A Coupled Model for Forecasting Spatiotemporal Variability of Regional Drought in the Mu Us Sandy Land Using a Meta-Heuristic Algorithm

Abstract

Vegetation plays a vital role in terrestrial ecosystems, and droughts driven by rising temperatures pose significant threats to vegetation health. This study investigates the evolution of vegetation drought from 2010 to 2024 and introduces a deep-learning-based forecasting model for analyzing regional spatial and temporal variations in drought. Extensive time-series remote-sensing data were utilized, and we integrated the Temperature–Vegetation Dryness Index (TVDI), Drought Severity Index (DSI), Evaporation Stress Index (ESI), and the Temperature–Vegetation–Precipitation Dryness Index (TVPDI) to develop a comprehensive methodology for extracting regional vegetation drought characteristics. To mitigate the effects of regional drought non-stationarity on predictive accuracy, we propose a coupling-enhancement strategy that combines the Whale Optimization Algorithm (WOA) with the Informer model, enabling more precise forecasting of long-term regional drought variations. Unlike conventional deep-learning models, this approach introduces rapid convergence and global search capabilities, utilizing a sparse self-attention mechanism that improves performance while reducing model complexity. The results demonstrate that: (1) compared to the traditional Transformer model, test accuracy is improved by 43%; (2) the WOA–Informer model efficiently handles multi-objective forecasting for extended time series, achieving MAE (Mean Absolute Error) ≤ 0.05, MSE (Mean Squared Error) ≤ 0.001, MSPE (Mean Squared Percentage Error) ≤ 0.01, and MAPE (Mean Absolute Percentage Error) ≤ 5%. This research provides advanced predictive tools and precise model support for long-term vegetation restoration efforts.

1. Introduction

Vegetation is crucial for the functioning of terrestrial ecosystems. Due to the effects of global warming and human activity, there is an increased occurrence of severe weather events, such as droughts, leading to a decrease in vegetation coverage [1]. The ongoing degradation of the ecology has extensive worldwide consequences. Time-series forecasting plays a crucial role in several sectors and is especially significant for extensive, long-term studies of time series [2]. This may aid in comprehending future patterns of change in the dry zone, therefore enhancing the sustainability of the environment [3]. Long-term time-series prediction is a crucial method for analyzing the spatial and temporal dryness of the vegetation in the area and assessing its influence on the biological environment.

Currently, the majority of vegetation-related indicators are specifically developed for addressing short-term issues [4]. Nevertheless, the model’s predictive capacity is severely constrained as the series length grows, which has hindered progress in long-term series prediction research [5]. Bishal Roy found that four supervised machine-learning algorithms, specifically support-vector regression, random forest, linear regression, and polynomial regression, successfully predicted vegetation indices over 1 year [6]. However, these algorithms were unable to accurately predict abrupt changes in vegetation indices that occurred during the middle of the year. Furthermore, machine learning has a significant disparity in accurately predicting lengthy time series, leading to suboptimal model performance and a substantial decrease in inference speed [7]. Deep learning is more effective in achieving superior outcomes for Natural Language Processing (NLP) tasks [8]. Nonetheless, the high cost of training makes it challenging to widely adopt deep-learning models. Presently, the self-attention mechanism is a limitation for predicting long time series, and the efficacy of both the self-attention mechanism and the Transformer design is a constraint for applying them to the problem of Long-Sequence Time-Series Forecasting (LSTF) [9]. The computational complexity of the self-attention mechanism is quadratic, meaning that as the length of the sequence rises, both the calculation time and memory needs increase at a square-level rate [10]. Additionally, it is possible to overlook the local information inside the sequence. Zhang et al. introduced an Informer model that addresses the memory bottleneck issue in deep learning and the challenge of long-time series prediction [11]. This model incorporates the ProbSparse self-attention mechanism and the self-attention distillation operation due to an issue of reduced performance. Hence, this work aims to use the enhanced Transformer–Informer model to address the issue of predicting lengthy time sequences.

