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The calibration of constitutive models from full-field data has recently gained increasing interest due to improvements in full-field measurement capabilities. In addition to the experimental characterization of novel materials, continuous structural health monitoring is another application that is of great interest. However, monitoring is usually associated with severe time constraints, difficult to meet with standard numerical approaches. Therefore, parametric physics-informed neural networks (PINNs) for constitutive model calibration from full-field displacement data are investigated. In an offline stage, a parametric PINN can be trained to learn a parameterized solution of the underlying partial differential equation. In the subsequent online stage, the parametric PINN then acts as a surrogate for the parameters-to-state map in calibration. We test the proposed approach for the deterministic least-squares calibration of a linear elastic as well as a hyperelastic constitutive model from noisy synthetic displacement data. We further carry out Markov chain Monte Carlo-based Bayesian inference to quantify the uncertainty. A proper statistical evaluation of the results underlines the high accuracy of the deterministic calibration and that the estimated uncertainty is valid. Finally, we consider experimental data and show that the results are in good agreement with a Finite Element Method-based calibration. Due to the fast evaluation of PINNs, calibration can be performed in near real-time. This advantage is particularly evident in many-query applications such as Markov chain Monte Carlo-based Bayesian inference.

Positive predictive value and negative predictive value are two widely used parameters to assess the clinical usefulness of a medical diagnostic test. When there are two diagnostic tests, it is recommendable to make a comparative assessment of the values of these two parameters after applying the two tests to the same subjects (paired samples). The objective is then to make individual or global inferences about the difference or the ratio of the predictive value of the two diagnostic tests. These inferences are usually based on complex and not very intuitive expressions, some of which have subsequently been reformulated. We define the two properties of symmetry which any inference method must verify - symmetry in diagnoses and symmetry in the tests -, we propose new inference methods, and we define them with simple expressions. All of the methods are compared with each other, selecting the optimal method: (a) to obtain a confidence interval for the difference or ratio; (b) to perform an individual homogeneity test of the two predictive values; and (c) to carry out a global homogeneity test of the two predictive values.

The main objective of this work is to explore the possibility of incorporating radiomic information from multiple lesions into survival models. We hypothesise that when more lesions are present, their inclusion can improve model performance, and we aim to find an optimal strategy for using multiple distinct regions in modelling. The idea of using multiple regions of interest (ROIs) to extract radiomic features for predictive models has been implemented in many recent works. However, in almost all studies, analogous regions were segmented according to particular criteria for all patients -- for example, the primary tumour and peritumoral area, or subregions of the primary tumour. They can be included in a model in a straightforward way as additional features. A more interesting scenario occurs when multiple distinct ROIs are present, such as multiple lesions in a regionally disseminated cancer. Since the number of such regions may differ between patients, their inclusion in a model is non-trivial and requires additional processing steps. We proposed several methods of handling multiple ROIs representing either ROI or risk aggregation strategy, compared them to a published one, and evaluated their performance in different classes of survival models in a Monte Carlo Cross-Validation scheme. We demonstrated the effectiveness of the methods using a cohort of 115 non-small cell lung cancer patients, for whom we predicted the metastasis risk based on features extracted from PET images in original resolution or interpolated to CT image resolution. For both feature sets, incorporating all available lesions, as opposed to a singular ROI representing the primary tumour, allowed for considerable improvement of predictive ability regardless of the model.

Super-resolution microscopy, or nanoscopy, enables the use of fluorescent-based molecular localization tools to study molecular structure at the nanoscale level in the intact cell, bridging the mesoscale gap to classical structural biology methodologies. Analysis of super-resolution data by artificial intelligence (AI), such as machine learning, offers tremendous potential for discovery of new biology, that, by definition, is not known and lacks ground truth. Herein, we describe the application of weakly supervised paradigms to super-resolution microscopy and its potential to enable the accelerated exploration of the nanoscale architecture of subcellular macromolecules and organelles.

In an attempt to provide an answer to the increasing criticism against p-values and to bridge the gap between statistical inference and prediction modelling, we introduce the probability of improved prediction (PIP). In general, the PIP is a probabilistic measure for comparing two competing models. Three versions of the PIP and several estimators are introduced and the relationships between them, p-values and the mean squared error are investigated. The performance of the estimators is assessed in a simulation study. An application shows how the PIP can support p-values to strengthen the conclusions or possibly point at issues with e.g. replicability.

