# Quantum Recurrent Neural Networks: Predicting the Dynamics of Oscillatory and Chaotic Systems
## 1 Introduction  
Ordinary differential equations (ODEs) are foundational mathematical models that play a critical role across diverse domains, such as physics [1], biology [2, 3], and economics [4]. The numerical resolution of ODE systems is pivotal for understanding the dynamics of these varied processes. To this end, numerous methodologies have been devised, including linear multistep methods [5, 6] and Runge-Kutta methods [7, 8]. Nevertheless, the practical application of these methods frequently encounters the complexity of systems characterized by nonlinear and time-varying elements, alongside a pronounced sensitivity to initial conditions. Such complexities [9] significantly challenge the efficacy of traditional analytical and numerical strategies, especially in terms of achieving desired levels of precision, stability, and computational efficiency. Given these considerations, there is a pressing need for more sophisticated techniques capable of navigating the intricate landscape of ODE systems. In recent years, recurrent neural networks (RNNs) have achieved significant breakthroughs in processing sequential data. RNNs have been used extensively in various domains, including but not limited to, time series data prediction [10, 11, 12], machine translation [13], and speech recognition [14]. Notably, RNNs have also played a critical role in scientific research [15, 16], an area of particular interest to us. Within this framework, RNNs and their specific variants, such as Long Short-Term Memory (LSTM) [17] and GatedRecurrent Units (GRUs) [18], have proven to be efficacious in learning or forecasting complex, nonlinear challenges. The intrinsic nature of RNNs, which are capable of capturing temporal dependencies and learning from sequential data, makes them particularly suited for modeling dynamical systems. Unlike conventional approaches that require explicit numerical schemes for integration over time, RNNs can implicitly learn the underlying dynamics from data, offering a way to bypass the complexities of directly solving ordinary differential equations (ODEs). This capability allows for the modeling of complex systems with high degrees of freedom and nonlinearity, providing a flexible and potentially more accurate alternative to traditional methods. Consequently, the use of RNNs in solving ODEs, especially systems of ODEs, represents a significant shift towards data-driven approaches in the prediction of dynamical [19; 20; 21; 22] and time series chaotic systems [23; 24; 25; 26]. Concurrently, quantum computing has been advancing rapidly, spearheaded by technology giants such as IBM [27], Google [28], PennyLane [29], and D-Wave Systems [30]. These quantum systems offer theoretical advantages for specific computational tasks and simulations of intricate quantum phenomena. However, the practical utility of current Noisy Intermediate-Scale Quantum (NISQ) devices is constrained by limitations in quantum error correction [31; 32; 33]. This limitation poses a significant challenge, as it restricts the reliability and scalability of quantum computations, impeding their broader application in complex problems. The advent of variational quantum algorithms and circuits, pioneered by [PERSON] et al. [34], has been a landmark development in the integration of quantum computing with machine learning. These algorithms leverage quantum entanglement to address fundamental machine learning challenges, such as function approximation and classification [34; 35]. More studies about variational quantum algorithms and circuits can be found here [36]. This innovation has facilitated the development of hybrid quantum-classical algorithms, adaptable for use on existing Noisy Intermediate-Scale Quantum (NISQ) devices. Such hybrid approaches have shown promise across various domains, including classification [37; 38], generative adversarial learning [39], and deep reinforcement learning [40].