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We present a novel fully implicit hybrid finite volume/finite element method for incompressible flows. Following previous works on semi-implicit hybrid FV/FE schemes, the incompressible Navier-Stokes equations are split into a pressure and a transport-diffusion subsystem. The first of them can be seen as a Poisson type problem and is thus solved efficiently using classical continuous Lagrange finite elements. On the other hand, finite volume methods are employed to solve the convective subsystem, in combination with Crouzeix-Raviart finite elements for the discretization of the viscous stress tensor. For some applications, the related CFL condition, even if depending only in the bulk velocity, may yield a severe time restriction in case explicit schemes are used. To overcome this issue an implicit approach is proposed. The system obtained from the implicit discretization of the transport-diffusion operator is solved using an inexact Newton-Krylov method, based either on the BiCStab or the GMRES algorithm. To improve the convergence properties of the linear solver a symmetric Gauss-Seidel (SGS) preconditioner is employed, together with a simple but efficient approach for the reordering of the grid elements that is compatible with MPI parallelization. Besides, considering the Ducros flux for the nonlinear convective terms we can prove that the discrete advection scheme is kinetic energy stable. The methodology is carefully assessed through a set of classical benchmarks for fluid mechanics. A last test shows the potential applicability of the method in the context of blood flow simulation in realistic vessel geometries.

We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite $n$ and $d$, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with $d$ and the bandwidth.

Background: Outcome measures that are count variables with excessive zeros are common in health behaviors research. There is a lack of empirical data about the relative performance of prevailing statistical models when outcomes are zero-inflated, particularly compared with recently developed approaches. Methods: The current simulation study examined five commonly used analytical approaches for count outcomes, including two linear models (with outcomes on raw and log-transformed scales, respectively) and three count distribution-based models (i.e., Poisson, negative binomial, and zero-inflated Poisson (ZIP) models). We also considered the marginalized zero-inflated Poisson (MZIP) model, a novel alternative that estimates the effects on overall mean while adjusting for zero-inflation. Extensive simulations were conducted to evaluate their the statistical power and Type I error rate across various data conditions. Results: Under zero-inflation, the Poisson model failed to control the Type I error rate, resulting in higher than expected false positive results. When the intervention effects on the zero (vs. non-zero) and count parts were in the same direction, the MZIP model had the highest statistical power, followed by the linear model with outcomes on raw scale, negative binomial model, and ZIP model. The performance of a linear model with a log-transformed outcome variable was unsatisfactory. When only one of the effects on the zero (vs. non-zero) part and the count part existed, the ZIP model had the highest statistical power. Conclusions: The MZIP model demonstrated better statistical properties in detecting true intervention effects and controlling false positive results for zero-inflated count outcomes. This MZIP model may serve as an appealing analytical approach to evaluating overall intervention effects in studies with count outcomes marked by excessive zeros.

When estimating a Global Average Treatment Effect (GATE) under network interference, units can have widely different relationships to the treatment depending on a combination of the structure of their network neighborhood, the structure of the interference mechanism, and how the treatment was distributed in their neighborhood. In this work, we introduce a sequential procedure to generate and select graph- and treatment-based covariates for GATE estimation under regression adjustment. We show that it is possible to simultaneously achieve low bias and considerably reduce variance with such a procedure. To tackle inferential complications caused by our feature generation and selection process, we introduce a way to construct confidence intervals based on a block bootstrap. We illustrate that our selection procedure and subsequent estimator can achieve good performance in terms of root mean squared error in several semi-synthetic experiments with Bernoulli designs, comparing favorably to an oracle estimator that takes advantage of regression adjustments for the known underlying interference structure. We apply our method to a real world experimental dataset with strong evidence of interference and demonstrate that it can estimate the GATE reasonably well without knowing the interference process a priori.

