Chromatography separation is a commonly used separation technique, and the chromatographic elution curve is an important basis for chromatographic separation design and calculation. Currently, chromatographic elution curves are usually obtained by solving chromatographic model. The chromatographic model is a mechanistic system of nonlinear, convection-dominated partial differential equations (PDEs). The solution methods of the model include finite difference method, finite element method, etc [[1], [2], [3], [4]]. The process of solving chromatographic models through these methods requires handling the stability or accuracy of iterative formats, and the solving process is relatively complex [[5], [6], [7], [8]]. The chromatographic model performs well when the adsorption isotherm equation is readily determinable. However, when the adsorption mechanism is complex and the isotherm equation cannot be accurately determined, the model's simulation accuracy becomes significantly compromised. To solve this problem, some researchers have replaced adsorption isotherms with artificial neural network (ANN) models to fit the corresponding adsorption relationships, and established Chromatography-ANN hybrid models [[9], [10], [11]]. Compared with the chromatographic model, these hybrid models have better fitting effects under complex adsorption relationships, but their solutions are more complex.

In recent years, with the development of artificial intelligence, some ANN methods have been gradually applied to chromatography process modeling, such as Physics-Informed Neural Network (PINN) and Graph Convolutional Network (GCN) [[12], [13], [14], [15], [16], [17], [18]]. Among these methods, PINN integrates the structure of neural networks with physical laws by incorporating the original PDEs and PDE constraints into the neural network's loss function. This enables the network training and numerical solution of chromatography models, essentially serving as an artificial neural network solving algorithm for chromatography models. GCN is a graph data model that can transform important operating variables of the chromatography process into graph structures and capture dependencies between graphs to achieve modeling and computation of the chromatography separation process. However, these models typically contain parameters on the order of hundreds and require substantial training data support, which hinders their application in conventional experimental research. Therefore, developing models with fewer parameters that can be driven by data from several experiments will significantly lower the adoption barrier and better meet practical needs.

Recurrent neural network (RNN) is a type of network capable of processing one-dimensional time series, excelling at capturing temporal dependencies within sequences. Through parameter sharing mechanisms, RNN maintains a fixed parameter count independent of input sequence length, making them applicable to time-series data of varying durations. The architecture is illustrated in Fig. 1. The RNN network consists of an input layer, a hidden layer, and an output layer. It takes sequential data as input and incorporates recurrent connections in the hidden layer. This allows the network to combine the current input with the previous hidden state (historical information) when processing the input, thereby capturing the influence of both current and historical inputs on the output. When future inputs do not affect the current output result, the hidden layer only has one forward layer, and the input values of the hidden layer come from the input layer and its own previous state values. However, when both past and future inputs affect the value of the output layer at the current time, the hidden layer includes both forward and backward layers, resulting in a Bidirectional recurrent neural network (BRNN) [[19], [20]]. During training, RNN is prone to vanishing and exploding gradients. These issues can be mitigated through methods such as gated unit design or recurrent transformation techniques [[21], [22]]. Alternatively, gradient-free optimization methods like genetic algorithms can be employed for network or parameter optimization [23].

The chromatographic separation process has certain similarities with RNN. The concentration change of solutes in a chromatographic column is a function of time and space. Within an individual theoretical plate of the chromatographic column, the concentration of solutes only changes with time under given experimental conditions, while RNN can process one-dimensional time series data. Therefore, the solute concentration changes inside the plate are calculated by RNN, and the calculation of chromatographic separation process can be achieved through plate theory and RNN. In this paper, we established an ANN model that can be used for two-component adsorption separation calculations. To reduce the number of unknown parameters in the ANN model, each layer in the model is given clear physical meanings, and the constraint relationships between model parameters are established through equations such as mass conservation. This enables the ANN model to achieve accurate calculations with a small number of experimental samples. The fitting effect of the single component and two-component competitive adsorption elution curves of the ANN model was evaluated under several experiments in Langmuir, Bi-Langmuir, and Sips equation chromatography systems.