Report
УЧЕТ ЗАКОНОВ СОХРАНЕНИЯ ПРИ НЕЙРОСЕТЕВОМ ПОДХОДЕ К ЧИСЛЕННОМУ РЕШЕНИЮ НЕЛИНЕЙНОГО УРАВНЕНИЯ ШРЕДИНГЕРА
| Title: | УЧЕТ ЗАКОНОВ СОХРАНЕНИЯ ПРИ НЕЙРОСЕТЕВОМ ПОДХОДЕ К ЧИСЛЕННОМУ РЕШЕНИЮ НЕЛИНЕЙНОГО УРАВНЕНИЯ ШРЕДИНГЕРА |
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| Publisher Information: | Проблемы информатики, 2023. |
| Publication Year: | 2023 |
| Subject Terms: | нелинейное уравнение Шредингера, законы сохранения, solitons, deep learning, nonlinear Schr�odinger equation, нейронные сети, conservation laws, глубокое обучение, neural networks, солитоны |
| Description: | We consider a possible modi�cation of a neural network approach to numerical solving of nonlinear partial di�erential 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 (PINN) 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 di�erential equations turns into the problem of minimizing the squared residuals over domain which is achieved by automatic di�erentiation and stochastic gradient descent. The proposed modi�cation 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 speci�c 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 Schr�odinger 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 �nd 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 di�erence from the predicted and reference solution (validation error). Reference functions for validation datasets are derived from the analytical expressions for the exact solutions. This modi�ed neural network approach is applied to the di�erent classes of analytic solutions of the nonlinear Schr�odinger equation: one soliton, interaction of two solitons (in breather form), �rst-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 solution, the regularizations improve not only conservation quality metrics, but also pointwise di�erence 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 signi�cantly 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 e�ect of conservative modi�cation of the deep learning approach for nonlinear partial di�erential equations is individual for di�erent systems and conserved quantities. Generalization ability of such neural networks should be further investigated and tested for di�erent problems. В работе рассматривается один из возможных вариантов модификации нейросетевого подхода к численному решению нелинейных уравнений в частных производных, у которых благодаря физическим свойствам описываемых явлений имеются интегралы движения. Представленный метод подразумевает учет и непосредственное использование соответствующих законов сохранения при построении и обучении нейронных сетей, аппроксимирующих решения такого класса задач, что позволяет улучшить характеристики и качество получаемых нелинейных регрессионных моделей. Более точное выполнение консервативных свойств физических систем для аппроксиматора обеспечивается регуляризацией функции потерь: добавлением невязки сохраняющейся величины нейросетевого решения. Данная концепция рассмотрена и апробирована на примере нелинейного уравнения Шредингера и двух его интегралов движения, отвечающих законам сохранения числа квантов и энергии. Для вычисления невязки этих сохраняющихся величин и реализации консервативной регуляризации функции потерь был использован метод плоскостей непрерывности (вычисление величин в фиксированные моменты времени). Полученные результаты показывают улучшение консервативных свойств, а также в некоторых случаях точности нейросетевого решения по сравнению с регрессионной моделью, построенной с помощью глубокого обучения без учета предложенной в работе модификации. |
| Document Type: | Research |
| DOI: | 10.24412/2073-0667-2023-2-5-20 |
| Rights: | CC BY |
| Accession Number: | edsair.doi...........e2ce7dc365f4fc252b8645f2e14bcbf8 |
| Database: | OpenAIRE |
| DOI: | 10.24412/2073-0667-2023-2-5-20 |
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