Fourier series time series prediction method based on Kolmogorov-Arnold theory

Abstract

This invention belongs to the field of time series data prediction technology, specifically providing a time series data prediction method based on Kolmogorov-Arnold representation theory and Fourier series. By utilizing the multi-layered nested structure in Kolmogorov-Arnold representation theory, high-dimensional nonlinear time series data is decomposed into multiple low-dimensional sub-problems. Combined with frequency domain analysis using Fourier series, efficient modeling and prediction of complex time series data are achieved. This invention utilizes Fourier series to perform frequency domain decomposition of time series data, extracting its main frequency components. Through the nonlinear mapping relationship in Kolmogorov-Arnold representation theory, a multi-layered nested prediction model is constructed, enabling the prediction of long-term trends and short-term fluctuations of complex nonlinear time series data. This effectively improves the generalization ability and prediction accuracy of the time series prediction model, while reducing the model's dependence on data scale and computational resources.

Description

Technical Field

[0001] This invention belongs to the field of time series data prediction technology, specifically involving a Fourier series time series prediction method based on Kolmogorov-Arnold theory. Background Technology

[0002] Time series forecasting is one of the core research directions in the field of data analysis. It involves mathematically modeling and analyzing historical time series data to predict the state or behavior of a system at a future point in time. Time series forecasting helps researchers and decision-makers understand and uncover the patterns, trends, and dynamic characteristics of data over time, facilitating subsequent planning, decision-making, and optimization. It has applications in multiple fields such as finance, meteorology, industrial production, and healthcare.

[0003] In time series forecasting methods, hybrid approaches that integrate mechanisms and data establish an intrinsic state-space representation of the data and learn model parameters using a data-driven approach, combining the advantages of theoretical analysis and data learning. These methods include autoregressive models, moving average models, and vector autoregressive models. However, for forecasting requirements demanding low computational complexity, high accuracy, and robustness, these methods perform poorly because they cannot achieve an accurate state space. Existing time series forecasting models, when dealing with complex nonlinear time series data, are limited by their fixed model structure and ability to capture the inherent complexity of the data. They lack sufficient flexibility to adapt to complex dynamics in the data, such as long-term dependencies and abrupt changes, leading to limitations in the accuracy and reliability of the forecast results. Summary of the Invention

[0004] The technical problem to be solved by this invention is to overcome the shortcomings of existing time series prediction methods, which cannot effectively handle complex nonlinear time series data, resulting in insufficient prediction accuracy and reliability. This invention provides a Fourier series time series prediction method based on Kolmogorov-Arnold theory.

[0005] The Fourier series time series prediction method based on Kolmogorov-Arnold theory includes the following steps: acquiring the original input data and generating a training sample set using the sliding window method; extracting frequency domain features using adaptive Fourier series decomposition; dynamically fusing Fourier feature components with different time resolutions and harmonic orders using a learnable weight matrix, compressing them to a low-dimensional space using a nonlinear activation function to form fused features; constructing a multi-layer nonlinear mapping network based on the Kolmogorov-Arnold representation theorem; the multi-layer nonlinear mapping network includes internal and external functions, the internal function nonlinearly encodes the input features using parameterized cosine basis functions, and the external function uses a fully connected network for aggregation encoding; inputting the time-frequency fused features into the multi-layer nonlinear mapping network, and optimizing the network's fundamental frequency parameters, phase offset, and weight coefficients through end-to-end training; inputting the time series data to be predicted into the multi-layer nonlinear mapping network, and achieving multi-step rolling prediction through a sliding window mechanism.

[0006] Furthermore, the original input data includes a multidimensional input sequence and a target output sequence. The multidimensional input sequence includes multiple input features at each time step, and the target output sequence includes output features at each time step. The method also includes: standardizing the input features of the multidimensional input sequence to eliminate dimensions; and normalizing the output features of the target output sequence.

