## 1 Introduction
[PERSON]'s research in 2021 [9] implemented machine learning techniques grounded on the fundamental factors of agricultural product futures, enhancing forecast precision and providing key contributions to the field. [PERSON] et al. (2017) [10] chose convolutional neural networks for anticipating stock prices and discovered they surpassed traditional multilayer perceptrons and support vector machines in their predictive prowess. [PERSON] et al. implement a deep neural network (DNN) in 2016 [11] for various commodity futures prices and reported uniquely accurate results. In the vein of leveraging sophisticated AI, [PERSON] et al. (2018) [12] incorporated LSTM, BPNN, and CNN in creating arbitrage models, with experiments on coking coal, iron ore, and rebar futures demonstrating LSTM's superior performance. [PERSON] and [PERSON]'s comparison in 2022 [13] among LASSO, Xgboost, BPNN, and LSTM for arbitrage effects underlined the outstanding forecasting abilities of deep learning models. [PERSON] et al., (2021) [14] proposed the Informer model based on the Transformer model and demonstrated the effectiveness of the model in enhancing the prediction capacity of long sequence time-series forecasting (LSTF) through experiments. [PERSON] et al. (2022) [15] designed a SCINet model based on TCN, which achieved higher accuracy on public datasets across multiple fields. Moreover, neural network-based generative techniques have also begun to be applied in the field of financial time series. [PERSON] et al. [16] proposed a generative adversarial network (GAN) structure based on LSTM and got a promising performance in the closing price prediction on the real data. It is becoming increasingly clear that AI, especially through deep learning and machine learning, offers tremendous promise for futures market prediction and navigation. These advanced methodologies adeptly tame the nonlinearity of financial markets, elevate forecasting accuracy to new heights, and furnish investors with sophisticated tools for crafting strategic decisions. Nonetheless, current predictive endeavors within financial studies commonly focus on the immediate future, aiming to determine the upcoming value or direction for a single metric, essentially for univariate time series projection. Researchers often design models that are calibrated solely to forecast a single point or a unidimensional trend in subsequent time steps. This traditional approach, however, does not suffice for arbitrage strategy construction, where a broader temporal and dimensional perspective is essential for evaluating multiple indicators over an upcoming time span. To navigate these complexities, this study introduces a convolutional long short-term memory (ConvLSTM) model geared toward predicting futures price spreads. This advanced model boasts capabilities for multistep and multidimensional forecasting, which aligns more closely with the nuanced demands of practical trading scenarios. However, network architectures like ConvLSTM pose challenges due to their complex decision-making process and limited transparency in how predictions are influenced [17], which can affect confidence and trust in their results. Additionally, choosing the right hyperparameters often depends on previous research or the subjective judgment of practitioners. Selecting the appropriate hyperparameters is essential for enhancing the network's structure, which improves its ability to generalize and fit data accurately. A significant scholarly effort is directed toward developing systematic methods to reduce subjective bias and identify the best set of hyperparameters. This study presents an innovative forecasting approach tailored for the futures arbitrage domain, utilizing a PSO Deep-ConvLSTM model--a sophisticated, multidimensional, multistep predictor that seamlessly integrates PSO with the ConvLSTM network. The primary aim is to enhance the accuracy of futures price spread forecasts, thereby amplifying the potential for profitable arbitrage strategies.