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This study introduces the divergence-conforming discontinuous Galerkin finite element method (DGFEM) for numerically approximating optimal control problems with distributed constraints, specifically those governed by stationary generalized Oseen equations. We provide optimal a priori error estimates in energy norms for such problems using the divergence-conforming DGFEM approach. Moreover, we thoroughly analyze $L^2$ error estimates for scenarios dominated by diffusion and convection. Additionally, we establish the new reliable and efficient a posteriori error estimators for the optimal control of the Oseen equation with variable viscosity. Theoretical findings are validated through numerical experiments conducted in both two and three dimensions.

Random fields are ubiquitous mathematical structures in physics, with applications ranging from thermodynamics and statistical physics to quantum field theory and cosmology. Recent works on information geometry of Gaussian random fields proposed mathematical expressions for the components of the metric tensor of the underlying parametric space, allowing the computation of the curvature in each point of the manifold. In this study, our hypothesis is that time irreversibility in Gaussian random fields dynamics is a direct consequence of geometric properties (curvature) of their parametric space. In order to validate this hypothesis, we compute the components of the metric tensor and derive the twenty seven Christoffel symbols of the metric to define the Euler-Lagrange equations, a system of partial differential equations that are used to build geodesic curves in Riemannian manifolds. After that, by the application of the fourth-order Runge-Kutta method and Markov Chain Monte Carlo simulation, we numerically build geodesic curves starting from an arbitrary initial point in the manifold. The obtained results show that, when the system undergoes phase transitions, the geodesic curve obtained by time reversing the computational simulation diverges from the original curve, showing a strange effect that we called the geodesic dispersion phenomenon, which suggests that time irreversibility in random fields is related to the intrinsic geometry of their parametric space.

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 present a method for end-to-end learning of Koopman surrogate models for optimal performance in control. In contrast to previous contributions that employ standard reinforcement learning (RL) algorithms, we use a training algorithm that exploits the potential differentiability of environments based on mechanistic simulation models. We evaluate the performance of our method by comparing it to that of other controller type and training algorithm combinations on a literature known eNMPC case study. Our method exhibits superior performance on this problem, thereby constituting a promising avenue towards more capable controllers that employ dynamic surrogate models.

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.

The numerical treatment of fluid-particle systems is a very challenging problem because of the complex coupling phenomena occurring between the two phases. Although accurate mathematical modelling is available to address this kind of application, the computational cost of the numerical simulations is very expensive. The use of the most modern high-performance computing infrastructures could help to mitigate such an issue but not completely fix it. In this work, we develop a non-intrusive data-driven reduced order model (ROM) for Computational Fluid Dynamics (CFD) - Discrete Element Method (DEM) simulations. The ROM is built using the proper orthogonal decomposition (POD) for the computation of the reduced basis space and the Long Short-Term Memory (LSTM) network for the computation of the reduced coefficients. We are interested in dealing both with system identification and prediction. The most relevant novelties rely on (i) a filtering procedure of the full-order snapshots to reduce the dimensionality of the reduced problem and (ii) a preliminary treatment of the particle phase. The accuracy of our ROM approach is assessed against the classic Goldschmidt fluidized bed benchmark problem. Finally, we also provide some insights about the efficiency of our ROM approach.

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.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.