[0007] Furthermore, adaptive Fourier series decomposition is used to extract frequency domain features; Fourier feature components with different time resolutions and harmonic orders are dynamically fused using a learnable weight matrix, and compressed to a low-dimensional space using a nonlinear activation function to form fused features. This includes the following methods: performing Fourier series decomposition on all features at each time step in each training sample in the training sample set to extract frequency domain features; dynamically weighting and fusing the same features at different time steps using a learnable weight matrix to obtain the Fourier feature components of the training samples; and compressing the Fourier feature components to a low-dimensional space using a nonlinear activation function to form fused features.

[0008] Furthermore, a multi-layer nonlinear mapping network is constructed, including the following steps: a multi-layer nonlinear mapping network is constructed based on a David Sprecher variant of the stacked multi-layer Kolmogorov-Arnold representation theorem, with each layer of the network containing multiple internal functions, which are adjusted by weight parameters, and translation parameters are introduced in each layer of the network; in the multi-layer nonlinear mapping network, the internal functions adopt the PReLU activation function, which is calculated based on the input tensor and slope parameters to process the tensor.

[0009] Furthermore, in the multi-layer nonlinear mapping network, the network layer of the external function includes two fully connected layers for aggregating features and introducing temporal dependencies. The output layer of the multi-layer nonlinear mapping network constrains the range of output values ​​within the interval [0,1] through the Sigmoid function.

[0010] Furthermore, the multi-layer nonlinear mapping network is optimized through an end-to-end training method, using mean squared error as the loss function. In the loss function, all samples are traversed, and the squared error of each sample is calculated based on the true value vector and the predicted value vector and accumulated. The average value is then calculated to obtain the MSE loss value.

[0011] Furthermore, in the training of the multi-layer nonlinear mapping network, mean squared error is used as the loss function, and the Adam optimizer is combined with an early stopping mechanism to ensure model convergence.

[0012] Furthermore, the time-series data prediction method is used for time-series data prediction of petroleum hydrocracking processes. The original input data is the processing data of petroleum hydrocracking processes. The original input data includes a multidimensional input sequence and a target output sequence. The input features of the multidimensional input sequence include total outlet temperature, reactor inlet temperature, reactor differential pressure, tower top pressure, water injection rate, and tower top reflux rate. The output feature of the target output sequence is the 10% diesel recovery temperature.

[0013] Beneficial Effects: This invention provides a time series data prediction method based on Kolmogorov-Arnold representation theory and Fourier series. By utilizing the multi-layered nested structure in Kolmogorov-Arnold representation theory, high-dimensional nonlinear time series data is decomposed into multiple low-dimensional sub-problems. Combined with frequency domain analysis using Fourier series, efficient modeling and prediction of complex time series data are achieved. This invention utilizes Fourier series to perform frequency domain decomposition of time series data, extracting its main frequency components. Through the nonlinear mapping relationship in Kolmogorov-Arnold representation theory, a multi-layered nested prediction model is constructed, enabling the prediction of long-term trends and short-term fluctuations of complex nonlinear time series data. This effectively improves the generalization ability and prediction accuracy of the time series prediction model while reducing the model's dependence on data scale and computational resources. Attached Figure Description

[0014] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0015] Figure 1 This is a schematic flowchart of the method steps of the present invention;

[0016] Figure 2 This is a schematic diagram of the time series prediction model structure of the present invention. Detailed Implementation

[0017] To make the above-mentioned objectives, features, and advantages of this application more apparent and understandable, the specific embodiments of this application are described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of this application. However, this application can be implemented in many other ways different from those described herein, and those skilled in the art can make similar modifications without departing from the spirit of this application. Therefore, this application is not limited to the specific embodiments disclosed below.

[0018] In the description of this application, it should be understood that the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "multiple" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0019] In this application, unless otherwise expressly specified and limited, the terms "installation," "connection," "joining," and "fixing," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication of two components or the interaction between two components, unless otherwise expressly limited. Those skilled in the art can understand the specific meaning of the above terms in this application according to the specific circumstances.

