A numerical solution of Schrödinger equation for the dynamics of early universe

Abstract

Artificial neural networks (ANNs) have attained widespread success across varied disciplines. This study is designated for looking into an application of an integrated intelligent computing paradigm concerning dynamics in the early Universe through numerical solutions to the Schrödinger equation. To arrive at this we leverage the Levenberg–Marquardt backpropagation networks (LMBNs) to probe cosmic evolution in the early Universe with the Friedmann–Lemaitre–Robertson–Walker (FLRW) metric for a flat minisuperspace model of the Universe in the background. This leads to bridging quantum mechanics and inflationary Universe dynamics conducing to quantum cosmology within the standard model. Wheeler–DeWitt equation corresponds to the time-independent Schrödinger equation obtained from the equations of motion for a single scalar field in flat spacetime with FLRW metric. Utilizing the ntstool the whole computing process is operated for simulation. To evaluate the accuracy and efficiency of the proposed scheme a comparative analysis is carried out. To construct continuous neural network mappings we employ the explicit Runge–Kutta method as the target parameter for generating datasets. To determine the solution datasets of different scenarios, the training, testing, and validation processes are employed to take advantage of these in the learning of neural network models established upon the backpropagation technique of Levenberg–Marquardt. By varying related parameters we develop three scenarios that produce nine cases, three for each. The data plots of performance, training state, error histogram, regression, time-series response, and error autocorrelation represent the visualization of the results. These plots show a complete case description by displaying all the necessary data values. The analysis of these plots is presented to validate all the cases. Performing the analysis by mean square error (MSE) validates the achieved accuracy of the results by validating and verifying neural networks. This work is motivated by the compelling need to develop innovative computational methods for solving complex cosmological questions to untangle the conundrums of the early universe. The attractive numerical solutions of the Schrödinger equation for the early Universe heralds a step towards quantum cosmology based on the interplay of the Wheeler–DeWitt equation and time-independent Schrödinger equation. There is an increasing trend to use computational methods to solve ordinary and partial differential equations with the help of code development in Matlab. For this purpose feed-forward artificial neural network is used for investigating the Schrödinger equation.