Comparative Analysis of Neural Networks and Numerical Methods for Solving Differential Equations

Abstract

In order to solve Time-Dependent Kohn-Sham (TDKS) equations in quantum systems, this thesis primarily uses analytical solutions, the Runge-Kutta 4th Order (RK4) method, and Physics-Informed Neural Networks (PINNs). Although analytical solutions provide exact results, they are applicable only to simplified or highly structured systems; this is because real-life problems tend to get more complicated [1, 8]. For small time steps in smooth systems, RK4 is a highly accurate method, though its performance degrades substantially for large, stiff, or non-linear equations that can become computationally intensive to iterate through recursively [7, 2]. PINNs offer a paradigm change by incorporating the governing equations intrinsically within the neural network architecture itself [3, 18]. That avoids iterative time-stepping, allowing for fast in-time simulations of general, high-dimensional, and nonlinear quantum circuits [9, 12]. Our results reveal that, while RK4 is effective for small, low-dimensional problems, when tackling more complex ones, PINNs triumph in terms of the scalability and robustness of the process [4, 11]. Moreover, PINNs not only demonstrate a huge potential to transform computational strategy in quantum mechanics but also pave the way to tackle the long-time dynamics and nonlinearity problems in quan tum many-body systems [10, 13