Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control

This paper proposes a unified family of learnable Koopman operator parameterizations that integrate linear dynamical systems theory with modern deep learning forecasting architectures. We introduce four learnable Koopman variants-scalar-gated, per-mode gated, MLP-shaped spectral mapping, and low-rank Koopman operators which generalize and interpolate between strictly stable Koopman operators and unconstrained linear latent dynamics. Our formulation enables explicit control over the spectrum, stability, and rank of the linear transition operator while retaining compatibility with expressive nonlinear backbones such as Patchtst, Autoformer, and Informer. We evaluate the proposed operators in a large-scale benchmark that also includes LSTM, DLinear, and simple diagonal State-Space Models (SSMs), as well as lightweight transformer variants. Experiments across multiple horizons and patch lengths show that learnable Koopman models provide a favorable bias-variance trade-off, improved conditioning, and more interpretable latent dynamics. We provide a full spectral analysis, including eigenvalue trajectories, stability envelopes, and learned spectral distributions. Our results demonstrate that learnable Koopman operators are effective, stable, and theoretically principled components for deep forecasting.

Asynchronous Federated Learning: A Scalable Approach for Decentralized Machine Learning

Federated Learning (FL) has emerged as a powerful paradigm for decentralized machine learning, enabling collaborative model training across diverse clients without sharing raw data. However, traditional FL approaches often face limitations in scalability and efficiency due to their reliance on synchronous client updates, which can result in significant delays and increased communication overhead, particularly in heterogeneous and dynamic environments. To address these challenges in this paper, we propose an Asynchronous Federated Learning (AFL) algorithm, which allows clients to update the global model independently and asynchronously. Our key contributions include a comprehensive convergence analysis of AFL in the presence of client delays and model staleness. By leveraging martingale difference sequence theory and variance bounds, we ensure robust convergence despite asynchronous updates. Assuming strongly convex local objective functions, we establish bounds on gradient variance under random client sampling and derive a recursion formula quantifying the impact of client delays on convergence. Furthermore, we demonstrate the practical applicability of AFL by training a decentralized Long Short-Term Memory (LSTM)-based deep learning model on the CMIP6 climate dataset, effectively handling non-IID and geographically distributed data. The proposed AFL algorithm addresses key limitations of traditional FL methods, such as inefficiency due to global synchronization and susceptibility to client drift. It enhances scalability, robustness, and efficiency in real-world settings with heterogeneous client populations and dynamic network conditions. Our results underscore the potential of AFL to drive advancements in distributed learning systems, particularly for large-scale, privacy-preserving applications in resource-constrained environments.