Deep-learning models frequently encounter difficulties in parameter optimization with insufficient data, particularly when addressing non-stationary data, resulting in suboptimal performance on test sets [12]. Attaining consistency in hyperparameter optimization is challenging. Meta-heuristic optimization strategies are essential for maximizing deep-learning parameters and tackling non-stationarity in long-term time-series predictions [13]. These methods provide three primary advantages: (1) ease of implementation, (2) capacity to circumvent local optima and identify global solutions, and (3) extensive application across many circumstances [14,15]. Numerous modern meta-heuristic algorithms emulate biological processes, including Genetic Algorithm (GA), Evolutionary Strategies (ES), Artificial Bee Colony (ABC), Particle Swarm Optimization (PSO), and Ant Colony Optimization (ACO) [16,17,18,19]. PSO, formulated by Kennedy et al., simulates the collective behavior of avian or aquatic species, employing collaboration and information dissemination to identify optimal solutions [20]. It encompasses both individual and collective optimal solutions, rendering it a frequently employed method in deep-learning optimization [21]. Likewise, ACO, created by Dorigo et al., emulates the pheromone trail-following behavior of ants to identify optimal pathways, leveraging positive feedback mechanisms [22]. Although both PSO and ACO provide rapid convergence and efficient global search abilities, they encounter difficulties in multi-objective optimization: PSO is susceptible to local optima, while ACO is significantly dependent on delicate parameter configurations [23,24,25]. Achieving equilibrium between exploration and exploitation continues to be a significant difficulty for the majority of meta-heuristic algorithms [26]. This study presents the WOA (Whale Optimization Algorithm), which emulates the feeding behavior of humpback whales, as an innovative solution to these issues. The system incorporates the hunting behavior of following prey using either a random or optimum search agent, as well as the use of spirals to emulate the mechanism of humpback whales’ bubble net assault [27]. The WOA algorithm is used to optimize the multi-objective prediction of the Informer model, which is proposed to address the issue of forecasting the spatial and temporal variability of a lengthy time series of regional vegetation drought indicators.

This work addresses the present issues of global warming and escalating drought by combining several approaches for analyzing trends in vegetation drought indices over extended periods. The analysis is conducted using the GEE (Google Earth Engine) cloud platform. Additionally, a multi-objective prediction model is proposed to optimize the parameters involved. The system’s objective is to comprehensively uncover the geographical and temporal variations in long-term droughts in vegetation at a regional level. This will enable the development of a reliable scientific prediction tool for the establishment of ecological environmental protection measures in arid and semi-arid areas worldwide. The principal objectives of this study are: (1) to examine the advantages of optimizing the parameters of the Informer model; (2) to evaluate the benefits of utilizing the WOA–Informer model for forecasting long-term, multi-objective drought-related vegetation indicators.

2. Materials and Methods

2.1. Study Area

The Mu Us Sandy Land is situated in Ordos City, Inner Mongolia Autonomous Region, within the semi-arid and desert regions of Northwest China, and it is one of the principal sandy locations in the country. The area encompasses over 29,800 square kilometers, with a mean elevation between 1065 m and 1623 m (see Figure 1) [28]. The study area is located inside the administrative confines of Ordos City, encompassing the Uxin Banner, Yijinhuoluo Banner, Otog Banner, and Otogqian Banner. The landscape features include elevated regions in the north and south, with lower areas in the east and west, resulting in a gradual decline from northwest to southeast. The region has a temperate arid and semi-arid climate, categorized as a temperate continental climate, characterized by an average annual temperature of 6 to 8.5 °C and average annual precipitation of 250 to 420 mm, predominantly occurring from July to September. The Mu Us Sandy Land encompasses a variety of natural environments, including mobile dunes, semi-permanent dunes, fixed dunes, low ridges, marshes, alluvial terraces, lakes, rivers, and streams [29]. Soil types are categorized as zonal and non-zonal soils, with chestnut–calcium soil predominating in the northeast and southwest, and brown–calcium soil being prominent in the west. The soils of Mu Us Sandy Land exhibit low organic matter, poor fertility, weak cohesiveness, and low water retention, contributing to desertification and the prevalence of mobile, semi-permanent, and fixed dunes [30]. In conclusion, the biological system of the Mu Us Sandy Land is delicate and dependent on scarce water resources. Consequently, forecasting geographical and temporal variations in drought is highly significant.

