Conservation laws in a neural network approach to numerical solving of the nonlinear Schrodinger equation

We consider a possible modification of a neural network approach to numerical solving of nonlinear partial differential equations (PDE), describing physical systems with integrals of motion. In this approach, we approximate solutions of the equations by deep neural networks, using physics-informed method. Physics-informed neural network (FINN) approach proposes nonlinear function approximators that integrate the observational data, initial and boundary conditions and description of physical system in form of PDE by embedding the corresponding residuals into the loss function of a neural network. Therefore, the problem of solving nonlinear differential equations turns into the problem of minimizing the squared residuals over domain which is achieved by automatic differentiation and stochastic gradient descent. The proposed modification of this method means consideration and implementation of corresponding conservation laws for training of the neural networks, and is expected to improve the physical properties of the trained nonlinear regression models. The purpose of this work is to modify a neural network using the conservation law constraint, such that the predicted solution will satisfy the continuity equation better and faster as well as speed up the rate of convergence and provide better accuracy. Improvement of the conservative properties of the approximation is provided by the specific loss function regularization: addition of the conserved quantities’ residuals to the loss function to train the neural network. To test this method, we considered one-dimensional nonlinear Schrodinger equation and its conservation laws in integral form. Number of quants and energy were used as conserved physical quantities. In our experiments, their values were calculated in several equidistant time moments and compared with reference to find the corresponding residuals and implement the conservation constraint in the loss function. Therefore, the average residuals of number of quants and energy for the prediction are considered as quality metrics in the problem, as well as pointwise difference from the predicted and reference solution (validation error). Reference functions for validation datasets are derived from the analytical expressions for the exact solutions. This modified neural network approach is applied to the different classes of analytic solutions of the nonlinear Schrodinger equation: one soliton, interaction of two solitons (in breather form), first-order rogue wave. For each solution, we apply three forms of the conservative regularization: quants’ number constraint, energy constraint and the sum of them. The training curves and predictions are compared with the solution obtained with the initial loss function (baseline). It is shown that introduction of the additional conservative constraints to loss function reduces the conserved quantities’ residuals for training and prediction in all cases. For the simplest one-soliton (c) Yu. Gurieva, E. Vasiliev, L. Smirnov, 2023 solution, the regularizations improve not only conservation quality metrics, but also pointwise difference with the reference in the same training time. The best result was obtained by the combination of constraints: validation error is reduced by more than three times. However, for more complex solution forms, such as two solitons and rogue wave, the results are not as good. The conservative constraints significantly change the form of loss function, so the training curves start to plateau, and the training process becomes more unstable. For the most complex two soliton interaction, it requires about two times more optimization steps to converge. The validation error is improved only for the energy constraint for both cases: for two-soliton solution, validation error is reduced by 13 %; for rogue wave, it is reduced by 67 %. Therefore, the effect of conservative modification of the deep learning approach for nonlinear partial differential equations is individual for different systems and conserved quantities. Generalization ability of such neural networks should be further investigated and tested for different problems.