FM-LLM: A frequency-enhanced mixture-of-experts framework for adapting LLMs to time series forecasting

Recent advances in Large Language Models (LLMs) have spurred cross-modal solutions for time-series forecasting. However, existing methods rely heavily on textual prompts for modality alignment-introducing nontrivial computational overhead and failing to leverage the rich spectral dynamics inherent in time-series data. To enable prompt-free, frequency-aware adaptation of frozen LLMs, we propose FM-LLM (Frequency-Enhanced Mixture-of-Experts for adapting LLMs to Time Series Forecasting), an autoregressive framework grounded in constrained asymmetric coupling. A Fourier Analysis Network (FAN)-based spectral token aligner injects structured harmonic representations directly into the frozen LLM with numerical compatibility. An asymmetric Mixture-of-Experts (MoE) decoder enforces role separation: shared experts with lightweight FAN layers reconstruct the global periodic backbone, while routed experts-restricted to standard FFNs-specialize in modeling non-periodic residual dynamics. A time-frequency hybrid loss function jointly optimizes temporal accuracy and spectral consistency, mitigating error accumulation during long-horizon autoregressive rollouts. Evaluated across eleven public benchmarks, FM-LLM achieves state-of-the-art performance on 59 out of 78 evaluation metrics. Compared to the strongest autoregressive LLM-based baseline, it delivers average improvements of 5.3% in MSE and 5.6% in MAE, with maximum gains reaching 8.0% for MSE and 8.4% for MAE. FM-LLM also demonstrates robust transferability, maintaining superior performance in 10% few-shot and zero-shot forecasting scenarios.