2.2. Data and Processing

Data were gathered in the Mu Us Sandy Land region (109°16′–109°26′ E, 38°46′–39°55′ N) in Erdos, Inner Mongolia, from 12 March 2023, to 18 July 2024. The data were acquired with a DJI Wizard 4 Pro multispectral drone, including a lens focal length of 20 mm, an image angle range of −90° to 30°, and a pixel size of 20 μm (Figure 2). The pixel dimension is 20 μm, and the photographic resolution is 400 × 3000 pixels. The UAV navigated an S-shaped trajectory at a height of 95–100 m. The high-resolution drone photographs were used to corroborate the model’s predictions by contrasting them with the terrestrial circumstances in the research region.

We derived the NDVI utilizing the NDVI image element dichotomous model from the LANDSAT/LT05/C02/T1_L2, LANDSAT/LE07/C02/T1_L2, and LANDSAT/LC08/C02/T1_L2 datasets, in conjunction with the MOD13A1, MOD13A2, and MOD13Q1 products of MODIS, as well as the LST products MOD11A2 and MOD13A2 as surface temperature variables. Subsequently, we computed NDVI, LST, TVDI, TVPDI, DSI, and ESI indices via the GEE platform (17 June 2024). The indices were used to measure changes in plant cover and drought conditions in the Mu Us Sandy Land from 2010 to 2024, using a spatial resolution of 500 m and a monthly temporal resolution. We picked the maximum pixel values from the monthly picture data for compositing to create time-series images over 15 years. Furthermore, to alleviate the uncertainty arising from the restricted number of picture acquisitions every year, we conducted pixel-by-pixel 95th-quantile compositing analysis on Landsat images obtained annually.

This research utilizes the GEE platform to produce vegetation cover photographs inside the study region. The primary computational procedure comprises [31]:

(1)

Specify the scope of the water source: indicate the limits of the region being studied.

(2)

Create time-series datasets by combining the LANDSAT/LT05/C02/T1_L2, LANDSAT/LE07/C02/T1_L2, and LANDSAT/LC08/C02/T1_L2 datasets. Apply a filter to choose data within a certain period and then remove any cloud cover present in the data.

(3)

Generate annual median image dataset: aggregate the produced picture datasets by year and compute the yearly median image for each period.

(4)

Water body mask processing: Utilize the mask function to exclude water bodies from the picture to minimize their impact on the estimate of plant cover.

Thus, in this research, the NDVI-based image element dichotomy model was used to assess the plant cover in the study area:

$$NDVI={\displaystyle \frac{NIR-R}{NIR+NIR}}$$

(1)

where NIR and R are the reflectance recorded by the sensor in the near-infrared and red-light regions, respectively.

The TVDI was computed using the following equation (Figure 3):

$$TVDI={\displaystyle \frac{LST-{LST}_{min}}{{LST}_{max}-{LST}_{min}}}$$

(2)

where LST_min refers to the wet boundary with the lowest land-surface temperature (LST) value, while LST_max refers to the dry boundary with the highest LST value. LST reflects the monthly land-surface temperature for the same month between 2000 and 2020.

The TVPDI is a composite indicator that is calculated using measurements of T_s, NDVI, and P. The amalgamation of these three variables encompasses the influence of temperature, the extent of plant cover, and the quantity of precipitation necessary for vegetation development, hence enabling the evaluation of drought conditions in the designated research region [32]. The equation for this is:

$${NT}_s={\displaystyle \frac{T_s-T_{smin}}{T_{smax}-T_{smin}}}$$

(3)

$$NNDVI={\displaystyle \frac{NDVI-{NDVI}_{min}}{{NDVI}_{max}-{NDVI}_{min}}}$$

(4)

$$NP={\displaystyle \frac{P-P_{min}}{P_{max}-P_{min}}}$$

(5)

The formula for TVPDI is as follows:

$$TVPDI=\sqrt{{\left({NT}_{smax}-{NT}_S\right)}^2+{\left(NNDVI-{NNDVI}_{min}\right)}^2+{\left(NP-{NP}_{min}\right)}^2}$$

(6)

where NT_smax is the maximum of NT_s, and NNDVI_min and NP_min are the minimum of the NNDVI and NP.