[0020] Reference Figure 1 and Figure 2 As shown, this embodiment provides a Fourier series time series prediction method based on Kolmogorov-Arnold theory, including the following method steps:

[0021] Step S1: Obtain the original input data and generate a training sample set using the sliding window method;

[0022] Step S2: Use adaptive Fourier series decomposition to extract frequency domain features; dynamically fuse Fourier feature components with different time resolutions and harmonic orders through a learnable weight matrix, and compress them to a low-dimensional space through a nonlinear activation function to form fused features;

[0023] Step S3: Based on the Kolmogorov-Arnold representation theorem, construct a multi-layer nonlinear mapping network; the multi-layer nonlinear mapping network includes an inner function and an outer function. The inner function performs nonlinear encoding on the input features through parameterized cosine basis functions, and the outer function adopts fully connected network aggregation encoding; input the time-frequency fusion features into the multi-layer nonlinear mapping network, and optimize the network's fundamental frequency parameters, phase offset, and weight coefficients through end-to-end training;

[0024] Step S4: Input the time series data to be predicted into a multi-layer nonlinear mapping network, and realize multi-step rolling prediction through a sliding window mechanism.

[0025] Specifically, the Kolmogorov–Arnold theorem states that any multivariate continuous function can be expressed as a finite combination of univariate continuous functions. For each multivariate continuous function f: [0,1] n →R can be represented as a superposition of continuous single-variable functions, specifically as follows:

[0026] ;

[0027] As a preferred embodiment, the David Sprecher variant of the Kolmogorov–Arnold representation of the theorem is chosen to save computational resources, specifically as follows:

[0028] ;

[0029] In this variant, a single internal function with appropriate parameter translation is used. Replace inner function This reduces the number of internal functions, simplifies the representation of multivariate continuous functions, reduces the number of parameters learned during model training, and optimizes the model structure.

[0030] In this embodiment, Fourier series is introduced to further enhance the model's ability to model complex time series data. The Fourier series is specifically represented as follows:

[0031] ;

[0032] in, Indicates the order of harmonics. Indicates the period of the signal. Indicates the harmonic amplitude. Indicates phase, The fundamental frequency is used to decompose any periodic signal into a superposition of finite or infinite simple harmonic waves.

[0033] In this embodiment, via baseband This method decomposes arbitrary periodic signals into a superposition of finite or infinite simple harmonic waves, possessing powerful frequency domain feature extraction capabilities. Furthermore, it introduces a finite, learnable cutoff frequency coefficient *s* to approximate an infinite series, finding the optimal balance between model complexity and fitting ability. It also utilizes a single internal function in the David Sprecher variant of the Kolmogorov–Arnold representation theorem. Replaced with a truncated finite Fourier series, it can be expressed as:

[0034] ;

[0035] Therefore, in this embodiment, the time series data prediction model is expressed as:

[0036] ;

[0037] Among them, weight parameters Used to adjust the contribution of each input variable to the final result. Represents the DC component of the Fourier series. Indicates the baseline value of the data. Indicates the first Amplitude of the first harmonic, cosine function Used to extract and characterize periodicity and frequency features in complex time-series data, it is a multivariate function. Supports accurate modeling of complex data. The value can be shifted for each summation term to adjust the final function value.

[0038] Specifically, in step S1, the original input data includes a multidimensional input sequence and a target output sequence. The multidimensional input sequence includes multiple input features at each time step, and the target output sequence includes output features at each time step. The steps also include: standardizing the input features of the multidimensional input sequence to eliminate dimensions; and normalizing the output features of the target output sequence to scale them to the [0,1] interval.

[0039] In this embodiment, the raw input data is collected through sensors or data interfaces and can be data from fields such as power load, meteorological parameters, or financial indicators. The multidimensional input sequence in the raw input data is represented as follows:

[0040] ;

[0041] Where n represents the number of input features at each time step. This represents the input features.

[0042] A sliding window is used to process the normalized data, with a window size of [size missing]. For the processed training data, the first Each training sample is represented as ,in , indicating from the first The time step to the The input feature sequence at each time step, It is the first The actual output at each time step.

[0043] Specifically, in step S2, for each training sample in the training sample set... All features at each time step Fourier series decomposition is performed to extract frequency domain features; a learnable weight matrix is ​​then used. The same features at different time steps are dynamically weighted and fused to obtain the Fourier feature components of the training samples; the Fourier feature components are compressed to a low-dimensional space through a nonlinear activation function to form fused features.