Deep learning methods find a solution to a boundary value problem by defining loss functions of neural networks based on governing equations, boundary conditions, and initial conditions. Furthermore, the authors show that when it comes to many engineering problems, designing the loss functions based on first-order derivatives results in much better accuracy, especially when there is heterogeneity and variable jumps in the domain \cite{REZAEI2022PINN}. The so-called mixed formulation for PINN is applied to basic engineering problems such as the balance of linear momentum and diffusion problems. In this work, the proposed mixed formulation is further extended to solve multi-physical problems. In particular, we focus on a stationary thermo-mechanically coupled system of equations that can be utilized in designing the microstructure of advanced materials. First, sequential unsupervised training, and second, fully coupled unsupervised learning are discussed. The results of each approach are compared in terms of accuracy and corresponding computational cost. Finally, the idea of transfer learning is employed by combining data and physics to address the capability of the network to predict the response of the system for unseen cases. The outcome of this work will be useful for many other engineering applications where DL is employed on multiple coupled systems of equations.

Dereverberation is often performed directly on the reverberant audio signal, without knowledge of the acoustic environment. Reverberation time, T60, however, is an essential acoustic factor that reflects how reverberation may impact a signal. In this work, we propose to perform dereverberation while leveraging key acoustic information from the environment. More specifically, we develop a joint learning approach that uses a composite T60 module and a separate dereverberation module to simultaneously perform reverberation time estimation and dereverberation. The reverberation time module provides key features to the dereverberation module during fine tuning. We evaluate our approach in simulated and real environments, and compare against several approaches. The results show that this composite framework improves performance in environments.

In iterative approaches to empirical game-theoretic analysis (EGTA), the strategy space is expanded incrementally based on analysis of intermediate game models. A common approach to strategy exploration, represented by the double oracle algorithm, is to add strategies that best-respond to a current equilibrium. This approach may suffer from overfitting and other limitations, leading the developers of the policy-space response oracle (PSRO) framework for iterative EGTA to generalize the target of best response, employing what they term meta-strategy solvers (MSSs). Noting that many MSSs can be viewed as perturbed or approximated versions of Nash equilibrium, we adopt an explicit regularization perspective to the specification and analysis of MSSs. We propose a novel MSS called regularized replicator dynamics (RRD), which simply truncates the process based on a regret criterion. We show that RRD is more adaptive than existing MSSs and outperforms them in various games. We extend our study to three-player games, for which the payoff matrix is cubic in the number of strategies and so exhaustively evaluating profiles may not be feasible. We propose a profile search method that can identify solutions from incomplete models, and combine this with iterative model construction using a regularized MSS. Finally, and most importantly, we reveal that the regret of best response targets has a tremendous influence on the performance of strategy exploration through experiments, which provides an explanation for the effectiveness of regularization in PSRO.

Platform trials offer a framework to study multiple interventions in a single trial with the opportunity of opening and closing arms. The use of a common control in platform trials can increase efficiency as compared to individual control arms or separate trials per treatment. However, the need for multiplicity adjustment as a consequence of common controls is currently a controversial debate among researchers, pharmaceutical companies, as well as regulators. We investigate the impact of a common control arm in platform trials on the type one error and power in comparison to what would have been obtained with a platform trial with individual control arms in a simulation study. Furthermore, we evaluate the impact on power in case multiplicity adjustment is required in a platform trial. In both study designs, the family-wise error rate (FWER) is inflated compared to a standard, two-armed randomized controlled trial when no multiplicity adjustment is applied. In case of a common control, the FWER inflation is smaller. In most circumstances, a platform trial with a common control is still beneficial in terms of sample size and power after multiplicity adjustment, whereas in some cases, the platform trial with a common control loses the efficiency gain. Therefore, we further discuss the need for adjustment in terms of a family definition or hypotheses dependencies.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

Video anomaly detection under weak labels is formulated as a typical multiple-instance learning problem in previous works. In this paper, we provide a new perspective, i.e., a supervised learning task under noisy labels. In such a viewpoint, as long as cleaning away label noise, we can directly apply fully supervised action classifiers to weakly supervised anomaly detection, and take maximum advantage of these well-developed classifiers. For this purpose, we devise a graph convolutional network to correct noisy labels. Based upon feature similarity and temporal consistency, our network propagates supervisory signals from high-confidence snippets to low-confidence ones. In this manner, the network is capable of providing cleaned supervision for action classifiers. During the test phase, we only need to obtain snippet-wise predictions from the action classifier without any extra post-processing. Extensive experiments on 3 datasets at different scales with 2 types of action classifiers demonstrate the efficacy of our method. Remarkably, we obtain the frame-level AUC score of 82.12% on UCF-Crime.