NEURAL NETWORK METHODS FOR SOLVING FORWARD AND INVERSE PROBLEMS - Yarosh, A. O.; Kudin, O. V. (Computer Science and Applied Mathematics, 2024)
The development of the approximaion methods for solving differential equations is important for many fields of science and technology. Numerical methods allow evaluation of the complex physical processes. These methods are necessary for computer modeling and simulation of technical systems. Classical methods of solving differential equations (collocation method, Galerkin method, Ritz method) require selection of the basis functions to construct an approximate solution. The wrong choice can lead to incorrect results. In addition, increasing the number of the basis functions to improve accuracy can lead to an increase in computational complexity, especially for large systems of differential equations. The use of physics informed neural networks for solving boundary value problems has several advantages compared to classical methods. First, neural networks allow approximation of complex physical processes without the need to select certain basis functions. Second, neural networks are able to automatically detect nonlinear dependencies in data, which makes them effective for modeling complex physical phenomena. In addition, neural networks can adapt to new data and change the conditions of the problem without the need to revise analytical approximations, which makes them more flexible and suitable for application in various fields of physics and engineering. Neural network methods are also effectively used to solve inverse problems. They allow the determination of system parameters or environmental properties based on measurements or observations. The unknown constants of the inverse problem to be determined are entered into the number of neural network parameters and optimized during training. In the paper, neural network architectures with physical information are developed for solving direct and inverse problems of the Burgers equation. Convergence is demonstrated on several numerical examples with different boundary conditions on the parameters of the problems.