[0044] The specific calculation method for the Fourier eigencomponents is as follows:

[0045] ;

[0046] in, Indicates the first A weighted frequency domain feature vector with 1 feature dimension. Represents the weight matrix No. Line number Column elements, Indicates the first Each feature dimension at time step The Fourier feature vectors. The frequency domain features obtained after Fourier series decomposition at each time step within the sliding window, ultimately yielding all training samples. Fourier characteristic components .

[0047] For external functions in time series prediction models By introducing the self-gated activation function Swish, the fused features are calculated. The specific calculation method is as follows:

[0048] ;

[0049] In this embodiment, the Swish activation function introduces nonlinear features into the time series prediction model, avoiding the gradient vanishing problem caused by the Sigmoid activation function. It also limits the output values ​​of negative inputs in the time series prediction model to a certain range, increasing stability at a deeper level. This solves the problem of determining activation using spline functions and grid search in the original KAN network, saving computational resources. The dynamically weighted and fused features are further processed by the nonlinear activation function. By compressing the data into a low-dimensional space, a joint representation is formed that integrates local fluctuation details and global trend changes. This frequency-time domain coupling feature not only preserves the periodicity of the original time series but also enhances the model's ability to capture multi-scale time series patterns through a dynamic fusion mechanism. This provides high-information-density input features for the subsequent Kolmogorov-Arnold mapping layer, thereby effectively supporting the modeling and prediction of complex nonlinear time series relationships.

[0050] Specifically, in step S3, a multi-layer nonlinear mapping network is constructed, including the following steps: a multi-layer nonlinear mapping network is constructed based on a David Sprecher variant of the stacked multi-layer Kolmogorov-Arnold representation theorem, with each layer of the network containing multiple internal functions. The internal function uses weight parameters Adjustments are made to capture the contribution of different input variables to the final result, and translation parameters are introduced into each layer of the network. This allows the model to learn at different time scales. In the multi-layer nonlinear mapping network, the internal function employs the PReLU activation function, which is calculated based on the input tensor and slope parameters to process the tensor.

[0051] As a further improvement to this embodiment, in the multi-layer nonlinear mapping network, the network layer of the external function includes two fully connected layers for aggregating features and introducing temporal dependencies. The output layer of the multi-layer nonlinear mapping network constrains the range of output values ​​within the interval [0,1] using the Sigmoid function. Specifically, refer to... Figure 2 As shown, the fused features are processed by a nonlinear activation function. The input is fed into the hidden layer, which consists of multiple stacked simplified KA layers (composed of inner and outer nonlinear activation functions and fully connected layers) built based on the Kolmogorov–Arnold representation theorem. These layers further extract higher-order temporal features and capture the nonlinear relationships and dynamic dependencies between the input features. Finally, the output layer uses a sigmoid activation function to constrain the prediction results to the range [0,1], mapping the aggregated features to the objective function. The estimated value enables high-precision nonlinear prediction of the input time series.

[0052] In this embodiment, the multilayer nonlinear mapping network is optimized through an end-to-end training method, using mean squared error (MSE) as the loss function. In this loss function, all samples are traversed, and the squared error of each sample is calculated based on the true value vector and the predicted value vector, and then accumulated. The average value is then calculated to obtain the MSE loss value. The specific calculation method is as follows:

[0053] ;

[0054] Where n represents the number of samples. Indicates the first The true value of each sample This represents the model's predicted value.

[0055] As a preferred embodiment, in the training of the multi-layer nonlinear mapping network, mean squared error is used as the loss function, and the Adam optimizer is combined with an early stopping mechanism to ensure model convergence.

[0056] Specifically, in step S4, the preprocessed time-series data to be predicted is input into the trained time-series prediction model, and multi-step rolling prediction is achieved through a sliding window mechanism. During the deployment phase, model performance is further optimized and adapted to data drift by fine-tuning model parameters and combining them with regularization methods. The final output is a prediction result that combines interpretable frequency domain features with nonlinear mapping capabilities, meeting the real-time requirements of industrial scenarios.

