Quantum Recurrent Neural Networks with Encoder-Decoder for Time-Dependent Partial Differential Equations

Quantum Recurrent Neural Networks with Encoder-Decoder for Time-Dependent Partial Differential Equations Yuan Chen 1, Abdul Khaliq 1,2, and Khaled M. Furati 3 1 Computational and Data Science Program, Middle Tennessee State University, Murfreesboro, 37132, TN, USA 2 Department of Mathematical Science, Middle Tennessee State University, Murfreesboro, 37132, TN, USA 3 Department of Mathematics, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia Nonlinear time-dependent partial differential equations are essential in modeling complex phenomena across diverse fields, yet they pose significant challenges due to their computational complexity, especially in higher dimensions. This study explores Quantum Recurrent Neural Networks within an encoder-decoder framework, integrating V ariational Quantum Circuits into Gated Recurrent Units and Long Short-T erm Memory networks. W e evaluate the algorithms on the Hamilton-Jacobi-Bellman equation, Burgers' equation, the Gray-Scott reaction-diffusion system, and the three dimensional Michaelis-Menten reaction-diffusion equation. The results demonstrate the superior performance of the quantum-based algorithms in capturing nonlinear dynamics, handling high-dimensional spaces, and providing stable solutions, highlighting their potential as an innovative tool in solving challenging and complex systems. 1 Introduction Partial differential equations (PDEs) are fundamental mathematical tools for modeling diverse phenomena in many fields such as physics, biology, chemistry, and economics. However, for many complex and high-dimensional PDEs, analytical solutions are often unattainable due to Yuan Chen: yc3y@mtmail.mtsu.edu To address this, numerical methods such as the finite-difference method (FDM) [1], finite-element method (FEM) [2], and finite-volume method (FVM) [3] have been developed to approximate solutions. These techniques have been effective in a variety of applications but face limitations in computational complexity, stability, and scalability, especially when applied to non-linear or high-dimensional problems.