The development of the Internet makes the e-commerce transaction scale in the total global trade share grow year by year, and cross-border e-commerce has become an important growth point of global trade by virtue of its unique advantages. In this paper, the ARIMA model is used to obtain the time series demand change of a cross-border e-commerce supply chain, and the results are input into the LSTM model to realize the construction of a cross-border e-commerce supply chain demand forecasting model. The ABC inventory classification method and economic lot ordering model are used as the basis for the establishment of cross-border e-commerce supply chain inventory control strategies and multi-cycle inventory control models. Taking the sales data of WT enterprise from May 2022 to May 2023 as an example, the effectiveness of the ARIMA-LSTM model in cross-border e-commerce supply chain demand forecasting is analyzed, and the inventory control optimization results of the multi-period inventory control model are verified. The relative error fluctuation range of supply chain demand forecasting of the ARIMA-LSTM model is between [-0.1,0.2], and the cross-border e-commerce supply chain’s monthly demand forecast MAPE value is only 0.0135. After using the inventory control optimization model, the annual average inventory is reduced by 178.42 tons, and the total cost of inventory is reduced by 0.09*108 yuan. Relying on neural networks can achieve accurate prediction of cross-border e-commerce supply chain demand and optimize cross-border e-commerce supply chain inventory.