The DSI used in this research was primarily computed using ET, PET, and NDVI. Thus, DSI was determined in this work using ET, PET from the MOD16A2 product, and NDVI from the MOD13A1 product [33]. The calculation may be determined using the following equation:

$$RT=ET/PET$$

(7)

$$Z_{RT}=\left(RT-\overline{RT}\right)/{\sigma }_{RT}$$

(8)

$$Z_{NDVI}=\left(NDVI-\overline{NDVI}\right)/{\sigma }_{NDVI}$$

(9)

$$Z=Z_{RT}+Z_{NDVI}$$

(10)

$$DSI=\left(Z-\overline{Z}\right)/{\sigma }_Z$$

(11)

where ET and PET are actual evapotranspiration and potential evapotranspiration, respectively, and RT is the ratio of ET to PET, which can be used as an indicator to evaluate the availability of terrestrial water resources and reflect the humid or arid state. RT and NDVI are the average values of RT and NDVI, respectively. σ_RT and σ_NDVI are the standard deviations of RT and NDVI, respectively. Z represents the sum of the normalized ratios of RT and NDVI; Z and σ_Z are the mean and standard deviation of Z.

In this work, the ESI was computed using ET from the MOD16A2 product. ET is primarily defined as 1 minus the ratio of actual to prospective ET [34]. The formula is simplified as follows:

$$ESI=1-{\displaystyle \frac{ET}{PET}}$$

(12)

where ET represents actual evapotranspiration and PET represents potential evapotranspiration.

We used ASTERGDEM 30 M resolution digital-elevation data and, specifically, chose Digital Elevation Model (DEM) data from a geospatial data cloud. Subsequently, we analyzed the DEM data with ArcGIS 10.8 to obtain precise elevation data.

2.3. Research Methodology

2.3.1. Whale Optimization Algorithm

Whales, being marine creatures that rely on obtaining oxygen from the ocean’s surface, possess double the number of spindle cells compared to mature individuals of other species. This abundance of spindle cells contributes to their significantly elevated levels of intelligence and empathy [35]. Humpback whales, as an exemplary species of cetaceans, use a distinctive feeding technique known as bubble nets, which involves the creation of “upward spirals” and “double loops” to ensnare krill or tiny fish near the water’s surface [36]. The Informer model was enhanced in this work by including the mathematical models of the WOA algorithm for surrounding prey, executing spiral bubble net feeding maneuvers, and hunting for prey.

(1)

Encircling prey

The WOA algorithm primarily emulates the hunting behavior of humpback whales to enhance the process of issue resolution by progressively approaching the present ideal answer. The provided equation represents this behavior:

$$\overrightarrow{D}=\left|\overrightarrow{C}.\overrightarrow{X^{\ast }}\left(t\right)-\overrightarrow{X}\left(t\right)\right|$$

(13)

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{X^{\ast }}\left(t\right)-\overrightarrow{A}.\overrightarrow{D}$$

(14)

where t represents the current iteration, $\overrightarrow{A}$ and $\overrightarrow{C}$ are coefficient vectors, X^∗ is the position vector of the best solution achieved so far, $\overrightarrow{X}$ is the position vector, | | denotes the absolute value, and · is an element-wise multiplication. It is important to note that X^∗ should be modified in every iteration if a superior solution is found.

$\overrightarrow{A}$ and $\overrightarrow{C}$ are calculated as follows:

$$\overrightarrow{A}=2\overrightarrow{a}·\overrightarrow{r}-\overrightarrow{a}$$

(15)

$$\overrightarrow{C}=2·\overrightarrow{r}$$

(16)

where $\overrightarrow{a}$ is linearly lowered from 2 to 0, while the vector $\overrightarrow{r}$ is a random vector within the range of 0 to 1.