[0057] In this embodiment, the time-series data prediction method is used for time-series data prediction of the petroleum hydrocracking process. The original input data is the processing data of the petroleum hydrocracking process. The original input data includes a multidimensional input sequence and a target output sequence. The input features of the multidimensional input sequence include total outlet temperature, reactor inlet temperature, reactor differential pressure, tower top pressure, water injection rate, and tower top reflux rate. The output feature of the target output sequence is the 10% diesel recovery temperature.

[0058] Specifically, in the hydrocracking process, multivariate time-series data within the reactor are collected in real time using industrial sensors as raw input data. This raw input data comprehensively reflects the operating status of the hydrocracking process, providing rich information for subsequent data analysis and model building. The raw input data was collected continuously for 20 months at 8-hour intervals, totaling 1800 samples. After normalization and standardization to eliminate dimensional differences, a sliding window method (window size of 3 steps) was used to construct a training sample set to predict the recovery temperature of the last time step in each sample. The data was then divided into a training set (70%), a validation set (15%), and a test set (15%) according to time series.

[0059] In model construction, the time segment of each input feature is first subjected to adaptive Fourier decomposition with a cutoff frequency of 5. The first three harmonic components (fundamental frequency, amplitude, and phase) are extracted. The frequency domain features of different time steps are dynamically fused through a learnable weight matrix to suppress sensor noise interference. The fused features are extended to 87 dimensions by the Swish activation function to form a joint frequency-time domain representation.

[0060] Subsequently, an intermediate layer is constructed based on the David Sprecher variant of the Kolmogorov-Arnold representation theory, with the PReLU activation function as the inner function. The fused features are then passed through a nonlinear activation function and input into the hidden layer to extract high-order temporal features. In the final output layer, a Sigmoid activation function is used to restrict the prediction results to the range [0,1]. This maps the aggregated features to the estimated value of the objective function, achieving high-precision nonlinear prediction of the original input data.

[0061] In a preferred embodiment, during training, mean squared error (MSE) is used as the loss function, combined with the Adam optimizer (initial learning rate η = 0.01, decreasing by 5% every 50 epochs), and an early stopping mechanism (termination if the validation set loss does not decrease for 10 consecutive epochs) ensures model convergence. In a preferred embodiment, the number of iterations is Nepoch = 500; the batch size is Nb = 32.

[0062] This embodiment also provides a computer device, including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the steps of the above-described time-series data prediction method.

[0063] This embodiment also provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the steps of the above-described time-series data prediction method.

[0064] This embodiment provides a time series data prediction method based on Kolmogorov-Arnold representation theory and Fourier series. By utilizing the multi-layered nested structure in Kolmogorov-Arnold representation theory, high-dimensional nonlinear time series data is decomposed into multiple low-dimensional sub-problems. Combined with frequency domain analysis using Fourier series, efficient modeling and prediction of complex time series data are achieved. This invention utilizes Fourier series to perform frequency domain decomposition of time series data, extracting its main frequency components. Through the nonlinear mapping relationships in Kolmogorov-Arnold representation theory, a multi-layered nested prediction model is constructed to predict the long-term trends and short-term fluctuations of complex nonlinear time series data. This effectively improves the generalization ability and prediction accuracy of the time series prediction model while reducing the model's dependence on data scale and computational resources.

[0065] In this embodiment, when the proposed time-series prediction method is applied to the hydrocracking process, compared with traditional time-series prediction methods such as LSTM and TCN, it not only provides higher accuracy prediction results but also meets the requirements of rapid response real-time control. By dynamically fusing frequency domain features and using nonlinear activation functions, the model also exhibits stronger robustness to noise.

