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.

This study proposes a deep learning model based on the combination of convolutional neural network (CNN) and bidirectional long short-term memory network (BiLSTM) for discriminant analysis of financial systemic risk. The model first uses CNN to extract local patterns of multidimensional features of financial markets, and then models the bidirectional dependency of time series through BiLSTM, to comprehensively characterize the changing laws of systemic risk in spatial features and temporal dynamics. The experiment is based on real financial data sets. The results show that the model is significantly superior to traditional single models (such as BiLSTM, CNN, Transformer, and TCN) in terms of accuracy, recall, and F1 score. The F1-score reaches 0.88, showing extremely high discriminant ability. This shows that the joint strategy of combining CNN and BiLSTM can not only fully capture the complex patterns of market data but also effectively deal with the long-term dependency problem in time series data. In addition, this study also explores the robustness of the model in dealing with data noise and processing high-dimensional data, providing strong support for intelligent financial risk management. In the future, the research will further optimize the model structure, introduce methods such as reinforcement learning and multimodal data analysis, and improve the efficiency and generalization ability of the model to cope with a more complex financial environment.

Federated Learning (FL) enables collaborative model training across distributed clients without data sharing, but its high computational and communication demands strain resource-constrained devices. While existing methods use dynamic pruning to improve efficiency by periodically adjusting sparse model topologies while maintaining sparsity, these approaches suffer from issues such as greedy adjustments, unstable topologies, and communication inefficiency, resulting in less robust models and suboptimal performance under data heterogeneity and partial client availability. To address these challenges, we propose Federated Robust pruning via combinatorial Thompson Sampling (FedRTS), a novel framework designed to develop robust sparse models. FedRTS enhances robustness and performance through its Thompson Sampling-based Adjustment (TSAdj) mechanism, which uses probabilistic decisions informed by stable, farsighted information instead of deterministic decisions reliant on unstable and myopic information in previous methods. Extensive experiments demonstrate that FedRTS achieves state-of-the-art performance in computer vision and natural language processing tasks while reducing communication costs, particularly excelling in scenarios with heterogeneous data distributions and partial client participation. Our codes are available at: https://github.com/Little0o0/FedRTS

We present a new Deep Neural Network (DNN) architecture capable of approximating functions up to machine accuracy. Termed Chebyshev Feature Neural Network (CFNN), the new structure employs Chebyshev functions with learnable frequencies as the first hidden layer, followed by the standard fully connected hidden layers. The learnable frequencies of the Chebyshev layer are initialized with exponential distributions to cover a wide range of frequencies. Combined with a multi-stage training strategy, we demonstrate that this CFNN structure can achieve machine accuracy during training. A comprehensive set of numerical examples for dimensions up to $20$ are provided to demonstrate the effectiveness and scalability of the method.

This study explores the application of generative adversarial networks in financial market supervision, especially for solving the problem of data imbalance to improve the accuracy of risk prediction. Since financial market data are often imbalanced, especially high-risk events such as market manipulation and systemic risk occur less frequently, traditional models have difficulty effectively identifying these minority events. This study proposes to generate synthetic data with similar characteristics to these minority events through GAN to balance the dataset, thereby improving the prediction performance of the model in financial supervision. Experimental results show that compared with traditional oversampling and undersampling methods, the data generated by GAN has significant advantages in dealing with imbalance problems and improving the prediction accuracy of the model. This method has broad application potential in financial regulatory agencies such as the U.S. Securities and Exchange Commission (SEC), the Financial Industry Regulatory Authority (FINRA), the Federal Deposit Insurance Corporation (FDIC), and the Federal Reserve.

With the global economic integration and the high interconnection of financial markets, financial institutions are facing unprecedented challenges, especially liquidity risk. This paper proposes a liquidity coverage ratio (LCR) prediction model based on the gated recurrent unit (GRU) network to help financial institutions manage their liquidity risk more effectively. By utilizing the GRU network in deep learning technology, the model can automatically learn complex patterns from historical data and accurately predict LCR for a period of time in the future. The experimental results show that compared with traditional methods, the GRU model proposed in this study shows significant advantages in mean absolute error (MAE), proving its higher accuracy and robustness. This not only provides financial institutions with a more reliable liquidity risk management tool but also provides support for regulators to formulate more scientific and reasonable policies, which helps to improve the stability of the entire financial system.