The WOA algorithm achieves the optimum solution by iteratively examining the positions of the agents and using Equations (15) and (16) to identify the route that is most similar to the ideal solution [37]. The WOA can investigate the nearest position to the best solution by using the random vector $\overrightarrow{r}$. Hence, the WOA achieves optimum exploration by iteratively altering the locations (X, Y) of the search agents to gradually approach the best solution, mimicking the behavior of a whale encircling its prey [38].

(2)

Bubble-net attacking method (exploitation phase)

The WOA can replicate two specific behaviors shown by whales while using a bubble net to catch their prey. This simulation allows WOA to effectively find the ideal solution. The shrink-wrap method is accomplished by reducing the value in Equation (15). In other words, the location of the exploring agent may be adjusted to any place that is a certain distance away from the ideal agent position [39]. The location update of the prey is achieved by imitating the spiral motion of humpback whales in the following manner.

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{D^{\prime }}·e^{bl}·\cos \left(2\pi l\right)+\overrightarrow{X^{\ast }}\left(t\right)$$

(17)

where $\overrightarrow{D^{\prime }}$ = $\left|\overrightarrow{X^{\ast }}\left(t\right)-\overrightarrow{X}\left(t\right)\right|$ and indicates the distance of the i th whale to the prey (best solution obtained so far), b is a constant for defining the shape of the logarithmic spiral, l is a random number in [−1, 1], and . is an element-by-element multiplication.

To more precisely ascertain the position of the ideal solution, we use a simulation of a whale moving in a circular pattern around the prey inside a restricted region. The equation for mathematical modeling is as follows:

$$\overrightarrow{X}\left(t+1\right)=\left\{\begin{array}{c}\overrightarrow{X^{\ast }}\left(t\right)-\overrightarrow{A}·\overrightarrow{D} if p<0.5\\ \overrightarrow{D^{\prime }}·e^{bl}·\cos \left(2\pi l\right)+\overrightarrow{X^{\ast }}\left(t\right) if p\ge 0.5\end{array}\right.$$

(18)

where p is a random number in [0, 1].

2.3.2. Search for Prey (Exploration Phase)

We obtain the best possible answer by representing the process of whale feeding as a stochastic search method that relies on the locations of other whales. The same technique relying on the fluctuation of $\overrightarrow{A}$ may be used to hunt for prey [40]. The search agent’s location is determined by randomly generated values of A that are higher than 1 or less than −1. When the absolute value of A exceeds 1, the WOA algorithm prioritizes exploration and conducts a comprehensive search throughout the solution space. The following equation represents the mathematical model:

$$\overrightarrow{D}=\left|\overrightarrow{C}·\overrightarrow{X_{rand}}-\overrightarrow{X}\right|$$

(19)

$$\overrightarrow{D}\left(t+1\right)=\left|\overrightarrow{X_{rand}}-\overrightarrow{A}·\overrightarrow{D}\right|$$

(20)

where $\overrightarrow{X_{rand}}$ is a random-position vector (a random whale) chosen from the current population.

The WOA algorithm commences with randomly assigned parameters and iteratively determines the location of the best solution [41]. The search agent is not affected by the continual lowering of the parameter, which controls the range of the best solution (Figure 4). The WOA algorithm can alternate between spiral or circular motion based on the parameter p [42]. The WOA algorithm halts when a certain termination condition is met. The termination criteria may be met by either achieving the maximum number of iterations, discovering a good solution, or satisfying other stated requirements.

The WOA method, being a global optimizer, effectively explores and utilizes the search space by combining stochastic and optimum solution searches. It obtains the globally optimal solution by utilizing the fluctuation of parameters and switching between motion modes.

