Physics-Informed Neural Networks (PINNs) represent a transformative approach to solving partial differential equation (PDE)-based boundary value problems by embedding physical laws into the learning process, addressing challenges such as non-physical solutions and data scarcity, which are inherent in traditional neural networks. This review analyzes critical challenges in PINN development, focusing on loss function design, geometric information integration, and their application in engineering modeling. We explore advanced strategies for constructing loss functions—including adaptive weighting, energy-based, and variational formulations—that enhance optimization stability and ensure physical consistency across multiscale and multiphysics problems. We emphasize geometry-aware learning through analytical representations—signed distance functions (SDFs), phi-functions, and R-functions—with complementary strengths: SDFs enable precise local boundary enforcement, whereas phi/R capture global multi-body constraints in irregular domains; in practice, hybrid use is effective for engineering problems. We also examine adaptive collocation sampling, domain decomposition, and hard-constraint mechanisms for boundary conditions to improve convergence and accuracy and discuss integration with commercial CAE via hybrid schemes that couple PINNs with classical solvers (e.g., FEM) to boost efficiency and reliability. Finally, we consider emerging paradigms—Physics-Informed Kolmogorov–Arnold Networks (PIKANs) and operator-learning frameworks (DeepONet, Fourier Neural Operator)—and outline open directions in standardized benchmarks, computational scalability, and multiphysics/multi-fidelity modeling for digital twins and design optimization.

The article analyses the methods of camouflage used in modern military conflicts. The article gives a historical overview of camouflage of military equipment and personnel since the beginning of the twentieth century. The mathematical and physical principles that should be taken into account in the manufacture of camouflage nets are discussed, the requirement for which is the difficulty in recognising and differentiating between an artificial surface and real terrain. The mathematical model of a camouflage net pattern is a fractal, a self-similar scale-invariant object of noninteger topological dimension. From the point of view of physics, the fractal dimension is a statistical value that demonstrates how densely a fractal fills a space. This means, in particular, that by dividing a photo of a real surface into cells and then counting the number of black cells in relation to all of them (in the simplest case of a photo of a winter forest, for example), you can find out the average indicator of the landscape's scale similarity, which can be reproduced on a camouflage grid. In the context of active military operations caused by russian aggression, for effective camouflage, it is proposed to use the concepts and methods of fractal geometry in the manufacture of camouflage means, in particular camouflage nets, to maximise the imitation of natural landscapes and structures that will not be easy to recognise.

This article deals with the urgent problem of introducing modern information technologies into the physics workshop, which increases the efficiency of the educational process and contributes to the formation of students' research competencies. The aim of the study is to analyze the capabilities of the Wolfram Mathematica software environment for processing and visualizing the results of interference experiments on the example of the laboratory work “Newton's Rings”. The research methods include theoretical analysis of interference phenomena, computer modeling in the Wolfram Mathematica software environment, and the use of interactive demonstrations of the Wolfram Demonstrations Project. An algorithm for processing experimental data is proposed, which allows the automation of calculations and improves their accuracy. The results of the study are presented in the form of a mathematical model of Newton's rings, which provides a visualization of the interference pattern and allows studying the dependence of its characteristics on various physical parameters of the system. Methodical recommendations for integrating Wolfram Mathematica software environment into a laboratory physics workshop have been developed. It is proved that the use of this software contributes to the development of skills in working with modern software tools, stimulates students' research activities, and provides interdisciplinary links between physics, mathematics, and computer science. The implementation of the proposed methodology allows the preparation of future specialists for professional activities in the context of the digital transformation of science and education. The paper highlights the pedagogical value of the Wolfram Demonstrations Project as an accessible and interactive supplement to theoretical instruction and physical experimentation. The results confirm that incorporating Wolfram Mathematica into the laboratory framework supports the development of digital competencies, critical thinking, and a research-oriented mindset. Ultimately, the implementation of this methodology prepares students for professional scientific work in an era of digital transformation in science and education.

Physically informed neural networks (PINNs) have proven to be a powerful tool for solving partial differential equations. However, their practical application is often limited to multi-objective optimization problems: the loss function consists of several components (e.g., residuals and boundary conditions) characterized by different physical scales and gradient magnitudes, which can lead to unstable convergence. Existing adaptive approaches, such as self-adaptive PINNs (SA-PINNs), address this problem by introducing additional trainable parameters, at the cost of significantly increasing computational costs. In this paper, we propose a computationally efficient adaptive loss balancing strategy that combines physical intuition with statistical balancing. The approach is based on three integrated steps: energy-based normalization of input and output variables to remove scale heterogeneity; analytical initialization of network parameters (pre-training) using approximate solutions to avoid bad local minima; and dynamic weighting of loss components inversely proportional to their current exponential moving averages. The effectiveness of the proposed strategy is tested on a two-dimensional steady-state heat conduction problem in a ring region with a singularity at the inner boundary. A comparative analysis with the SA-PINNs approach demonstrates that the proposed method provides faster training while maintaining a comparable level of accuracy. The results confirm that the adaptive weighting mechanism is able to automatically identify and prioritize “hard” regions with high gradients (e.g., near geometric discontinuities) without resorting to complex minimum-maximum optimization procedures. The proposed strategy is general within the class of PINN-based formulations and can be integrated into a wide range of existing PINN architectures.