Mathematical Foundations of Neural Tangents and Infinite-Width Networks

We investigate the mathematical foundations of neural networks in the infinite-width regime through the Neural Tangent Kernel (NTK). We propose the NTK-Eigenvalue-Controlled Residual Network (NTK-ECRN), an architecture integrating Fourier feature embeddings, residual connections with layerwise scaling, and stochastic depth to enable rigorous analysis of kernel evolution during training. Our theoretical contributions include deriving bounds on NTK dynamics, characterizing eigenvalue evolution, and linking spectral properties to generalization and optimization stability. Empirical results on synthetic and benchmark datasets validate the predicted kernel behavior and demonstrate improved training stability and generalization. This work provides a comprehensive framework bridging infinite-width theory and practical deep-learning architectures.

Geometric-Stochastic Multimodal Deep Learning for Predictive Modeling of SUDEP and Stroke Vulnerability

Sudden Unexpected Death in Epilepsy (SUDEP) and acute ischemic stroke are life-threatening conditions involving complex interactions across cortical, brainstem, and autonomic systems. We present a unified geometric-stochastic multimodal deep learning framework that integrates EEG, ECG, respiration, SpO2, EMG, and fMRI signals to model SUDEP and stroke vulnerability. The approach combines Riemannian manifold embeddings, Lie-group invariant feature representations, fractional stochastic dynamics, Hamiltonian energy-flow modeling, and cross-modal attention mechanisms. Stroke propagation is modeled using fractional epidemic diffusion over structural brain graphs. Experiments on the MULTI-CLARID dataset demonstrate improved predictive accuracy and interpretable biomarkers derived from manifold curvature, fractional memory indices, attention entropy, and diffusion centrality. The proposed framework provides a mathematically principled foundation for early detection, risk stratification, and interpretable multimodal modeling in neural-autonomic disorders.

Persistent Topological Structures and Cohomological Flows as a Mathematical Framework for Brain-Inspired Representation Learning

This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the evolution of cochain maps over dynamic simplicial complexes, enabling representations that capture invariants across temporal, spatial, and functional brain states. The proposed architecture integrates algebraic topology with differential geometry to construct cohomological operators that generalize gradient-based learning within a homological landscape. Synthetic data with controlled topological signatures and real neural datasets are jointly analyzed using persistent homology, sheaf cohomology, and spectral Laplacians to quantify stability, continuity, and structural preservation. Empirical results demonstrate that the model achieves superior manifold consistency and noise resilience compared to graph neural and manifold-based deep architectures, establishing a coherent mathematical foundation for topology-driven representation learning.