Quantifying numerical data involves addressing two key challenges: first, determining whether the data can be naturally quantified, and second, identifying the numerical intervals or ranges of values that correspond to specific value classes, referred to as "quantums," which represent statistically meaningful states. If such quantification is feasible, continuous streams of numerical data can be transformed into sequences of "symbols" that reflect the states of the system described by the measured parameter. People often perform this task intuitively, relying on common sense or practical experience, while information theory and computer science offer computable metrics for this purpose. In this study, we assess the applicability of metrics based on information compression and the Silhouette coefficient for quantifying numerical data. We also investigate the extent to which these metrics correlate with one another and with what is commonly referred to as "human intuition." Our findings suggest that the ability to classify numeric data values into distinct categories is associated with a Silhouette coefficient above 0.65 and a Dip Test below 0.5; otherwise, the data can be treated as following a unimodal normal distribution. Furthermore, when quantification is possible, the Silhouette coefficient appears to align more closely with human intuition than the "normalized centroid distance" method derived from information compression perspective.

This article introduces Perception-Informed Neural Networks (PrINNs), a framework designed to incorporate perception-based information into neural networks, addressing both systems with known and unknown physics laws or differential equations. Moreover, PrINNs extend the concept of Physics-Informed Neural Networks (PINNs) and their variants, offering a platform for the integration of diverse forms of perception precisiation, including singular, probability distribution, possibility distribution, interval, and fuzzy graph. In fact, PrINNs allow neural networks to model dynamical systems by integrating expert knowledge and perception-based information through loss functions, enabling the creation of modern data-driven models. Some of the key contributions include Mixture of Experts Informed Neural Networks (MOEINNs), which combine heterogeneous expert knowledge into the network, and Transformed-Knowledge Informed Neural Networks (TKINNs), which facilitate the incorporation of meta-information for enhanced model performance. Additionally, Fuzzy-Informed Neural Networks (FINNs) as a modern class of fuzzy deep neural networks leverage fuzzy logic constraints within a deep learning architecture, allowing online training without pre-training and eliminating the need for defuzzification. PrINNs represent a significant step forward in bridging the gap between traditional physics-based modeling and modern data-driven approaches, enabling neural networks to learn from both structured physics laws and flexible perception-based rules. This approach empowers neural networks to operate in uncertain environments, model complex systems, and discover new forms of differential equations, making PrINNs a powerful tool for advancing computational science and engineering.

The challenges posed by epidemics and pandemics are immense, especially if the causes are novel. This article introduces a versatile open-source simulation framework designed to model intricate dynamics of infectious diseases across diverse population centres. Taking inspiration from historical precedents such as the Spanish flu and COVID-19, and geographical economic theories such as Central place theory, the simulation integrates agent-based modelling to depict the movement and interactions of individuals within different settlement hierarchies. Additionally, the framework provides a tool for decision-makers to assess and strategize optimal distribution plans for limited resources like vaccines or cures as well as to impose mobility restrictions.