However, modern deep forecasting models often fail to capture these recurring patterns due to spectral bias and a lack of frequency-aware inductive priors. Motivated by this gap, we propose a simple yet effective method that enhances long-term forecasting by explicitly modeling periodicity through spectral initialization and frequency-constrained optimization. Specifically, we extract dominant low-frequency components via Fast Fourier Transform (FFT)-guided coordinate descent, initialize sinusoidal embeddings with these components, and employ a two-speed learning schedule to preserve meaningful frequency structure during training. Our approach is model-agnostic and integrates seamlessly into existing Transformer-based architectures. Extensive experiments across diverse real-world benchmarks demonstrate consistent performance gains--particularly at long horizons--highlighting the benefits of injecting spectral priors into deep temporal models for robust and interpretable long-range forecasting. Moreover, on synthetic data, our method accurately recovers ground-truth frequencies, further validating its interpretability and effectiveness in capturing latent periodic patterns.

Accurate time series forecasting, predicting future values based on past data, is crucial for diverse industries. Many current time series methods decompose time series into multiple sub-series, applying different model architectures and training with an end-to-end overall loss for forecasting. However, this raises a question: does this overall loss prioritize the importance of critical sub-series within the decomposition for the better performance? To investigate this, we conduct a study on the impact of overall loss on existing time series methods with sequence decomposition. Our findings reveal that overall loss may introduce bias in model learning, hindering the learning of the prioritization of more significant sub-series and limiting the forecasting performance. To address this, we propose a hybrid loss framework combining the global and component losses. This framework introduces component losses for each sub-series alongside the original overall loss. It employs a dual min-max algorithm to dynamically adjust weights between the overall loss and component losses, and within component losses. This enables the model to achieve better performance of current time series methods by focusing on more critical sub-series while still maintaining a low overall loss. We integrate our loss framework into several time series methods and evaluate the performance on multiple datasets. Results show an average improvement of 0.5-2% over existing methods without any modifications to the model architectures.

Accurate exchange rate prediction is fundamental to financial stability and international trade, positioning it as a critical focus in economic and financial research. Traditional forecasting models often falter when addressing the inherent complexities and non-linearities of exchange rate data. This study explores the application of advanced deep learning models, including LSTM, CNN, and transformer-based architectures, to enhance the predictive accuracy of the RMB/USD exchange rate. Utilizing 40 features across 6 categories, the analysis identifies TSMixer as the most effective model for this task. A rigorous feature selection process emphasizes the inclusion of key economic indicators, such as China-U.S. trade volumes and exchange rates of other major currencies like the euro-RMB and yen-dollar pairs. The integration of grad-CAM visualization techniques further enhances model interpretability, allowing for clearer identification of the most influential features and bolstering the credibility of the predictions. These findings underscore the pivotal role of fundamental economic data in exchange rate forecasting and highlight the substantial potential of machine learning models to deliver more accurate and reliable predictions, thereby serving as a valuable tool for financial analysis and decision-making.

Utilizing market forecasts is pivotal in optimizing portfolio selection strategies. We introduce DeepClair, a novel framework for portfolio selection. DeepClair leverages a transformer-based time-series forecasting model to predict market trends, facilitating more informed and adaptable portfolio decisions. To integrate the forecasting model into a deep reinforcement learning-driven portfolio selection framework, we introduced a two-step strategy: first, pre-training the time-series model on market data, followed by fine-tuning the portfolio selection architecture using this model. Additionally, we investigated the optimization technique, Low-Rank Adaptation (LoRA), to enhance the pre-trained forecasting model for fine-tuning in investment scenarios. This work bridges market forecasting and portfolio selection, facilitating the advancement of investment strategies.

The safe and stable operation of power systems is greatly challenged by the high variability and randomness of wind power in large-scale wind-power-integrated grids. Wind power forecasting is an effective solution to tackle this issue, with wind speed forecasting being an essential aspect. In this paper, a Graph-attentive Frequency-enhanced Spatial-Temporal Wind Speed Forecasting model based on graph attention and frequency-enhanced mechanisms, i.e., GFST-WSF, is proposed to improve the accuracy of short-term wind speed forecasting. The GFST-WSF comprises a Transformer architecture for temporal feature extraction and a Graph Attention Network (GAT) for spatial feature extraction. The GAT is specifically designed to capture the complex spatial dependencies among wind speed stations to effectively aggregate information from neighboring nodes in the graph, thus enhancing the spatial representation of the data. To model the time lag in wind speed correlation between adjacent wind farms caused by geographical factors, a dynamic complex adjacency matrix is formulated and utilized by the GAT. Benefiting from the effective spatio-temporal feature extraction and the deep architecture of the Transformer, the GFST-WSF outperforms other baselines in wind speed forecasting for the 6-24 hours ahead forecast horizon in case studies.

A core issue in multi-agent federated reinforcement learning is defining how to aggregate insights from multiple agents. This is commonly done by taking the average of each participating agent's model weights into one common model (FedAvg). We instead propose FedFormer, a novel federation strategy that utilizes Transformer Attention to contextually aggregate embeddings from models originating from different learner agents. In so doing, we attentively weigh the contributions of other agents with respect to the current agent's environment and learned relationships, thus providing a more effective and efficient federation. We evaluate our methods on the Meta-World environment and find that our approach yields significant improvements over FedAvg and non-federated Soft Actor-Critic single-agent methods. Our results compared to Soft Actor-Critic show that FedFormer achieves higher episodic return while still abiding by the privacy constraints of federated learning. Finally, we also demonstrate improvements in effectiveness with increased agent pools across all methods in certain tasks. This is contrasted by FedAvg, which fails to make noticeable improvements when scaled.

Although Transformer-based methods have significantly improved state-of-the-art results for long-term series forecasting, they are not only computationally expensive but more importantly, are unable to capture the global view of time series (e.g. overall trend). To address these problems, we propose to combine Transformer with the seasonal-trend decomposition method, in which the decomposition method captures the global profile of time series while Transformers capture more detailed structures. To further enhance the performance of Transformer for long-term prediction, we exploit the fact that most time series tend to have a sparse representation in well-known basis such as Fourier transform, and develop a frequency enhanced Transformer. Besides being more effective, the proposed method, termed as Frequency Enhanced Decomposed Transformer ({\bf FEDformer}), is more efficient than standard Transformer with a linear complexity to the sequence length. Our empirical studies with six benchmark datasets show that compared with state-of-the-art methods, FEDformer can reduce prediction error by $14.8\%$ and $22.6\%$ for multivariate and univariate time series, respectively. the code will be released soon.