SOLVING NONLINEAR BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS USING PHYSICS-INFORMED NEURAL NETWORKS: ANALYSIS OF COLLOCATION POINT SELECTION STRATEGIES

The use of differential equations remains relevant for studying processes in various applied fields, where they have proven to be an effective tool for describing the motion of mechanical and oscillatory systems, modeling electrical circuits, heat and mass transfer processes, chemical kinetics, and biologicalphenomena such as the spread of diseases and population dynamics. The solution of boundary value problems based on the corresponding equations makes it possible to determine the state of an object over time or at different points in space, as well as to analyze its stability under changes in model parameters or external conditions. The complexity and adequacy of a mathematical model depend on the type of differential equation used. To account for complex patterns, nonlinear differential equations or their systems are typically employed. However, despite their widespread use in applied research, exact analytical solutions for such equations have been obtained only in particular cases, while in most situations they are difficult or impossible to derive. Under these conditions, numerical solution methods are applied, allowing one to obtain approximate solutions of boundary value problems on a set of points (a grid) with a prescribed accuracy. The effective use of numerical methods is associated with computational tools that combine high accuracy with acceptable computational cost. At the same time, the development of computer technologies has enabled not only the automation of classical numerical algorithms but also the emergence of new approaches, in particular those based on neural networks (Physics-Informed Neural Networks, PINNs). One of the key issues in the use of PINNs is the assessment of the accuracy of the obtained solutions. In this work, the quality of solutions obtained by PINNs for certain nonlinear second-order differential equations with boundary conditions (using the classical Bratu problem as an example) is investigated by comparing them with analytical (exact) and numerical solutions. Conclusions are drawn based on a comparison of uniform and adaptive selection strategies for PINN collocation points.