# Predictive Modeling of Light-Matter Interaction in One Dimension: A Dynamic Deep Learning Approach
## 1 Introduction  
Modeling the behavior of electromagnetic waves in free space and also in complex media is well-known to be an involved procedure that requires the concurrent solution of [PERSON]'s equations with the Schrodinger wave equation in order to fully capture the wave behavior under time-varying electron dynamics [1; 2; 3; 4]. Various computational methods, such as the Finite-Difference Time-Domain (FDTD) Method, Finite Element Method, Method of Moments, Discontinuous Galerkin Method, etc., have been proposed for solving the Maxwell equations in simple and complex media [5; 6; 7]. These methods vary in their success in capturing electromagnetic wave behavior in spatial and temporal domains through media that are simple and complex (bi-anisotropic, in-homogenous, nonlinear, etc.). Under an interaction medium with slowly varying electron dynamics, solving the Maxwell equation is usually sufficient, in which case the electron dynamics can be modeled using the rate equations. The most commonly used method for dealing with complex media in microwave theory, optics, and photonics is the FDTD method. The FDTD method,besides being theoretically simple, is straightforward to implement computationally and can be applied to model wave dynamics in almost all kinds of complex media [5]. However, the FDTD method is limited in its ability to model geometrically complex media or media with rough surfaces and significant shape irregularities. In this study, we will focus on using the FDTD method for modeling the light-matter interaction (LMI) process in nonlinear dispersive media, typically encountered in optics/photonics. To model the behavior of light, we use the wave equation derived from [PERSON]'s equations by substituting the equation for [PERSON]'s law into the equation for [PERSON]'s law [5; 7]. The dynamics of the electrons are accounted for via the polarization term (also involved in the wave equation) [8; 9; 10; 11; 12], which is modeled using the Lorentz dispersion equations [5; 6]. The wave equation and the Lorentz dispersion equations are coupled to each other through the polarization term [7]. Under a particle-based treatment of the electrons, the electron dynamics are modeled using the rate equations, helping to identify the population of electrons at each energy level in the medium [5]. The set of all coupled differential equations is modeled based on the FDTD discretization. Importantly, the stability of the numerical solutions relies heavily on the preciseness of the spatial and temporal FDTD discretization. If the discretization step size is too large in the temporal domain, then the solutions can become unstable, meaning that the model can produce non-physical values that jump to infinity [5; 6; 7]. Hence, for the stability of the numerical solutions, it is very important to choose a sufficiently small temporal step size. Note that the temporal step size must be selected in accordance with the spatial step size. This is because one would be interested in capturing every detail of the propagating light wave through the spatial domain of the involved medium, and for that, the spatial step size should be selected as small (how small depends on the medium properties and the desired resolution of the light wave). Once an adequately small spatial step size is determined, the corresponding temporal step size must satisfy the Courant stability criterion, which states that the temporal step size must be smaller than the spatial step size divided by the speed of light \(c\) in free space \(\Delta t<\Delta x/c\)[5; 6; 7]. Given this condition, one is left with the conclusion that in order to simulate and predict the light behavior in complex media with high spatial resolution over a long duration, one must select the temporal and spatial step sizes to be very small, which leads to an enormous computational cost [10; 11; 12]. Here a major problem occurs because the wavelength of optical waves (mid-infrared-visible-ultraviolet-X-ray) is quite small, often less than 1 \(\upmu\)m [13; 14].