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Physics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to provide a comprehensive review of currently available results on the numerical analysis of PINNs and related models that constitute the backbone of physics-informed machine learning. We provide a unified framework in which analysis of the various components of the error incurred by PINNs in approximating PDEs can be effectively carried out. We present a detailed review of available results on approximation, generalization and training errors and their behaviour with respect to the type of the PDE and the dimension of the underlying domain. In particular, we elucidate the role of the regularity of the solutions and their stability to perturbations in the error analysis. Numerical results are also presented to illustrate the theory. We identify training errors as a key bottleneck which can adversely affect the overall performance of various models in physics-informed machine learning.

Some of the important geodetic time series used in various Earth science disciplines are provided without uncertainty estimates. This can affect the validity of conclusions based on such data. However, an efficient uncertainty quantification algorithm to tackle this problem is currently not available. Here we present a methodology to approximate the aleatoric uncertainty in time series, called Bayesian Hamiltonian Monte Carlo Autoencoders (BaHaMAs). BaHaMAs is based on three elements: (1) self-supervised autoencoders that learn the underlying structure of the time series, (2) Bayesian machine learning that accurately quantifies the data uncertainty, and (3) Monte Carlo sampling that follows the Hamiltonian dynamics. The method can be applied in various fields in the Earth sciences. As an example, we focus on Atmospheric and Oceanic Angular Momentum time series (AAM and OAM, respectively), which are typically provided without uncertainty information. We apply our methodology to 3-hourly AAM and OAM time series and quantify the uncertainty in the data from 1976 up to the end of 2022. Furthermore, since Length of Day (LOD) is a geodetic time series that is closely connected to AAM and OAM and its short-term prediction is important for various space-geodetic applications, we show that the use of the derived uncertainties alongside the time series of AAM and OAM improves the prediction performance of LOD on average by 17% for different time spans. Finally, a comparison with alternative uncertainty quantification baseline methods, i.e., variational autoencoders and deep ensembles, reveals that BaHaMAs is more accurate in quantifying uncertainty. Graphical Abstract

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm of measure-valued solutions for these problems. We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

The difference between observed and modelled precession/nutation reveals unmodelled signals commonly referred to as Celestial Pole Offsets (CPO), denoted by dX and dY. CPO are currently observed only by Very Long Baseline Interferometry (VLBI), but there is nearly 4 weeks of latency by which the data centers provide the most accurate, final CPO series. This latency problem necessitates predicting CPO for high-accuracy, real-time applications that require information regarding Earth rotation, such as spacecraft navigation. Even though the International Earth Rotation and Reference Systems Service (IERS) provides so-called rapid CPO, they are usually less accurate and therefore, may not satisfy the requirements of the mentioned applications. To enhance the quality of CPO predictions, we present a new methodology based on Neural Additive Models (NAMs), a class of interpretable machine learning algorithms. We formulate the problem based on long short-term memory neural networks and derive simple analytical relations for the quantification of prediction uncertainty and feature importance, thereby enhancing the intelligibility of predictions made by machine learning. We then focus on the short-term prediction of CPO with a forecasting horizon of 30 days. We develop an operational framework that consistently provides CPO predictions. Using the CPO series of Jet Propulsion Laboratory as the input to the algorithm, we show that NAMs predictions improve the IERS rapid products on average by 57% for dX and 25% for dY under fully operational conditions. Our predictions are both accurate and overcome the latency issue of final CPO series and thus, can be used in real-time applications. Graphical Abstract

Highly efficient fluorescent and biocompatible europium doped sodium zinc molybdate (NZMOE) nanoprobes were successfully synthesized via Polyol method. Non-radiative defect centres get reduced with Li + co-doping in NZMOE nanoprobes. XRD spectra and Rietveld refinement confirmed successful incorporation of lithium ion and crystallinity was also improved with Li + co-doping. The shape of phosphor is rod shaped, as determined by TEM. Significant enhancement in photoluminescence intensity was observed with 266, 395 and 465 nm excitations. Profound red emission was recorded for 5 at% Li + co-doped NZMOE nanoprobes with 266 nm excitation. It shows high asymmetry ratio (~15), color purity (94.90%) and good quantum efficiency (~70%). Judd Ofelt parameters have been calculated to measure intensity parameters and radiative transition rates. In order to measure biocompatibility of the nanoprobes, cytotoxicity assays were performed with HePG2 cells. The fluorescence emitted from phosphor material treated HePG2 cells was also measured by Laser Scanning Confocal Microscopy. The bright red fluorescence in HePG2 cells treated with very low concentration (20 μg/ml) of phosphor material indicates that it could be a promising phosphor for biological detection or bio-imaging.

