We study early prediction of convergence outcomes in parameterized root-finding problems. The analysis focuses on whether short prefixes of solver trajectories contain useful information about convergence within prescribed iteration horizons, without modifying the underlying numerical scheme. To avoid relying on profile-derived targets, we use solver-level success fractions Y20, Y50, and Y100, defined as the fractions of trajectories that satisfy the convergence criterion within 20, 50, and 100 iterations, respectively. Within this setting, we compare several families of early solver-derived features, including residual-based, step-based, and trajectory-derived quantities. We also distinguish between direct-prefix predictors, which use the first N values of a feature family, and scalar summary predictors, in which the same prefix is compressed into a single descriptor. Numerical experiments on controlled parameterized root-finding benchmarks show that simple residual- and step-based summaries provide the strongest within-base predictors among the tested feature families. In particular, mean and median log residual and log step summaries are informative even at short prefixes. More elaborate trajectory-derived descriptors, including the kNN–LLE proxy, are less effective as standalone predictors in the present experiments. The results also show that prediction difficulty depends strongly on the benchmark structure: Dataset 1 leads to near-saturated prediction performance at very short prefixes, whereas the oscillatory Dataset 2A provides a more challenging and discriminative case. In this latter setting, compact scalar summaries at N=10 can occasionally match or slightly outperform the corresponding full-prefix predictors, suggesting that scalar compression may suppress part of the transient variability present in the full-prefix representation. Cross-base transfer is substantially more difficult than within-base prediction, indicating that the observed predictive relationships are not uniformly portable across the considered benchmark problems. Overall, the study suggests that, in controlled parameterized settings of the type examined here, convergence outcomes can often be predicted from simple and computationally inexpensive early trajectory summaries. The results should be interpreted as a feature-based early convergence outcome prediction study, rather than as a general diagnostic methodology for nonlinear solvers.

Advanced statistics, often referred to as advanced statistical analysis, pertains to the use of refined statistical methods to examine data, derive insights, and formulate significant conclusions. It advances simple descriptive statistics and investigates connections, patterns, and trends in more detailed manners. Advanced statistics frequently includes multivariate analyses that examine interactions among various variables, Bayesian statistics, causal inference, non-parametric and high-dimensional statistics, estimation theory, optimality criteria, and hypothesis testing. All these areas provide the theoretical foundations for methodologies that are now proving extremely necessary for analyzing data as complex as that relating to biological networks. This paper aims to present possible schemes for integrating advanced statistical methods for inference, analysis and comparison of biological networks, emphasizing the difficulties that these methods, although highly sophisticated, still face, as identified by the study of extensive literature reported in the paper itself.

In recent years, a growing number of artificial intelligence (AI)–driven approaches have been developed to elucidate chemico-biological interactions associated with DNA damage and oxidative stress. Deep learning–based techniques, in particular, have demonstrated substantial potential within molecular biology and toxicology. As a result, researchers and clinicians alike hold high expectations that AI-enabled tools will soon make meaningful contributions to our understanding of the molecular and cellular mechanisms governing DNA damage and repair. In this article, we present a concise yet comprehensive overview of the computational methodologies underpinning contemporary deep learning approaches. We examine their capacity to support DNA damage assessment by revealing mechanistic insights into damage induction and response pathways. Particular emphasis is placed on deep learning techniques designed to enhance the analysis of complex biological data, including the automated detection and quantification of DNA damage from comet assay images and microscopy-based platforms. Furthermore, we critically assess the extent to which a gap exists between the expectations of researchers, biologists, and clinicians and the current practical capabilities of AI technologies in this domain. Finally, we offer a forward-looking perspective on how this gap might be narrowed, outlining key methodological, data-driven, and translational challenges that must be addressed to fully realize the potential of AI in DNA damage and repair research.

We propose an interpretable AI-assisted reliability diagnostic framework for parameterized root-finding schemes based on kNN-LLE proxy stability profiling and multi-horizon early prediction. The approach augments a numerical solver with a lightweight predictive layer that estimates solver reliability from short prefixes of iteration dynamics, enabling early identification of stable and unstable parameter regimes. For each configuration in the parameter space, raw and smoothed proxy profiles of a largest Lyapunov exponent (LLE) estimator are constructed, from which contractivity-based reliability scores summarizing finite-time convergence are derived. Machine learning models predict the reliability score from early segments of the proxy profile, allowing the framework to determine when solver dynamics become diagnostically informative. Experiments on a two-parameter parallel root-finding scheme show reliable prediction after only a few iterations: the best models achieve R^2=0.48 at horizon T=1, improve to R^2=0.67 by T=3, and exceed R^2=0.89 before the characteristic minimum-location scale of the stability profile. Prediction accuracy increases to R^2=0.96 at larger horizons, with mean absolute errors around 0.03, while inference costs remain negligible (microseconds per sample). The framework provides interpretable stability indicators and supports early decisions during solver execution, such as continuing, restarting, or adjusting parameters.