[0066] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0067] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. A Fourier series time-series prediction method based on Kolmogorov-Arnold theory, used for time-series data prediction in petroleum hydrocracking processes. The input features of the multidimensional input sequence in the time-series data prediction include the total outlet temperature and reactor inlet temperature, and the output features of the target output sequence include diesel recovery temperature. Its key feature is... The method includes the following steps: obtaining the original input data, generating a training sample set using the sliding window method, and extracting frequency domain features using adaptive Fourier series decomposition; Fourier feature components with different time resolutions and harmonic orders are dynamically fused using a learnable weight matrix and compressed into a low-dimensional space using a nonlinear activation function to form fused features. A multi-layer nonlinear mapping network is constructed based on the Kolmogorov-Arnold representation theorem. A multi-layer nonlinear mapping network is also constructed based on a David Sprecher variant of the stacked multi-layer Kolmogorov-Arnold representation theorem. The Sprecher variant replaces the single inner function with a truncated finite Fourier series, containing multiple inner functions at each layer of the network. These inner functions are adjusted by weight parameters, and translation parameters are introduced at each layer. In the multi-layer nonlinear mapping network, the inner functions employ the PReLU activation function, which is calculated based on the input tensor and slope parameters. The multi-layer nonlinear mapping network includes inner and outer functions. The inner functions nonlinearly encode the input features using parameterized cosine basis functions, while the outer functions use fully connected network aggregation encoding. Time-frequency fusion features are input into the multi-layer nonlinear mapping network, and the network's fundamental frequency parameters, phase shift, and weight coefficients are optimized through end-to-end training. The time-series data to be predicted is input into the multi-layer nonlinear mapping network, and multi-step rolling prediction is achieved through a sliding window mechanism.

2. The Fourier series time series prediction method based on Kolmogorov-Arnold theory according to claim 1, characterized in that, The original input data includes a multidimensional input sequence and a target output sequence. The multidimensional input sequence includes multiple input features at each time step, and the target output sequence includes output features at each time step. It also includes: standardizing the input features of the multidimensional input sequence to eliminate dimensions; and normalizing the output features of the target output sequence.

3. The Fourier series time series prediction method based on Kolmogorov-Arnold theory according to claim 1, characterized in that, Frequency domain features are extracted using adaptive Fourier series decomposition; By dynamically fusing Fourier feature components with different time resolutions and harmonic orders using a learnable weight matrix, and compressing them into a low-dimensional space using a nonlinear activation function, a fused feature is formed. The method includes: performing Fourier series decomposition on all features at each time step in each training sample in the training sample set to extract frequency domain features; The same features at different time steps are dynamically weighted and fused using a learnable weight matrix to obtain the Fourier feature components of the training samples; the Fourier feature components are compressed to a low-dimensional space through a nonlinear activation function to form fused features.

4. The Fourier series time series prediction method based on Kolmogorov-Arnold theory according to claim 1, characterized in that, In the multi-layer nonlinear mapping network, the network layer of the external function includes two fully connected layers for aggregating features and introducing temporal dependencies. The output layer of the multi-layer nonlinear mapping network constrains the range of output values ​​within the interval [0,1] through the Sigmoid function.

5. The Fourier series time series prediction method based on Kolmogorov-Arnold theory according to claim 1, characterized in that, The multi-layer nonlinear mapping network is optimized through an end-to-end training method. The mean squared error is used as the loss function. In the loss function, all samples are traversed, and the squared error of each sample is calculated based on the true value vector and the predicted value vector and accumulated. The average value is then calculated to obtain the MSE loss value.

6. The Fourier series time series prediction method based on Kolmogorov-Arnold theory according to claim 5, characterized in that, In the training of the multi-layer nonlinear mapping network, mean squared error is used as the loss function, and the Adam optimizer is combined with an early stopping mechanism to ensure model convergence.

7. The Fourier series time series prediction method based on Kolmogorov-Arnold theory according to claim 1, characterized in that, The time-series data prediction method is used for time-series data prediction of petroleum hydrocracking process. The original input data is the processing data of petroleum hydrocracking process. The original input data includes a multidimensional input sequence and a target output sequence. The input features of the multidimensional input sequence include total outlet temperature, reactor inlet temperature, reactor differential pressure, tower top pressure, water injection rate, and tower top reflux rate. The output feature of the target output sequence is the diesel 10% recovery temperature.

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