Neural network approach to solving the problem of self-action of wave fields in nonlinear media

We consider the problem of propagation of optical impulses in media with Kerr nonlinearity. As a mathematical model describing an optical pulse propagation process, we chose a generalized parabolic equation, which in dimensionless variables has the form of a one-dimensional modi ed Nonlinear Schr odinger Equation. We trained a fully connected neural network with various optimization functions and did experiments with network con guration and hyperparameters optimization. The conducted experiments have shown the promise of using the quasi-Newtonian L-BGFS optimization function over rst-order optimization functions in this problem. The article presents the results of experiments on training the model using various optimization functions: SDG, RMSProp, Adam, L-BGFS, the last optimizer allows you to train the network about an order faster. We also consider the size of the training sample required to train the model. From the obtained training results, we can conclude that due to the randomly uniform selection of points from the area using the Latin hypercube, it is enough to make a train sample of 10 15 % of the dataset, this will correspond to a step of about 0.12 in z and 0.022 in τ compared to 0.039 in z and 0.008 in τ in a regular grid obtained by numerical methods. In low-dimensional problems, the use of machine learning is not always appropriate, since training takes much more time than solving the problem using direct numerical simulation. However, as the complexity of the system increases, due to the increase in the number of unknown variables, a huge superiority of machine learning methods is expected due to fast calculation using an already trained network. Also open and interesting for future research is the issue of fast retraining of an already trained model for a problem with new parameters