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

TL;DR

This paper introduces Quantum Recurrent Neural Networks with encoder-decoder architecture, leveraging variational quantum circuits to efficiently solve complex, high-dimensional, time-dependent partial differential equations across various scientific domains.

Contribution

It presents a novel quantum neural network framework integrating variational quantum circuits into recurrent units for modeling nonlinear PDEs, demonstrating improved efficiency and stability.

Findings

Superior performance in capturing nonlinear dynamics

Effective handling of high-dimensional data

Stable solutions for complex PDEs

Abstract

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 Variational Quantum Circuits into Gated Recurrent Units and Long Short-Term Memory networks. Using this architecture, the model efficiently compresses high-dimensional spatiotemporal data into a compact latent space, facilitating more efficient temporal evolution. We 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.