Traditional low-rank approximation is a powerful tool to compress the huge data matrices that arise in simulations of partial differential equations (PDE), but suffers from high computational cost and requires several passes over the PDE data. The compressed data may also lack interpretability thus making it difficult to identify feature patterns from the original data. To address this issue, we present an online randomized algorithm to compute the interpolative decomposition (ID) of large-scale data matrices in situ. Compared to previous randomized IDs that used the QR decomposition to determine the column basis, we adopt a streaming ridge leverage score-based column subset selection algorithm that dynamically selects proper basis columns from the data and thus avoids an extra pass over the data to compute the coefficient matrix of the ID. In particular, we adopt a single-pass error estimator based on the non-adaptive Hutch++ algorithm to provide real-time error approximation for determining the best coefficients. As a result, our approach only needs a single pass over the original data and thus is suitable for large and high-dimensional matrices stored outside of core memory or generated in PDE simulations. We also provide numerical experiments on turbulent channel flow and ignition simulations, and on the NSTX Gas Puff Image dataset, comparing our algorithm with the offline ID algorithm to demonstrate its utility in real-world applications.

In uncertainty quantification, variance-based global sensitivity analysis quantitatively determines the effect of each input random variable on the output by partitioning the total output variance into contributions from each input. However, computing conditional expectations can be prohibitively costly when working with expensive-to-evaluate models. Surrogate models can accelerate this, yet their accuracy depends on the quality and quantity of training data, which is expensive to generate (experimentally or computationally) for complex engineering systems. Thus, methods that work with limited data are desirable. We propose a diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression to train a polynomial dimensional decomposition surrogate of the output that minimizes the number of training data. The new method first computes a sparse Lasso solution and uses it to define the cost function. A subsequent D-MORPH regression minimizes the difference between the D-MORPH and Lasso solution. The resulting D-MORPH based surrogate is more robust to input variations and more accurate with limited training data. We illustrate the accuracy and computational efficiency of the new surrogate for global sensitivity analysis using mathematical functions and an expensive-to-simulate model of char combustion. The new method is highly efficient, requiring only 15% of the training data compared to conventional regression.

Grokking is the phenomenon where neural networks NNs initially fit the training data and later generalize to the test data during training. In this paper, we empirically provide a frequency perspective to explain the emergence of this phenomenon in NNs. The core insight is that the networks initially learn the less salient frequency components present in the test data. We observe this phenomenon across both synthetic and real datasets, offering a novel viewpoint for elucidating the grokking phenomenon by characterizing it through the lens of frequency dynamics during the training process. Our empirical frequency-based analysis sheds new light on understanding the grokking phenomenon and its underlying mechanisms.

Inference for functional linear models in the presence of heteroscedastic errors has received insufficient attention given its practical importance; in fact, even a central limit theorem has not been studied in this case. At issue, conditional mean estimates have complicated sampling distributions due to the infinite dimensional regressors, where truncation bias and scaling issues are compounded by non-constant variance under heteroscedasticity. As a foundation for distributional inference, we establish a central limit theorem for the estimated conditional mean under general dependent errors, and subsequently we develop a paired bootstrap method to provide better approximations of sampling distributions. The proposed paired bootstrap does not follow the standard bootstrap algorithm for finite dimensional regressors, as this version fails outside of a narrow window for implementation with functional regressors. The reason owes to a bias with functional regressors in a naive bootstrap construction. Our bootstrap proposal incorporates debiasing and thereby attains much broader validity and flexibility with truncation parameters for inference under heteroscedasticity; even when the naive approach may be valid, the proposed bootstrap method performs better numerically. The bootstrap is applied to construct confidence intervals for centered projections and for conducting hypothesis tests for the multiple conditional means. Our theoretical results on bootstrap consistency are demonstrated through simulation studies and also illustrated with a real data example.

Since the weak convergence for stochastic processes does not account for the growth of information over time which is represented by the underlying filtration, a slightly erroneous stochastic model in weak topology may cause huge loss in multi-periods decision making problems. To address such discontinuities Aldous introduced the extended weak convergence, which can fully characterise all essential properties, including the filtration, of stochastic processes; however was considered to be hard to find efficient numerical implementations. In this paper, we introduce a novel metric called High Rank PCF Distance (HRPCFD) for extended weak convergence based on the high rank path development method from rough path theory, which also defines the characteristic function for measure-valued processes. We then show that such HRPCFD admits many favourable analytic properties which allows us to design an efficient algorithm for training HRPCFD from data and construct the HRPCF-GAN by using HRPCFD as the discriminator for conditional time series generation. Our numerical experiments on both hypothesis testing and generative modelling validate the out-performance of our approach compared with several state-of-the-art methods, highlighting its potential in broad applications of synthetic time series generation and in addressing classic financial and economic challenges, such as optimal stopping or utility maximisation problems.