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In this contribution we apply an adaptive model hierarchy, consisting of a full-order model, a reduced basis reduced order model, and a machine learning surrogate, to parametrized linear-quadratic optimal control problems. The involved reduced order models are constructed adaptively and are called in such a way that the model hierarchy returns an approximate solution of given accuracy for every parameter value. At the same time, the fastest model of the hierarchy is evaluated whenever possible and slower models are only queried if the faster ones are not sufficiently accurate. The performance of the model hierarchy is studied for a parametrized heat equation example with boundary value control.

We consider the numerical behavior of the fixed-stress splitting method for coupled poromechanics as undrained regimes are approached. We explain that pressure stability is related to the splitting error of the scheme, not the fact that the discrete saddle point matrix never appears in the fixed-stress approach. This observation reconciles previous results regarding the pressure stability of the splitting method. Using examples of compositional poromechanics with application to geological CO$_2$ sequestration, we see that solutions obtained using the fixed-stress scheme with a low order finite element-finite volume discretization which is not inherently inf-sup stable can exhibit the same pressure oscillations obtained with the corresponding fully implicit scheme. Moreover, pressure jump stabilization can effectively remove these spurious oscillations in the fixed-stress setting, while also improving the efficiency of the scheme in terms of the number of iterations required at every time step to reach convergence.

Deep neural networks used for reconstructing sparse-view CT data are typically trained by minimizing a pixel-wise mean-squared error or similar loss function over a set of training images. However, networks trained with such pixel-wise losses are prone to wipe out small, low-contrast features that are critical for screening and diagnosis. To remedy this issue, we introduce a novel training loss inspired by the model observer framework to enhance the detectability of weak signals in the reconstructions. We evaluate our approach on the reconstruction of synthetic sparse-view breast CT data, and demonstrate an improvement in signal detectability with the proposed loss.

Non-Hermitian topological phases can produce some remarkable properties, compared with their Hermitian counterpart, such as the breakdown of conventional bulk-boundary correspondence and the non-Hermitian topological edge mode. Here, we introduce several algorithms with multi-layer perceptron (MLP), and convolutional neural network (CNN) in the field of deep learning, to predict the winding of eigenvalues non-Hermitian Hamiltonians. Subsequently, we use the smallest module of the periodic circuit as one unit to construct high-dimensional circuit data features. Further, we use the Dense Convolutional Network (DenseNet), a type of convolutional neural network that utilizes dense connections between layers to design a non-Hermitian topolectrical Chern circuit, as the DenseNet algorithm is more suitable for processing high-dimensional data. Our results demonstrate the effectiveness of the deep learning network in capturing the global topological characteristics of a non-Hermitian system based on training data.

Under a generalised estimating equation analysis approach, approximate design theory is used to determine Bayesian D-optimal designs. For two examples, considering simple exchangeable and exponential decay correlation structures, we compare the efficiency of identified optimal designs to balanced stepped-wedge designs and corresponding stepped-wedge designs determined by optimising using a normal approximation approach. The dependence of the Bayesian D-optimal designs on the assumed correlation structure is explored; for the considered settings, smaller decay in the correlation between outcomes across time periods, along with larger values of the intra-cluster correlation, leads to designs closer to a balanced design being optimal. Unlike for normal data, it is shown that the optimal design need not be centro-symmetric in the binary outcome case. The efficiency of the Bayesian D-optimal design relative to a balanced design can be large, but situations are demonstrated in which the advantages are small. Similarly, the optimal design from a normal approximation approach is often not much less efficient than the Bayesian D-optimal design. Bayesian D-optimal designs can be readily identified for stepped-wedge cluster randomised trials with binary outcome data. In certain circumstances, principally ones with strong time period effects, they will indicate that a design unlikely to have been identified by previous methods may be substantially more efficient. However, they require a larger number of assumptions than existing optimal designs, and in many situations existing theory under a normal approximation will provide an easier means of identifying an efficient design for binary outcome data.

The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess fundamental conjugacy properties. When used as priors for the vector of parameters in general probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although such a core result has led to important advancements in Bayesian inference and computation, its applicability beyond likelihoods associated with fully-observed, discretized, or censored realizations from multivariate Gaussian models remains yet unexplored. This article covers such an important gap by proving that the wider family of multivariate unified skew-elliptical (SUE) distributions, which extends SUNs to more general perturbations of elliptical densities, guarantees conjugacy for broader classes of models, beyond those relying on fully-observed, discretized or censored Gaussians. Such a result leverages the closure under linear combinations, conditioning and marginalization of SUE to prove that such a family is conjugate to the likelihood induced by general multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement substantially enlarges the set of models that enable conjugate Bayesian inference to general formulations arising from elliptical and skew-elliptical families, including the multivariate Student's t and skew-t, among others.

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.

Untargeted metabolomic profiling through liquid chromatography-mass spectrometry (LC-MS) measures a vast array of metabolites within biospecimens, advancing drug development, disease diagnosis, and risk prediction. However, the low throughput of LC-MS poses a major challenge for biomarker discovery, annotation, and experimental comparison, necessitating the merging of multiple datasets. Current data pooling methods encounter practical limitations due to their vulnerability to data variations and hyperparameter dependence. Here we introduce GromovMatcher, a flexible and user-friendly algorithm that automatically combines LC-MS datasets using optimal transport. By capitalizing on feature intensity correlation structures, GromovMatcher delivers superior alignment accuracy and robustness compared to existing approaches. This algorithm scales to thousands of features requiring minimal hyperparameter tuning. Applying our method to experimental patient studies of liver and pancreatic cancer, we discover shared metabolic features related to patient alcohol intake, demonstrating how GromovMatcher facilitates the search for biomarkers associated with lifestyle risk factors linked to several cancer types.

Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this paper, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.

Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to simulate complex multiphysics problems with accelerated performance times. However, mesh-based GNNs require a large number of message-passing (MP) steps and suffer from over-smoothing for problems involving very fine mesh. In this work, we develop a multiscale mesh-based GNN framework mimicking a conventional iterative multigrid solver, coupled with adaptive mesh refinement (AMR), to mitigate challenges with conventional mesh-based GNNs. We use the framework to accelerate phase field (PF) fracture problems involving coupled partial differential equations with a near-singular operator due to near-zero modulus inside the crack. We define the initial graph representation using all mesh resolution levels. We perform a series of downsampling steps using Transformer MP GNNs to reach the coarsest graph followed by upsampling steps to reach the original graph. We use skip connectors from the generated embedding during coarsening to prevent over-smoothing. We use Transfer Learning (TL) to significantly reduce the size of training datasets needed to simulate different crack configurations and loading conditions. The trained framework showed accelerated simulation times, while maintaining high accuracy for all cases compared to physics-based PF fracture model. Finally, this work provides a new approach to accelerate a variety of mesh-based engineering multiphysics problems