Toxoplasmosis is caused by Toxoplasma gondii and in immunocompromised patients it may lead to seizures, encephalitis or death. The conserved enzyme prolyl-tRNA synthetase (PRS) is a validated druggable target in Toxoplasma gondii but the traditional ‘single target–single drug’ approach has its caveats. Here, we describe two potent inhibitors namely halofuginone (HFG) and a novel ATP mimetic (L95) that bind to Toxoplasma gondii PRS simultaneously at different neighbouring sites to cover all three of the enzyme substrate subsites. HFG and L95 act as one triple-site inhibitor in tandem and form an unusual ternary complex wherein HFG occupies the 3’-end of tRNA and the L-proline (L-pro) binding sites while L95 occupies the ATP pocket. These inhibitors exhibit nanomolar IC 50 and EC 50 values independently, and when given together reveal an additive mode of action in parasite inhibition assays. This work validates a novel approach and lays a structural framework for further drug development based on simultaneous targeting of multiple pockets to inhibit druggable proteins.

We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of trainable parameters. These parameters are determined in an o ine training process by (approximately) minimizing suitable (possibly nonconvex) loss functions by (stochastic) gradient descent methods. The proposed algorithm is designed to be always consistent with the underlying di erential equation. Numerical experiments involving both linear and non-linear ODE and PDE model problems demonstrate a significant gain in computational e ciency over standard numerical methods.

Two-fluid ideal plasma equations are a generalized form of the ideal MHD equations in which electrons and ions are considered as separate species. The design of efficient numerical schemes for the these equations is complicated on account of their non-linear nature and the presence of stiff source terms, especially for high charge to mass ratios and for low Larmor radii. In this article, we design entropy stable finite difference schemes for the two-fluid equations by combining entropy conservative fluxes and suitable numerical diffusion operators. Furthermore, to overcome the time step restrictions imposed by the stiff source terms, we devise time-stepping routines based on implicit-explicit (IMEX)-Runge Kutta (RK) schemes. The special structure of the two-fluid plasma equations is exploited by us to design IMEX schemes in which only local (in each cell) linear equations need to be solved at each time step. Benchmark numerical experiments are presented to illustrate the robustness and accuracy of these schemes.

Gastrin is secreted following a rise in gastric pH, leading to gastric acid secretion. Sleeve gastrectomy (SG), a bariatric surgery where 80% of the gastric corpus is excised, presents a challenge for gastric pH homeostasis. Using histology, and single-cell RNA sequencing of the gastric epithelium in 12 women, we observed that SG is associated with an increase in a sub-population of acid-secreting parietal cells that overexpress respiratory enzymes and an increase in histamine-secreting enterochromaffin-like cells (ECLs). ECLs of SG-operated patients overexpressed genes coding for biosynthesis of neuropeptides and serotonin. Mathematical modeling showed that pH homeostasis by gastrin is analogous to non-linear proportional and integral control, that drives adaptation of the epithelium to acid-secretion demand. Quantitative model predictions were validated in patients. The results demonstrate human gastric epithelium remodeling following SG at the molecular and cellular levels, and more generally how trophic hormones enable robust adaptation of tissue function to meet physiological demand. Sleeve gastrectomy is a common bariatric surgery in which most of the gastric corpus is removed. Here, the authors show the adaptation of the gastric mucosa to surgery in patients, and how it facilitates maintenance of gastric pH homeostasis through a proportional-integral feedback control.

The prolyl-tRNA synthetase (PRS) is a validated drug target for febrifugine and its synthetic analog halofuginone (HFG) against multiple apicomplexan parasites including Plasmodium falciparum and Toxoplasma gondii . Here, a novel ATP-mimetic centered on 1-(pyridin-4-yl) pyrrolidin-2-one (PPL) scaffold has been validated to bind to Toxoplasma gondii PRS and kill toxoplasma parasites. PPL series exhibited potent inhibition at the cellular ( T . gondii parasites) and enzymatic ( Tg PRS) levels compared to the human counterparts. Cell-based chemical mutagenesis was employed to determine the mechanism of action via a forward genetic screen. Tg- resistant parasites were analyzed with wild-type strain by RNA-seq to identify mutations in the coding sequence conferring drug resistance by computational analysis of variants. DNA sequencing established two mutations, T477A and T592S, proximal to terminals of the PPL scaffold and not directly in the ATP, tRNA, or L-pro sites, as supported by the structural data from high-resolution crystal structures of drug-bound enzyme complexes. These data provide an avenue for structure-based activity enhancement of this chemical series as anti-infectives.