition of collapsibility in terms of standardization, and we show that a measure of association is collapsible if and only if all of its contour lines are straight. We illustrate these ideas using data from a study conducted in Newcastle upon Tyne, United Kingdom, where the causal effect of smoking on 20-year mortality was confounded by age. We conclude that causal inference should be taught using geometry before using regression models.

In this study, we investigate the Shallow Water Equations incorporating source terms accounting for Manning friction and a non-flat bottom topology. Our primary focus is on developing and validating numerical schemes that serve a dual purpose: firstly, preserving all steady states within the model, and secondly, maintaining the late-time asymptotic behavior of solutions, which is governed by a diffusion equation and coincides with a long time and stiff friction limit. Our proposed approach draws inspiration from a penalization technique adopted in {\it{[Boscarino et. al, SIAM Journal on Scientific Computing, 2014]}}. By employing an additive implicit-explicit Runge-Kutta method, the scheme can ensure a correct asymptotic behavior for the limiting diffusion equation, without suffering from a parabolic-type time step restriction which often afflicts multiscale problems in the diffusive limit. Numerical experiments are performed to illustrate high order accuracy, asymptotic preserving, and asymptotically accurate properties of the designed schemes.

In this paper we present a complete framework for the energy-stable simulation of stratified incompressible flow in channels, using the one-dimensional two-fluid model. Building on earlier energy-conserving work on the basic two-fluid model, our new framework includes diffusion, friction, and surface tension. We show that surface tension can be added in an energy-conserving manner, and that diffusion and friction have a strictly dissipative effect on the energy. We then propose spatial discretizations for these terms such that a semi-discrete model is obtained that has the same conservation properties as the continuous model. Additionally, we propose a new energy-stable advective flux scheme that is energy-conserving in smooth regions of the flow and strictly dissipative where sharp gradients appear. This is obtained by combining, using flux limiters, a previously developed energy-conserving advective flux with a novel first-order upwind scheme that is shown to be strictly dissipative. The complete framework, with diffusion, surface tension, and a bounded energy, is linearly stable to short wavelength perturbations, and exhibits nonlinear damping near shocks. The model yields smoothly converging numerical solutions, even under conditions for which the basic two-fluid model is ill-posed. With our explicit expressions for the dissipation rates, we are able to attribute the nonlinear damping to the different dissipation mechanisms, and compare their effects.

In this study, we develop a novel framework to assess health risks due to heat hazards across various localities (zip codes) across the state of Maryland with the help of two commonly used indicators i.e. exposure and vulnerability. Our approach quantifies each of the two aforementioned indicators by developing their corresponding feature vectors and subsequently computes indicator-specific reference vectors that signify a high risk environment by clustering the data points at the tail-end of an empirical risk spectrum. The proposed framework circumvents the information-theoretic entropy based aggregation methods whose usage varies with different views of entropy that are subjective in nature and more importantly generalizes the notion of risk-valuation using cosine similarity with unknown reference points.

In this paper, we study multimodal coreference resolution, specifically where a longer descriptive text, i.e., a narration is paired with an image. This poses significant challenges due to fine-grained image-text alignment, inherent ambiguity present in narrative language, and unavailability of large annotated training sets. To tackle these challenges, we present a data efficient semi-supervised approach that utilizes image-narration pairs to resolve coreferences and narrative grounding in a multimodal context. Our approach incorporates losses for both labeled and unlabeled data within a cross-modal framework. Our evaluation shows that the proposed approach outperforms strong baselines both quantitatively and qualitatively, for the tasks of coreference resolution and narrative grounding.

In this work, we present a modification of the phase-field tumor growth model given in [26] that leads to bounded, more physically meaningful, volume fraction variables. In addition, we develop an upwind discontinuous Galerkin (DG) scheme preserving the mass conservation, pointwise bounds and energy stability of the continuous model. Finally, some computational tests in accordance with the theoretical results are introduced. In the first test, we compare our DG scheme with the finite element (FE) scheme related to the same time approximation. The DG scheme shows a well-behavior even for strong cross-diffusion effects in contrast with FE where numerical spurious oscillations appear. Moreover, the second test exhibits the behavior of the tumor-growth model under different choices of parameters and also of mobility and proliferation functions.

Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.