2.3.3. Informer

Currently, short-term time-series forecasting has a restricted range and fails to identify trends over extended periods, making it less efficient in managing intricate system dynamics (see Figure 5). Conversely, long-term forecasting may discern trends, cyclical variations, seasonal oscillations, and enduring nonlinear patterns, making it more appropriate for multi-objective regional precision forecasting. This work employs a deep-learning model to construct an encoder–decoder architecture for addressing the LSTF issue. The flowchart illustrating this design can be seen in Figure 6. Below is the primary data of the model.

The efficient self-attention technique is described using tuple inputs, namely inputs consisting of queries, keys, and values [43]. The attention of the interrogated individual is defined as a kernel smoother in probabilistic form, as shown in Equation:

$$A\left(Q,K,V\right)={\sum }_j{\displaystyle \frac{k\left(q_i,k_j\right)}{{\sum }_{l}k\left(q_i,k_l\right)}}{v}_j=E_p\left(k_j|q_j\right)\left[v_j\right]$$

(21)

where $p\left(k_j\right|q_i)=k({\mathrm{q}}_i,{\mathrm{k}}_j)/{\sum }_{l}k\left(q_i,k_l\right)$ and k(q_i, k_j) selects the asymmetric exponential kernel $exp\left(q_i{k_j}^T/\sqrt{d}\right)$.

To handle the partial sparsity of the probability distribution in the self-attention mechanism, a sparse approach with high correlation is used to sparsify all p(k_j|q_i). Additionally, each cell is designed to attend to the preceding cell in an exponentially rising way. In addition to the prior information, sparse alignment is included to address the issue of individual attention mechanisms [44].

ProbSparse self-focus detects the most significant crucial aspects:

$$A\left(Q,K,V\right)=Softmax\left({\displaystyle \frac{{\overline{Q}K}^T}{\sqrt{d}}}\right)v$$

(22)

where $\overline{Q}$ is a sparse matrix that has the same dimension as q, and it only includes the Top-u queries based on the sparsity measurement M(q,K).

Measurements of the highest average value:

$$\overline{M}\left(q_i,K\right)={max}_j\left\{{\displaystyle \frac{q_i{k_i}^T}{\sqrt{d}}}\right\}-{\displaystyle \frac{1}{L_K}}{\sum }_{j=1}^{L_K}{\displaystyle \frac{q_i{k_i}^T}{\sqrt{d}}}$$

(23)

where q_i is the query length, k_i is the bond length, L is query and key length, $\overline{M}\left(q_i,K\right)$ is the i th query the sparsity measurement of q_i and key set K.

The encoder of the Informer model consists of two identical polytopic attention layers. These layers are specifically designed to capture highly correlated relationships between different elements of a long sequence [45]. They also prevent the prediction process from decaying too rapidly by dynamically decoding markers and using generations to expand in NLP:

$$X_{j+1}^t=MaxPool\left(ELU\left(Convld\left({\left[X_j^t\right]}_{AB}\right)\right)\right)$$

(24)

where [·]_AB is Attention Block, Conv1d(·) is a 1D convolution filter applied in the time dimension, ELU(·) uses the ELU activation function, $X_j^t$ is the list entries, and Max-pool is the maximum pooling layer.

Figure 6 provides a comprehensive explanation. Due to its inherent quality of prioritizing dominating features, the ProbSparse self-attention mechanism can effectively identify and focus on the most important elements, minimizing the input sequence as much as feasible [46]. The ProbSparse algorithm utilizes CNN techniques while undergoing the process of refinement:

$$X_{de}^t=Concat\left(X_{token}^t,X_o^t\right)\in {\mathbb{R}}^{\left(L_{token}+L_y\right)\times d_{model}}$$

(25)

where $X_{token}^t\in {\mathbb{R}}^{L_{token}\times d_{model}}$ is the start token, $X_o^t\in {\mathbb{R}}^{L_y\times d_{model}}$ is a placeholder for the target sequence (set scalar as 0), and Concat is the output size.

