Improved TsMixer time mixing based on low-rank adaptation and its application in flotation grade prediction

Flotation is a crucial step in non-ferrous metal beneficiation, and fluctuations in concentrate grade can significantly impact subsequent smelting energy consumption and production organization. Accurately predicting the copper grade of rougher concentrate is challenging due to the multivariate, strongly coupled, time-lag, and significant cumulative effects of flotation data, as well as its strong instantaneous fluctuations and noise errors. To address this issue, this paper uses the TsMixer time-series prediction model with a fully multilayer perceptron architecture as a baseline and introduces the low-rank adaptation concept to structurally modify the time mixing mechanism. Four TsMixer time mixing variants based on low-rank adaptation (TsMixer-LoRA1～LoRA3, TsMixer-LR) are designed, and the impact of the training/freezing strategy of the base weights and the low-rank order on model performance is investigated. Experiments are conducted using real flotation data, incorporating multivariate inputs including flotation process variables and foam visual features. Historical 60 min data is used to predict the grade for the next 15 min, and comparisons are made with models such as LSTM, GRU, TimesNet, FEDformer, PatchTST, and iTransformer. The results show that applying a low-rank constraint to the temporal mixing layer effectively improves the model′s generalization performance; TsMixer-LR achieves the best overall performance and exhibits a clear optimal rank interval. Further parameter scale analysis reveals that TsMixer-LR not only achieves the best prediction results at medium to low rank orders, but its total parameter count remains lower than or close to the benchmark TsMixer, demonstrating a better balance between parameter scale and prediction accuracy. Mechanistic analysis indicates that the low-rank bottleneck prompts the model to preferentially learn the dominant temporal structure, suppressing overfitting to instantaneous fluctuations and noise perturbations, thereby enhancing prediction stability and robustness.