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The prototypical diffuse-interface model that describes multi-component flows is the Navier-Stokes Cahn-Hilliard model (NSCH). Over the last decades many NSCH models have appeared that claim to describe the same physical phenomena, yet are distinct from one another. In a recent article [M.F.P. ten Eikelder, K.G. van der Zee, I. Akkerman, and D. Schillinger, Math. Mod. Meth. Appl. S. 33, pp 175-221, 2023.] we have established a unified framework of virtually all NSCH models. The framework reveals that there is only a single consistent NSCH model that naturally emanates from the underlying mixture theory. In the current article we present, verify and validate this novel consistent NSCH model by means of numerical simulation. To this purpose we discretize a divergence-free velocity formulation of the NSCH model using divergence-conforming isogeometric spaces. We compare computations of our consistent model to results of existing models from literature. The predictive capability of the numerical methodology is demonstrated via three-dimensional computations of a rising bubble and the contraction of a liquid filament that compare well with experimental data.

In Computational Fluid Dynamics (CFD), coarse mesh simulations offer computational efficiency but often lack precision. Applying conventional super-resolution to these simulations poses a significant challenge due to the fundamental contrast between downsampling high-resolution images and authentically emulating low-resolution physics. The former method conserves more of the underlying physics, surpassing the usual constraints of real-world scenarios. We propose a novel definition of super-resolution tailored for PDE-based problems. Instead of simply downsampling from a high-resolution dataset, we use coarse-grid simulated data as our input and predict fine-grid simulated outcomes. Employing a physics-infused UNet upscaling method, we demonstrate its efficacy across various 2D-CFD problems such as discontinuity detection in Burger's equation, Methane combustion, and fouling in Industrial heat exchangers. Our method enables the generation of fine-mesh solutions bypassing traditional simulation, ensuring considerable computational saving and fidelity to the original ground truth outcomes. Through diverse boundary conditions during training, we further establish the robustness of our method, paving the way for its broad applications in engineering and scientific CFD solvers.

We propose Tied-LoRA, a simple paradigm utilizes weight tying and selective training to further increase parameter efficiency of the Low-rank adaptation (LoRA) method. Our investigations include all feasible combinations parameter training/freezing in conjunction with weight tying to identify the optimal balance between performance and the number of trainable parameters. Through experiments covering a variety of tasks and two base language models, we provide analysis revealing trade-offs between efficiency and performance. Our experiments uncovered a particular Tied-LoRA configuration that stands out by demonstrating comparable performance across several tasks while employing only 13~\% percent of parameters utilized by the standard LoRA method.

In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam\'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.

In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation.

In this paper I propose a generative model of supervised learning that unifies two approaches to supervised learning, using a concept of a correct loss function. Addressing two measurability problems, which have been ignored in statistical learning theory, I propose to use convergence in outer probability to characterize the consistency of a learning algorithm. Building upon these results, I extend a result due to Cucker-Smale, which addresses the learnability of a regression model, to the setting of a conditional probability estimation problem. Additionally, I present a variant of Vapnik-Stefanuyk's regularization method for solving stochastic ill-posed problems, and using it to prove the generalizability of overparameterized supervised learning models.

In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.

In this paper, we consider a nonlinear PDE system governed by a parabolic heat equation coupled in a nonlinear way with a hyperbolic momentum equation describing the behavior of a displacement field coupled with a nonlinear elliptic equation based on an internal damage variable. We present a numerical scheme based on a low-order Galerkin finite element method (FEM) for the space discretization of the time-dependent nonlinear PDE system and an implicit finite difference method (FDM) to discretize in the direction of the time variable. Moreover, we present a priori estimates for the exact and discrete solutions for the pointwise-in-time $L^2$-norm. Based on the a priori estimates, we rigorously prove the convergence of the solutions of the fully discretized system to the exact solutions. Denoting the properties of the internal parameters, we find the order of convergence concerning the discretization parameters.

Big data is ubiquitous in practices, and it has also led to heavy computation burden. To reduce the calculation cost and ensure the effectiveness of parameter estimators, an optimal subset sampling method is proposed to estimate the parameters in marginal models with massive longitudinal data. The optimal subsampling probabilities are derived, and the corresponding asymptotic properties are established to ensure the consistency and asymptotic normality of the estimator. Extensive simulation studies are carried out to evaluate the performance of the proposed method for continuous, binary and count data and with four different working correlation matrices. A depression data is used to illustrate the proposed method.

Quantum supervised learning, utilizing variational circuits, stands out as a promising technology for NISQ devices due to its efficiency in hardware resource utilization during the creation of quantum feature maps and the implementation of hardware-efficient ansatz with trainable parameters. Despite these advantages, the training of quantum models encounters challenges, notably the barren plateau phenomenon, leading to stagnation in learning during optimization iterations. This study proposes an innovative approach: an evolutionary-enhanced ansatz-free supervised learning model. In contrast to parametrized circuits, our model employs circuits with variable topology that evolves through an elitist method, mitigating the barren plateau issue. Additionally, we introduce a novel concept, the superposition of multi-hot encodings, facilitating the treatment of multi-classification problems. Our framework successfully avoids barren plateaus, resulting in enhanced model accuracy. Comparative analysis with variational quantum classifiers from the technology's state-of-the-art reveal a substantial improvement in training efficiency and precision. Furthermore, we conduct tests on a challenging dataset class, traditionally problematic for conventional kernel machines, demonstrating a potential alternative path for achieving quantum advantage in supervised learning for NISQ era.