This study introduces a training-free conditional diffusion model for learning unknown stochastic differential equations (SDEs) using data. The proposed approach addresses key challenges in computational efficiency and accuracy for modeling SDEs by utilizing a score-based diffusion model to approximate their stochastic flow map. Unlike the existing methods, this technique is based on an analytically derived closed-form exact score function, which can be efficiently estimated by Monte Carlo method using the trajectory data, and eliminates the need for neural network training to learn the score function. By generating labeled data through solving the corresponding reverse ordinary differential equation, the approach enables supervised learning of the flow map. Extensive numerical experiments across various SDE types, including linear, nonlinear, and multi-dimensional systems, demonstrate the versatility and effectiveness of the method. The learned models exhibit significant improvements in predicting both short-term and long-term behaviors of unknown stochastic systems, often surpassing baseline methods like GANs in estimating drift and diffusion coefficients.

This paper takes the graph neural network as the technical framework, integrates the intrinsic connections between enterprise financial indicators, and proposes a model for enterprise credit risk assessment. The main research work includes: Firstly, based on the experience of predecessors, we selected 29 enterprise financial data indicators, abstracted each indicator as a vertex, deeply analyzed the relationships between the indicators, constructed a similarity matrix of indicators, and used the maximum spanning tree algorithm to achieve the graph structure mapping of enterprises; secondly, in the representation learning phase of the mapped graph, a graph neural network model was built to obtain its embedded representation. The feature vector of each node was expanded to 32 dimensions, and three GraphSAGE operations were performed on the graph, with the results pooled using the Pool operation, and the final output of three feature vectors was averaged to obtain the graph's embedded representation; finally, a classifier was constructed using a two-layer fully connected network to complete the prediction task. Experimental results on real enterprise data show that the model proposed in this paper can well complete the multi-level credit level estimation of enterprises. Furthermore, the tree-structured graph mapping deeply portrays the intrinsic connections of various indicator data of the company, and according to the ROC and other evaluation criteria, the model's classification effect is significant and has good "robustness".

We present a numerical method for learning unknown nonautonomous stochastic dynamical system, i.e., stochastic system subject to time dependent excitation or control signals. Our basic assumption is that the governing equations for the stochastic system are unavailable. However, short bursts of input/output (I/O) data consisting of certain known excitation signals and their corresponding system responses are available. When a sufficient amount of such I/O data are available, our method is capable of learning the unknown dynamics and producing an accurate predictive model for the stochastic responses of the system subject to arbitrary excitation signals not in the training data. Our method has two key components: (1) a local approximation of the training I/O data to transfer the learning into a parameterized form; and (2) a generative model to approximate the underlying unknown stochastic flow map in distribution. After presenting the method in detail, we present a comprehensive set of numerical examples to demonstrate the performance of the proposed method, especially for long-term system predictions.

Despite real-time planners exhibiting remarkable performance in autonomous driving, the growing exploration of Large Language Models (LLMs) has opened avenues for enhancing the interpretability and controllability of motion planning. Nevertheless, LLM-based planners continue to encounter significant challenges, including elevated resource consumption and extended inference times, which pose substantial obstacles to practical deployment. In light of these challenges, we introduce AsyncDriver, a new asynchronous LLM-enhanced closed-loop framework designed to leverage scene-associated instruction features produced by LLM to guide real-time planners in making precise and controllable trajectory predictions. On one hand, our method highlights the prowess of LLMs in comprehending and reasoning with vectorized scene data and a series of routing instructions, demonstrating its effective assistance to real-time planners. On the other hand, the proposed framework decouples the inference processes of the LLM and real-time planners. By capitalizing on the asynchronous nature of their inference frequencies, our approach have successfully reduced the computational cost introduced by LLM, while maintaining comparable performance. Experiments show that our approach achieves superior closed-loop evaluation performance on nuPlan's challenging scenarios.