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Decision making and learning in the presence of uncertainty has attracted significant attention in view of the increasing need to achieve robust and reliable operations. In the case where uncertainty stems from the presence of adversarial attacks this need is becoming more prominent. In this paper we focus on linear and nonlinear classification problems and propose a novel adversarial training method for robust classifiers, inspired by Support Vector Machine (SVM) margins. We view robustness under a data driven lens, and derive finite sample complexity bounds for both linear and non-linear classifiers in binary and multi-class scenarios. Notably, our bounds match natural classifiers' complexity. Our algorithm minimizes a worst-case surrogate loss using Linear Programming (LP) and Second Order Cone Programming (SOCP) for linear and non-linear models. Numerical experiments on the benchmark MNIST and CIFAR10 datasets show our approach's comparable performance to state-of-the-art methods, without needing adversarial examples during training. Our work offers a comprehensive framework for enhancing binary linear and non-linear classifier robustness, embedding robustness in learning under the presence of adversaries.

The present study introduces an advanced multi-physics and multi-scale modeling approach to investigate in silico colon motility. We introduce a generalized electromechanical framework, integrating cellular electrophysiology and smooth muscle contractility, thus advancing a first-of-its-kind computational model of laser tissue soldering after incision resection. The proposed theoretical framework comprises three main elements: a microstructural material model describing intestine wall geometry and composition of reinforcing fibers, with four fiber families, two active-conductive and two passive; an electrophysiological model describing the propagation of slow waves, based on a fully-coupled nonlinear phenomenological approach; and a thermodynamical consistent mechanical model describing the hyperelastic energetic contributions ruling tissue equilibrium under diverse loading conditions. The active strain approach was adopted to describe tissue electromechanics by exploiting the multiplicative decomposition of the deformation gradient for each active fiber family and solving the governing equations via a staggered finite element scheme. The computational framework was fine-tuned according to state-of-the-art experimental evidence, and extensive numerical analyses allowed us to compare manometric traces computed via numerical simulations with those obtained clinically in human patients. The model proved capable of reproducing both qualitatively and quantitatively high or low-amplitude propagation contractions. Colon motility after laser tissue soldering demonstrates that material properties and couplings of the deposited tissue are critical to reproducing a physiological muscular contraction, thus restoring a proper peristaltic activity.

We endeavour to estimate numerous multi-dimensional means of various probability distributions on a common space based on independent samples. Our approach involves forming estimators through convex combinations of empirical means derived from these samples. We introduce two strategies to find appropriate data-dependent convex combination weights: a first one employing a testing procedure to identify neighbouring means with low variance, which results in a closed-form plug-in formula for the weights, and a second one determining weights via minimization of an upper confidence bound on the quadratic risk.Through theoretical analysis, we evaluate the improvement in quadratic risk offered by our methods compared to the empirical means. Our analysis focuses on a dimensional asymptotics perspective, showing that our methods asymptotically approach an oracle (minimax) improvement as the effective dimension of the data increases.We demonstrate the efficacy of our methods in estimating multiple kernel mean embeddings through experiments on both simulated and real-world datasets.

Entropy conditions play a crucial role in the extraction of a physically relevant solution for a system of conservation laws, thus motivating the construction of entropy stable schemes that satisfy a discrete analogue of such conditions. TeCNO schemes (Fjordholm et al. 2012) form a class of arbitrary high-order entropy stable finite difference solvers, which require specialized reconstruction algorithms satisfying the sign property at each cell interface. Recently, third-order WENO schemes called SP-WENO (Fjordholm and Ray, 2016) and SP-WENOc (Ray, 2018) have been designed to satisfy the sign property. However, these WENO algorithms can perform poorly near shocks, with the numerical solutions exhibiting large spurious oscillations. In the present work, we propose a variant of the SP-WENO, termed as Deep Sign-Preserving WENO (DSP-WENO), where a neural network is trained to learn the WENO weighting strategy. The sign property and third-order accuracy are strongly imposed in the algorithm, which constrains the WENO weight selection region to a convex polygon. Thereafter, a neural network is trained to select the WENO weights from this convex region with the goal of improving the shock-capturing capabilities without sacrificing the rate of convergence in smooth regions. The proposed synergistic approach retains the mathematical framework of the TeCNO scheme while integrating deep learning to remedy the computational issues of the WENO-based reconstruction. We present several numerical experiments to demonstrate the significant improvement with DSP-WENO over the existing variants of WENO satisfying the sign property.

Dynamical systems across the sciences, from electrical circuits to ecological networks, undergo qualitative and often catastrophic changes in behavior, called bifurcations, when their underlying parameters cross a threshold. Existing methods predict oncoming catastrophes in individual systems but are primarily time-series-based and struggle both to categorize qualitative dynamical regimes across diverse systems and to generalize to real data. To address this challenge, we propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries based on the extraction of topologically invariant features. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a wide range of applications. Our convolutional attention method is trained with data augmentations that encourage the learning of topological invariants which can be used to detect bifurcation boundaries in unseen systems and to design models of biological systems like oscillatory gene regulatory networks. We further demonstrate our method's use in analyzing real data by recovering distinct proliferation and differentiation dynamics along pancreatic endocrinogenesis trajectory in gene expression space based on single-cell data. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.

We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial "deflation" step to the standard generic chaining argument. The resulting tail bound is the sum of the complexity of the "deflated function class" in terms of a generalization of Talagrand's $\gamma$ functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cram\'{e}r functions. We also provide certain approximations for the mentioned seminorm when the function class lies in a given (exponential type) Orlicz space, that can be used to make the complexity term and the deviation term more explicit.

Electrohydrodynamics is a discipline that studies the interaction between fluid motion and electric field. Finite element method, finite difference method and other numerical simulations are effective numerical calculation methods for electrofluid dynamics models. In this paper, the finite element format of the electrofluid dynamics model is established, and the second-order convergence accuracy of the format is achieved through time filtering method. Finally, a numerical example is given to verify the convergence.

Existing statistical methods for the analysis of micro-randomized trials (MRTs) are designed to estimate causal excursion effects using data from a single MRT. In practice, however, researchers can often find previous MRTs that employ similar interventions. In this paper, we develop data integration methods that capitalize on this additional information, leading to statistical efficiency gains. To further increase efficiency, we demonstrate how to combine these approaches according to a generalization of multivariate precision weighting that allows for correlation between estimates, and we show that the resulting meta-estimator possesses an asymptotic optimality property. We illustrate our methods in simulation and in a case study involving two MRTs in the area of smoking cessation.

The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric variables of solutions. For the corresponding spatial approximations, an independently refined finite element mesh is used for each polynomial coefficient. The method relies on multilevel expansions of input random fields and achieves error reduction with uniform rate. In particular, the saturation property for the refinement process is ensured by the algorithm. The results are illustrated by numerical experiments, including cases with random fields of low regularity.

The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham-Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal $L^2$-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.