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We present a new high-order accurate spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order $hp$-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series expansion, which traditionally have been used within the shifted boundary method. In this setting, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The resulting solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small-cut-cells. Dirichlet, Neumann, and general Robin boundary conditions are enforced weakly through: i) a generalized Nitsche's method and ii) a generalized Aubin's method. For this, a consistent asymptotic preserving formulation of the embedded Robin formulations is presented. We present several numerical experiments and analysis of the algorithmic properties of the different weak formulations. With this, we include convergence studies under polynomial, $p$, increase of the basis functions, mesh, $h$, refinement, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. This is done to assess the influence of errors across variational formulations, polynomial order, mesh size, and mappings between the true and surrogate boundaries.

Generative diffusion models have achieved spectacular performance in many areas of generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We argue that this lead to a form of instability that lies at the heart of their generative capabilities and that can be described by a set of mean field critical exponents. We conclude by analyzing recent work connecting diffusion models and associative memory networks in view of the thermodynamic formulations.

An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich structure to account for a wide range of geometries that can be modelled after an energy landscape. Second, many standard data analysis tools initially developed for data in Euclidean space can also be generalised to data on a Riemannian manifold. In the context of protein dynamics, a conceptual challenge comes from the lack of a suitable smooth manifold and the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a novel local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold of point clouds modulo rigid body group actions and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. It performs exceptionally well on coarse-grained molecular dynamics simulated data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

The evaluation of clustering algorithms can involve running them on a variety of benchmark problems, and comparing their outputs to the reference, ground-truth groupings provided by experts. Unfortunately, many research papers and graduate theses consider only a small number of datasets. Also, the fact that there can be many equally valid ways to cluster a given problem set is rarely taken into account. In order to overcome these limitations, we have developed a framework whose aim is to introduce a consistent methodology for testing clustering algorithms. Furthermore, we have aggregated, polished, and standardised many clustering benchmark dataset collections referred to across the machine learning and data mining literature, and included new datasets of different dimensionalities, sizes, and cluster types. An interactive datasets explorer, the documentation of the Python API, a description of the ways to interact with the framework from other programming languages such as R or MATLAB, and other details are all provided at <//clustering-benchmarks.gagolewski.com>.

Neuromorphic computing relies on spike-based, energy-efficient communication, inherently implying the need for conversion between real-valued (sensory) data and binary, sparse spiking representation. This is usually accomplished using the real valued data as current input to a spiking neuron model, and tuning the neuron's parameters to match a desired, often biologically inspired behaviour. We developed a tool, the WaLiN-GUI, that supports the investigation of neuron models and parameter combinations to identify suitable configurations for neuron-based encoding of sample-based data into spike trains. Due to the generalized LIF model implemented by default, next to the LIF and Izhikevich neuron models, many spiking behaviors can be investigated out of the box, thus offering the possibility of tuning biologically plausible responses to the input data. The GUI is provided open source and with documentation, being easy to extend with further neuron models and personalize with data analysis functions.

In recent decades, the use of optical detection systems for meteor studies has increased dramatically, resulting in huge amounts of data being analyzed. Automated meteor detection tools are essential for studying the continuous meteoroid incoming flux, recovering fresh meteorites, and achieving a better understanding of our Solar System. Concerning meteor detection, distinguishing false positives between meteor and non-meteor images has traditionally been performed by hand, which is significantly time-consuming. To address this issue, we developed a fully automated pipeline that uses Convolutional Neural Networks (CNNs) to classify candidate meteor detections. Our new method is able to detect meteors even in images that contain static elements such as clouds, the Moon, and buildings. To accurately locate the meteor within each frame, we employ the Gradient-weighted Class Activation Mapping (Grad-CAM) technique. This method facilitates the identification of the region of interest by multiplying the activations from the last convolutional layer with the average of the gradients across the feature map of that layer. By combining these findings with the activation map derived from the first convolutional layer, we effectively pinpoint the most probable pixel location of the meteor. We trained and evaluated our model on a large dataset collected by the Spanish Meteor Network (SPMN) and achieved a precision of 98\%. Our new methodology presented here has the potential to reduce the workload of meteor scientists and station operators and improve the accuracy of meteor tracking and classification.

We systematically analyze optimization dynamics in deep neural networks (DNNs) trained with stochastic gradient descent (SGD) and study the effect of learning rate $\eta$, depth $d$, and width $w$ of the neural network. By analyzing the maximum eigenvalue $\lambda^H_t$ of the Hessian of the loss, which is a measure of sharpness of the loss landscape, we find that the dynamics can show four distinct regimes: (i) an early time transient regime, (ii) an intermediate saturation regime, (iii) a progressive sharpening regime, and (iv) a late time ``edge of stability" regime. The early and intermediate regimes (i) and (ii) exhibit a rich phase diagram depending on $\eta \equiv c / \lambda_0^H $, $d$, and $w$. We identify several critical values of $c$, which separate qualitatively distinct phenomena in the early time dynamics of training loss and sharpness. Notably, we discover the opening up of a ``sharpness reduction" phase, where sharpness decreases at early times, as $d$ and $1/w$ are increased.

A design optimization framework for process parameters of additive manufacturing based on finite element simulation is proposed. The finite element method uses a coupled thermomechanical model developed for fused deposition modeling from the authors' previous work. Both gradient-based and gradient-free optimization methods are proposed. The gradient-based approach, which solves a PDE-constrained optimization problem, requires sensitivities computed from the fully discretized finite element model. We show the derivation of the sensitivities and apply them in a projected gradient descent algorithm. For the gradient-free approach, we propose two distinct algorithms: a local search algorithm called the method of local variations and a Bayesian optimization algorithm using Gaussian processes. To illustrate the effectiveness and differences of the methods, we provide two-dimensional design optimization examples using all three proposed algorithms.

Opportunistic pharmacokinetic (PK) studies have sparse and imbalanced clinical measurement data, and the impact of sample time errors is an important concern when seeking accurate estimates of treatment response. We evaluated an approximate Bayesian model for individualized pharmacokinetics in the presence of time recording errors (TREs), considering both a short and long infusion dosing pattern. We found that the long infusion schedule generally had lower bias in estimates of the pharmacodynamic (PD) endpoint relative to the short infusion schedule. We investigated three different design strategies for their ability to mitigate the impact of TREs: (i) shifting blood draws taken during an active infusion to the post-infusion period, (ii) identifying the best next sample time by minimizing bias in the presence of TREs, and (iii) collecting additional information on a subset of patients based on estimate uncertainty or quadrature-estimated variance in the presence of TREs. Generally, the proposed strategies led to a decrease in bias of the PD estimate for the short infusion schedule, but had a negligible impact for the long infusion schedule. Dosing regimens with periods of high non-linearity may benefit from design modifications, while more stable concentration-time profiles are generally more robust to TREs with no design modifications.

Deep learning has achieved widespread success in medical image analysis, leading to an increasing demand for large-scale expert-annotated medical image datasets. Yet, the high cost of annotating medical images severely hampers the development of deep learning in this field. To reduce annotation costs, active learning aims to select the most informative samples for annotation and train high-performance models with as few labeled samples as possible. In this survey, we review the core methods of active learning, including the evaluation of informativeness and sampling strategy. For the first time, we provide a detailed summary of the integration of active learning with other label-efficient techniques, such as semi-supervised, self-supervised learning, and so on. Additionally, we also highlight active learning works that are specifically tailored to medical image analysis. In the end, we offer our perspectives on the future trends and challenges of active learning and its applications in medical image analysis.