Long-range time series forecasting remains challenging, as it requires capturing non-stationary and multi-scale temporal dependencies while maintaining noise robustness, efficiency, and stability. Transformer-based architectures such as Autoformer and Informer improve generalization but suffer from quadratic complexity and degraded performance on very long time horizons. State space models, notably S-Mamba, provide linear-time updates but often face unstable training dynamics, sensitivity to initialization, and limited robustness for multivariate forecasting. To address such challenges, we propose the Quantum-Optimized Selective State Space Model (Q-SSM), a hybrid quantum-optimized approach that integrates state space dynamics with a variational quantum gate. Instead of relying on expensive attention mechanisms, Q-SSM employs a simple parametrized quantum circuit (RY-RX ansatz) whose expectation values regulate memory updates adaptively. This quantum gating mechanism improves convergence stability, enhances the modeling of long-term dependencies, and provides a lightweight alternative to attention. We empirically validate Q-SSM on three widely used benchmarks, i.e., ETT, Traffic, and Exchange Rate. Results show that Q-SSM consistently improves over strong baselines (LSTM, TCN, Reformer), Transformer-based models, and S-Mamba. These findings demonstrate that variational quantum gating can address current limitations in long-range forecasting, leading to accurate and robust multivariate predictions.

The problem of Time-series Forecasting is generally addressed by recurrent, Transformer-based and the recently proposed Mamba-based architectures. However, existing architectures generally process their input at a single temporal scale, which may be sub-optimal for many tasks where information changes over multiple time scales. In this paper, we introduce a novel architecture called Multi-scale Mamba (ms-Mamba) to address this gap. ms-Mamba incorporates multiple temporal scales by using multiple Mamba blocks with different sampling rates ($Δ$s). Our experiments on many benchmarks demonstrate that ms-Mamba outperforms state-of-the-art approaches, including the recently proposed Transformer-based and Mamba-based models.

State space models, such as Mamba, have recently garnered attention in time series forecasting due to their ability to capture sequence patterns. However, in electricity consumption benchmarks, Mamba forecasts exhibit a mean error of approximately 8\%. Similarly, in traffic occupancy benchmarks, the mean error reaches 18\%. This discrepancy leaves us to wonder whether the prediction is simply inaccurate or falls within error given spread in historical data. To address this limitation, we propose a method to quantify the predictive uncertainty of Mamba forecasts. Here, we propose a dual-network framework based on the Mamba architecture for probabilistic forecasting, where one network generates point forecasts while the other estimates predictive uncertainty by modeling variance. We abbreviate our tool, Mamba with probabilistic time series forecasting, as Mamba-ProbTSF and the code for its implementation is available on GitHub (https://github.com/PessoaP/Mamba-ProbTSF). Evaluating this approach on synthetic and real-world benchmark datasets, we find Kullback-Leibler divergence between the learned distributions and the data--which, in the limit of infinite data, should converge to zero if the model correctly captures the underlying probability distribution--reduced to the order of $10^{-3}$ for synthetic data and $10^{-1}$ for real-world benchmark, demonstrating its effectiveness. We find that in both the electricity consumption and traffic occupancy benchmark, the true trajectory stays within the predicted uncertainty interval at the two-sigma level about 95\% of the time. We end with a consideration of potential limitations, adjustments to improve performance, and considerations for applying this framework to processes for purely or largely stochastic dynamics where the stochastic changes accumulate, as observed for example in pure Brownian motion or molecular dynamics trajectories.

Predicting Bitcoin price remains a challenging problem due to the high volatility and complex non-linear dynamics of cryptocurrency markets. Traditional time-series models, such as ARIMA and GARCH, and recurrent neural networks, like LSTMs, have been widely applied to this task but struggle to capture the regime shifts and long-range dependencies inherent in the data. In this work, we propose CryptoMamba, a novel Mamba-based State Space Model (SSM) architecture designed to effectively capture long-range dependencies in financial time-series data. Our experiments show that CryptoMamba not only provides more accurate predictions but also offers enhanced generalizability across different market conditions, surpassing the limitations of previous models. Coupled with trading algorithms for real-world scenarios, CryptoMamba demonstrates its practical utility by translating accurate forecasts into financial outcomes. Our findings signal a huge advantage for SSMs in stock and cryptocurrency price forecasting tasks.

