Abstract: Hyperbolic neural networks (HNNs) have shown remarkable success in representing hierarchical and tree-like structures, yet most existing work relies on the Poincaré ball and hyperboloid models. While these models admit closed-form Riemannian operators, their constrained nature potentially leads to numerical instabilities, especially near model boundaries. In this work, we explore the Proper Velocity (PV) manifold, an unconstrained representation of hyperbolic space rooted in Einstein’s special relativity, as a stable alternative. We first establish the complete Riemannian toolkit of the PV space. Building on this foundation, we introduce Proper Velocity Neural Networks (PVNNs) with core layers including Multinomial Logistic Regression (MLR), Fully Connected (FC), convolutional, activation, and batch normalization layers. Extensive experiments across four domains, namely numerical stability, graph node classification, image classification, and genomic sequence learning, demonstrate the stability and effectiveness of PVNNs.

Abstract: Credit risk refers to the possibility that a borrower fails to meet contractual repayment obligations, posing potential losses to lenders. This study aims to enhance post-loan default prediction in credit risk management by constructing a time-series modeling framework based on repayment behavior data, enabling the capture of repayment risks that emerge after loan issuance. To achieve this objective, a Residual Enhanced Encoder Bidirectional Long Short-Term Memory (ResE-BiLSTM) model is proposed, in which the attention mechanism is responsible for discovering long-range correlations, while the residual connections ensure the preservation of distant information. This design mitigates the tendency of conventional recurrent architectures to overemphasize recent inputs while underrepresenting distant temporal information in long-term dependency modeling. Using the real-world large-scale Freddie Mac Single-Family Loan-Level Dataset, the model is evaluated on 44 independent cohorts and compared with five baseline models, including Long Short-Term Memory (LSTM), Bidirectional LSTM (BiLSTM), Gated Recurrent Unit (GRU), Convolutional Neural Network (CNN), and Recurrent Neural Network (RNN) across multiple evaluation metrics. The experimental results demonstrate that ResE-BiLSTM achieves superior performance on key indicators such as F1 and AUC, with average values of 0.92 and 0.97, respectively, and demonstrates robust performance across different feature window lengths and resampling settings. Ablation experiments and SHapley Additive exPlanations (SHAP)-based interpretability analyses further reveal that the model captures non-monotonic temporal importance patterns across key financial features. This study advances time-series–based anomaly detection for credit risk prediction by integrating global and local temporal learning. The findings offer practical value for financial institutions and risk management practitioners, while also providing methodological insights and a transferable modeling paradigm for future research on credit risk assessment.

Abstract: This study addresses a critical challenge in time series anomaly detection: enhancing the predictive capability of loan default models more than three months in advance to enable early identification of default events, helping financial institutions implement preventive measures before risk events materialize. Existing methods have significant drawbacks, such as their lack of accuracy in early predictions and their dependence on training and testing within the same year and specific time frames. These issues limit their practical use, particularly with out-of-time data. To address these, the study introduces two innovative architectures, GRU-KAN and LSTM-KAN, which merge Kolmogorov–Arnold Networks (KAN) with Gated Recurrent Units (GRU) and Long Short-Term Memory (LSTM) networks. The proposed models were evaluated against the baseline models (LSTM, GRU, LSTM-Attention, and LSTM-Transformer) in terms of accuracy, precision, recall, F1 and AUC in different lengths of feature window, sample sizes, and early prediction intervals. The results demonstrate that the proposed model achieves a prediction accuracy of over 92% three months in advance and over 88% eight months in advance, significantly outperforming existing baselines.