Abstract:Forecasting systems in science must be accurate, physically consistent, and certifiably reliable. Most existing models address prediction, constraint enforcement, and verification separately, limiting scalability and interpretability. We introduce GeoCert, a geometric AI framework that unifies forecasting, physical reasoning, and formal verification within a single differentiable computation. GeoCert formulates forecasting as evolution along a hyperbolic manifold, where negative curvature induces contraction dynamics, intrinsic robustness, and logarithmic-time certification. A hierarchical constraint architecture separates universal physical laws from domain-specific dynamics, enabling certified generalization across energy, climate, finance, and transportation systems. GeoCert achieves state-of-the-art accuracy while reducing computational cost by 97.5% and maintaining better certification rates. By embedding verification into the geometry of learning, GeoCert transforms forecasting from empirical approximation to formally verified inference, offering a scalable foundation for trustworthy, reproducible, and physically grounded scientific AI.
Abstract:Time series prediction plays a pivotal role across diverse domains such as finance, healthcare, energy systems, and environmental modeling. However, existing approaches often struggle to balance efficiency, scalability, and accuracy, particularly when handling long-range dependencies and irregularly sampled data. To address these challenges, we propose MODE, a unified framework that integrates Low-Rank Neural Ordinary Differential Equations (Neural ODEs) with an Enhanced Mamba architecture. As illustrated in our framework, the input sequence is first transformed by a Linear Tokenization Layer and then processed through multiple Mamba Encoder blocks, each equipped with an Enhanced Mamba Layer that employs Causal Convolution, SiLU activation, and a Low-Rank Neural ODE enhancement to efficiently capture temporal dynamics. This low-rank formulation reduces computational overhead while maintaining expressive power. Furthermore, a segmented selective scanning mechanism, inspired by pseudo-ODE dynamics, adaptively focuses on salient subsequences to improve scalability and long-range sequence modeling. Extensive experiments on benchmark datasets demonstrate that MODE surpasses existing baselines in both predictive accuracy and computational efficiency. Overall, our contributions include: (1) a unified and efficient architecture for long-term time series modeling, (2) integration of Mamba's selective scanning with low-rank Neural ODEs for enhanced temporal representation, and (3) substantial improvements in efficiency and scalability enabled by low-rank approximation and dynamic selective scanning.