Image edges are essential for understanding and processing the content of the acquired scene, but a clear edge detection is not always possible. Unfavorable environmental conditions, poor lighting, incorrect camera settings and/or vibrations may produce blurry or dark images with unclear content and details. Local spatial filters are integrated into many hardware and software as sharpening tools, but choosing the most adequate filter and its parameters is usually non trivial. This work proposes a new filter, whose kernel is computed by comparing over a pre-defined window the image Prewitt gradient with an image contrast measure inspired by Retinex theory. The experiments, carried out on public real-world images with different edge visibility, show that the proposed image-aware filter efficaciously increases the edge visibility with low computational cost and performs better than a standard Laplacian filter. As an usage example, the sharpening filter is here applied to medical images of retinal fundus: improving the clarity and detail of these images is essential for visualizing and analysing anatomical structures, identifying abnormalities, and assisting in diagnosis and treatments. Finally, a hardware architecture of the sharpening filter, partially integrating on-chip the gradient and contrast computation, is outlined. Such an integration could be beneficial for enhancing in real-time the quality of pictures captured by devices with limited power resources, with an average consumption of similar to 28 mu W and similar to 30 kb of embedded memory in case of a color VGA image. The data obtained in filter validation experiments show a marked increase in the Prewitt's gradient magnitudes in 100% of cases and in the edge thickness in a percentage ranging from 73.3% to 100% of cases, without visibly affecting the image naturalness.

In mathematics, a physical network (e.g. biological network, social network, IT network, communication network) is usually represented by a graph. The determination of the metric space (also referred to as latent geometry) of the graph and the disposition of its nodes on it provide important information on the reaction propensity and consequently on the possible presence of bottlenecks in a system of interacting molecules, such as it happens in pharmacokinetics. To determine the latent geometry and the coordinates of nodes, it is necessary to have the dissimilarity or distance matrix of the network, an input that is not always easy to measure in experiments. The main result of this study is the mathematical and computational procedure for determining the distance/dissimilarity matrix between nodes and for identifying the latent network geometry from experimental time series of node concentrations. Specifically, we show how this matrix can be calculated from the transfer entropy between nodes, which is a measure of the flow of information between nodes and thus indirectly of the reaction propensity between them. We implemented a procedure of spectral graph embedding to embed the distance/dissimilarity matrix in flat and curved metric spaces, and consequently to determine the optimal latent geometry of the network. The distances between nodes in the metric space describing the latent geometry can be analyzed to identify bottlenecks in the reaction system. As a case study for this procedure, we consider the pharmacokinetics of atorvastatin, as described by recent studies and experimental time data. The method of determining distances between nodes from temporal measurements of node concentrations through the calculation of transfer entropy makes it possible to incorporate the information of kinetics (inherent in the time series) in the construction of the distance/dissimilarity matrix, and, consequently, in the determination of the network latent geometry, a characterisation of the network itself that is intimately connected to its dynamics, but which has so far been scarcely investigated and taken into account. The results on the case study of the pharmacokinetics of atorvastatin corroborate the usability and reliability of the method within certain limits of the experimental errors on the data.

Understanding the mechanics of climate changes on time and space scales is a difficult undertaking. According to the current literature on the subject, this endeavour is unfeasible in the near future, yet it is critical for forecasting both short-term weather and long-term climate change. Many problems impede the study of earth climate data, including non-stationarity (e.g., abrupt vs. slow variations), chaotic dynamics, non-linear dynamics, high-dimensionality, and natural vs. anthropogenic effects. The analytical approaches of descriptive and inferential statistics could benefit from network analysis techniques for investigating a complicated system like the climate. This study suggests an analysis that could help much more complex and refined, and ultimately enormously more informative, analyses could be based on an calculation of the centrality measures of an imaginary graph connecting the different continents and climatic zones of the Earth.

