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.

A problem of packing unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. The circle is considered in a general sense, as a set of points that are all the same distance (not necessary Euclidean) from a given point. An integer formulation is proposed using a grid approximating the container and considering the nodes of the grid as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0–1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. The resulting binary problem is then solved by commercial software. Valid inequalities are proposed to strengthening the original formulation. Nesting circles inside one another is considered tacking into account the thickness of the circles. Numerical results on packing circles, ellipses, rhombuses and octagons are presented to demonstrate the efficiency of the proposed approach.

Packing irregular 3D objects in a cuboid of minimum volume is considered. Each object is composed of a number of convex shapes, such as oblique and right circular cylinders, cones and truncated cones. New analytical tools are introduced to state placement constraints for oblique shapes. Using the phi-function technique, optimized packing is reduced to a nonlinear programming problem. Novel solution approach is provided and illustrated by numerical examples.

Lagrangian constraint relaxation and the corresponding bounds for the optimal value of an original optimization problem are examined. Techniques for the refinement of the classical Lagrangian bounds are investigated in the case where the complementary slackness conditions are not fulfilled because either the original formulation is nonconvex or the Lagrange multipliers are nonoptimal. Examples are given of integer and convex problems for which the modified bounds improve the classical Lagrangian bounds.

The concept of the sparsest packing is introduced in this paper. The sparsest packing is aimed to place the objects in the container as distant as possible. More specifically, the minimal Euclidean distance between the objects, as well as, between the objects and the boundary of the container is maximised. This new problem statement is motivated by modern clean and energy-saving technologies such as ultrasonic hardening and finishing by detonating gas mixtures. The sparsest packing of two-dimensional objects in a circular container is considered subject to balancing conditions. The objects may have regular or irregular shapes bounded by arcs and line segments. Using the phi-function technique a mathematical model is formulated and a corresponding nonlinear programming problem is stated. A solution algorithm is proposed and computational results are presented to illustrate the approach.