LAUSR

In soft sensing, quality indicators are predicted by the signals of the discrete process variables in the time domain, in which the time delay exists in the process variables and quality indicators. Traditionally, discrete variables are used to estimate quality indicators directly through the time-series model, in which the sliding window is used to extract time-series features of the process signals. As discrete variables can be served as samples of a continuous function, this article proposes a soft sensing method whose inputs follow a continuous function. The continuous function is estimated by discrete variables and used to estimate quality indicators. Meanwhile, in the time-series model, the features extracted by the signals within the sliding window are inconsistent with the ones with all the historical signals, and thus, the time-series model lacks interpretability. To make a consistent prediction when the length of the time-series signals changes, this article proposes the windowed convolutional transformation to extract the features of the evaluated continuous function of discrete inputs. Besides, the proposed windowed convolutional transformation can automatically deal with the time delay in the process variables and quality indicators. To effectively calculate the gradients of the windowed convolutional transformation, a memory-efficient gradient estimation method for neural ordinary differential equations with inputs is designed. Finally, a soft-sensing method based on windowed convolutional transformation is proposed.