Physics provides mathematical formalisations of phenomena, such as oscillatory motion, that prove particularly useful and effective in describing characteristics of network nodes such as robustness (or conversely, vulnerability) to perturbations. At the same time, neural networks constitute tools for solving differential equations without the use of training data and without discretizing the integration domain. In particular, neural networks can prove to be efficient in calculating the numerical solution of systems of non-linear and stiff differential equations, cases in which traditional methods can be computationally cumbersome and accumulate significantly large errors. In this study, we propose a neural network activation function model that includes the vibration centrality of the physical network nodes whose dynamics we wish to simulate. We show how this can be particularly useful for the simulation of oscillating systems and analyse the case study of cellular glycolytic oscillations and the challenges that systems like that pose.

Abstract Physics Informed Neural Networks (PINNs) are a type of function approximators that use both data-driven supervised neural networks to learn the model of the dynamics of a physical system, and mathematical equations of the physical laws governing that system. PINNs have the benefit of being data-driven to train a model, but also of being able to assure consistency with the physics, and to extrapolate accurately beyond the range of data that currently accessible. As a result, PINNs can provide models that are more reliable while using less data. Specifically, the PINNs objective is to learn the solutions of a systems of equations using supervised learning on the available data and incorporating the knowledge of physical laws and constraints into the training process. However, solving single differential equations with a PINN may be relatively simple, solving systems of coupled differential equations may not be so simple. In this study, I present a neural network model specialized in solving differential equations of enzyme kinetics that has the main characteristic of being a demonstrative simple case of coupled equations system. The study focuses mainly on the theoretical aspects of the definition of a physics-informed loss function and shows a case study that highlights the challenges still to be overcome in solving systems of coupled differential equations.

In this chapter, we report the R scripts implemented to perform identifiabilityIdentifiability analysis and regression, mentioned in the previous chapters. The input files are only available upon request to the author.

Regression is used for explaining the relationship between a variable Y called response, and a set of one or more variables $$X_1, X_2, \dots , X_N$$ called explanatory variables. The variable selectionVariable selection is the process that choose a reduced number of explanatory variables to describe a response variable in a regression models. Variable selectionVariable selection is used to (i) make the model easier to interpret, removing redundant non-informative variables; (ii) reduce the size of the problem to enable algorithms to run faster; and (iii) reduce the overfittingOverfitting and, consequently, make the model predictive.

The concepts of complexityComplexity and networks are recurrent in modern systems biology. They are intimately linked to the very nature of biological processes governed by mathematically complex laws and orchestrated by thousands of interactions among thousands of molecular components. In this chapter, we explain what it means that a system is complex, what are the mathematical tools and the abstract data structures that we can use to describe a complex system, and finally what challenges the scientific community must face today to deduce a mathematical or computational model from observations experimental.

Graphs are used as a model of complex relationships among data in biological science since the advent of systems biology in the early 2000. In particular, graph data analysis and graph data mining play an important role in biology interaction networks, where recent techniques of artificial intelligence, usually employed in other type of networks (e.g., social, citations, and trademark networks) aim to implement various data mining tasks including classification, clustering, recommendation, anomaly detection, and link prediction. The commitment and efforts of artificial intelligence research in network biology are motivated by the fact that machine learning techniques are often prohibitively computational demanding, low parallelizable, and ultimately inapplicable, since biological network of realistic size is a large system, which is characterised by a high density of interactions and often with a non-linear dynamics and a non-Euclidean latent geometry. Currently, graph embedding emerges as the new learning paradigm that shifts the tasks of building complex models for classification, clustering, and link prediction to learning an informative representation of the graph data in a vector space so that many graph mining and learning tasks can be more easily performed by employing efficient non-iterative traditional models (e.g., a linear support vector machine for the classification task). The great potential of graph embedding is the main reason of the flourishing of studies in this area and, in particular, the artificial intelligence learning techniques. In this mini review, we give a comprehensive summary of the main graph embedding algorithms in light of the recent burgeoning interest in geometric deep learning.

Recent research emphasises the significance of identifying the latent geometry—also known as the geometry underlying a complex network—which is determined by the manifold class, curvature, and dimension. Geometry’s explanation of the organisational principles of complex systems and the potential that network nodes might be categorised according to their distances in this geometry are what confers geometry its significance. In this study, we analysed the network of genes resulting differentially expressed as a consequence of the modulation of protein tyrosine phosphatase receptor type G (PTPRG) in chronic myeloid leukaemia (CML) cell model K562 transcriptome. We found that the latent geometry of this network is hyperbolic, and that clustering according to the angular coordinates of its nodes in hyperbolic space classifies genes by functional pathway classes.

In this chapter, we introduce the basics of differential and integral calculus that are useful for understanding the meaning of differential equation and differential equation solution—in the domain of the real numbers—used in the following to model the dynamics of biological systems and chemical kinetics.