The rapid development of artificial intelligence and deep learning has created new opportunities for research in biomathematics and medical mathematics. This talk presents our recent advances in two directions. First, a Neural Ordinary Differential Equation (NODE)-based framework for local manifold approximation of nonlinear dynamical systems is introduced. The approach learns the underlying vector field directly from observational data, enabling the reconstruction and prediction of local system manifolds. A universal approximation theorem for NODE-based manifold learning is established, and extensive numerical experiments validate the proposed method. Applications to infectious disease and population dynamics further demonstrate its effectiveness in capturing the evolution of complex biological systems. Second, a Deep Siamese Residual Support Vector Machine (DSRSVM) is proposed for medical prediction tasks. The model combines siamese pre-training with residual SVM fine-tuning in an end-to-end framework, where the SVM loss is propagated through the deep network during training. Theoretical convergence guarantees are established, and experiments on publicly available medical datasets show improved accuracy, recall, and F1 scores compared with conventional deep learning–SVM hybrid methods. These results highlight the potential of deep learning for both nonlinear biological system modeling and disease prediction, illustrating how AI-driven mathematical tools can support the analysis of complex biological processes and medical decision-making.