Машинне навчання на основi PINN для моделювання внутрiшнiх хвиль у напiвнескiнченних рiдинах

This study investigates the application of Physics-Informed Neural Networks (PINNs) for modeling wave processes at the interface between two incompressible fluids of differing densities. As a first step, the linear formulation of the problem is considered, which admits an analytical solution based on a spectral method involving Fourier decomposition of the initial perturbation. This solution serves as a benchmark for testing and validating the accuracy of the PINN predictions.The implementation is carried out in Python using specialized libraries such as TensorFlow, NumPy, SciPy, and Matplotlib, which provide both efficient deep learning frameworks and tools for solving mathematical physics problems numerically. The approach integrates artificial intelligence with domain-specific knowledge in hydrodynamics, enabling the construction of interpretable and physically consistent models. Particular attention is given to the organization of the computational experiment, automation of visualizations, and storage of intermediate results for further analysis. The PINN model includes a loss function that encodes the governing equations and boundary conditions, and the training is conducted on randomly sampled points across the spatio-temporal domain. The influence of network architecture and training parameters on solution accuracy is examined. Visualization of loss function evolution and predicted wave profiles provides insight into convergence behavior and physical plausibility of the solutions.A comparative analysis between the PINN-based and analytical solutions across different time instances is presented, revealing phase shifts and amplitude deviations. The model demonstrates good agreement at early times and a gradual accumulation of errors as time progresses—an expected feature of this class of methods. The results confirm the feasibility of applying the PINN framework to linear hydrodynamic problems, laying the groundwork for future extensions to weakly and strongly nonlinear regimes, including studies of wave stability and nonlinear wave dynamics.