Mamba has recently emerged as a promising alternative to Transformers, offering near-linear complexity in processing sequential data. However, while channels in time series (TS) data have no specific order in general, recent studies have adopted Mamba to capture channel dependencies (CD) in TS, introducing a sequential order bias. To address this issue, we propose SOR-Mamba, a TS forecasting method that 1) incorporates a regularization strategy to minimize the discrepancy between two embedding vectors generated from data with reversed channel orders, thereby enhancing robustness to channel order, and 2) eliminates the 1D-convolution originally designed to capture local information in sequential data. Furthermore, we introduce channel correlation modeling (CCM), a pretraining task aimed at preserving correlations between channels from the data space to the latent space in order to enhance the ability to capture CD. Extensive experiments demonstrate the efficacy of the proposed method across standard and transfer learning scenarios. Time series (TS) forecasting is prevalent in various fields, including weather (Angryk et al., 2020), traffic (Cirstea et al., 2022), and energy (Dudek et al., 2021). While Transformers (Vaswani et al., 2017) have been widely employed for this task due to their ability to capture long-term dependencies in sequences (Wen et al., 2022), their quadratic computational complexity causes substantial computational overhead, limiting their practicality in real-world applications.

Multivariate Time series forecasting is crucial in domains such as transportation, meteorology, and finance, especially for predicting extreme weather events. State-of-the-art methods predominantly rely on Transformer architectures, which utilize attention mechanisms to capture temporal dependencies. However, these methods are hindered by quadratic time complexity, limiting the model's scalability with respect to input sequence length. This significantly restricts their practicality in the real world. Mamba, based on state space models (SSM), provides a solution with linear time complexity, increasing the potential for efficient forecasting of sequential data. In this study, we propose UmambaTSF, a novel long-term time series forecasting framework that integrates multi-scale feature extraction capabilities of U-shaped encoder-decoder multilayer perceptrons (MLP) with Mamba's long sequence representation. To improve performance and efficiency, the Mamba blocks introduced in the framework adopt a refined residual structure and adaptable design, enabling the capture of unique temporal signals and flexible channel processing. In the experiments, UmambaTSF achieves state-of-the-art performance and excellent generality on widely used benchmark datasets while maintaining linear time complexity and low memory consumption.

In multivariate time-series forecasting (MTSF), extracting the temporal correlations of the input sequences is crucial. While popular Transformer-based predictive models can perform well, their quadratic computational complexity results in inefficiency and high overhead. The recently emerged Mamba, a selective state space model, has shown promising results in many fields due to its strong temporal feature extraction capabilities and linear computational complexity. However, due to the unilateral nature of Mamba, channel-independent predictive models based on Mamba cannot attend to the relationships among all variables in the manner of Transformer-based models. To address this issue, we combine fast-attention with Mamba to introduce a novel framework named FMamba for MTSF. Technically, we first extract the temporal features of the input variables through an embedding layer, then compute the dependencies among input variables via the fast-attention module. Subsequently, we use Mamba to selectively deal with the input features and further extract the temporal dependencies of the variables through the multi-layer perceptron block (MLP-block). Finally, FMamba obtains the predictive results through the projector, a linear layer. Experimental results on eight public datasets demonstrate that FMamba can achieve state-of-the-art performance while maintaining low computational overhead.

In the realm of time series forecasting (TSF), it is imperative for models to adeptly discern and distill hidden patterns within historical time series data to forecast future states. Transformer-based models exhibit formidable efficacy in TSF, primarily attributed to their advantage in apprehending these patterns. However, the quadratic complexity of the Transformer leads to low computational efficiency and high costs, which somewhat hinders the deployment of the TSF model in real-world scenarios. Recently, Mamba, a selective state space model, has gained traction due to its ability to process dependencies in sequences while maintaining near-linear complexity. For TSF tasks, these characteristics enable Mamba to comprehend hidden patterns as the Transformer and reduce computational overhead compared to the Transformer. Therefore, we propose a Mamba-based model named Simple-Mamba (S-Mamba) for TSF. Specifically, we tokenize the time points of each variate autonomously via a linear layer. A bidirectional Mamba layer is utilized to extract inter-variate correlations and a Feed-Forward Network is set to learn temporal dependencies. Finally, the generation of forecast outcomes through a linear mapping layer. Experiments on thirteen public datasets prove that S-Mamba maintains low computational overhead and achieves leading performance. Furthermore, we conduct extensive experiments to explore Mamba's potential in TSF tasks. Our code is available at https://github.com/wzhwzhwzh0921/S-D-Mamba.