This research evaluates the effectiveness of this strategy by using four different loss functions: MAE, MSE, RMSE, and MAPE:

$$e_{MAE}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n|y_i-{\widehat{y}}_i|$$

(26)

$$e_{MSE}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{|y_i-{\widehat{y}}_i|}^2$$

(27)

$$e_{RMSE}=\sqrt{{\displaystyle \frac{1}{n{\sum }_{i=1}^n{\left(|y_i-{\widehat{y}}_i|\right)}^2}}}$$

(28)

$$e_{MAPE}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n|y_i-{\widehat{y}}_i|∕y_i$$

(29)

where y_i is the true value of the original data; ${\widehat{y}}_i$ is the predicted value of the original data; n is the test sample size; and i is the test-sample point-sequence number.

The WOA–Informer model combines the WOA with the Informer model to provide automated hyperparameter tuning for time-series forecasting. WOA is used to identify and optimize the critical hyperparameters of the Informer model. Utilizing WOA’s global search capacity, the WOA–Informer model can ascertain the ideal hyperparameter combinations for various time-series jobs, thereby enhancing prediction accuracy. WOA modifies the placements of the solutions according to these assessment outcomes, thereby converging towards the best hyperparameter configuration. The WOA–Informer model integrates the robust efficacy of the Informer for long-sequence time-series forecasting with the global optimization proficiency of WOA. The primary benefits are as follows: The model effectively manages long-sequence time-series forecasting by using Informer’s sparse self-attention mechanism and hierarchical sampling techniques. WOA allows the WOA–Informer model to adjust to various time-series data formats, hence removing the intricacies of human hyperparameter tweaking and efficiently circumventing local optima. The model may autonomously optimize hyperparameters using WOA’s exploration and exploitation processes, identifying the optimal configuration for certain tasks. This amalgamation greatly improves the model’s predictive powers and flexibility.

2.4. Experiment

2.4.1. Can the WOA Enhance the Functionality and Performance of Informer?

We contend that the use of WOA enhances Informer’s capacity to accurately capture the fluctuations in regional non-stationary vegetation drought indices, provided that there is an adequate amount of training data available. To substantiate this, we constructed Transformer, Informer, and WOA–Informer models for the nine sites in this study, utilizing TVDI, DSI, ESI, and TVPDI data from 1 January 2010, to 12 March 2023, (a total of 636 samples) as training input, and forecasting TVDI, DSI, ESI, and TVPDI data for the period from April 2023 to April 2024 for accuracy assessment.

2.4.2. The Need for WOA to Enhance the Informer Model

The linked WOA–Informer strategy is believed to be a viable solution for mitigating the issue of parameter adjustment in the Informer model. WOA employs a random selection process to update the location of the search agent and the best solution, enabling it to conduct a global search and prevent being trapped in a local optimum solution. Furthermore, the adaptive parameter-tuning method included in the WOA algorithm aids the Informer model in identifying the most suitable parameters for various datasets and tasks. The WOA algorithm replicates the feeding behaviors of humpback whales, including the confined encircling and spiral movement. By combining these two behaviors, WOA can effectively obtain the most ideal solution. Implementing this varied search technique may enhance the Informer model’s ability to adjust to intricate data patterns. Additionally, it allows for the acquisition of a pre-trained model that is very near to the ideal answer while simultaneously lowering the time required for modeling. About the notion of zones for forecasting the changes in vegetation drought indices in dry and semi-dry regions, we developed three ways for dividing these areas into zones. The purpose was to confirm the importance of WOA in enhancing the Informer model. Therefore, we used distinct zoning strategies for each site to validate performance versus time cost. The research area’s data volume was established as 172 months (688 samples from January 2010 to April 2024) to optimize the Informer model. This was done to identify the optimal parameters for various datasets and jobs to enhance the performance of the Informer model.

3. Results

3.1. Advantages of WOA for Optimizing Informer Model Parameters

Our research demonstrates that the Informer model significantly enhances the dynamic prediction of non-stationary data when enough long-time series data are available. The ProbSparse self-attention mechanism identifies Q-values with a more informational contribution based on the probabilistic sparsity assumption, hence enhancing the efficiency of long-series data processing and mitigating the internal bottleneck issue in the Transformer model. Simultaneously, the WOA method may dynamically modify the hyperparameters, allowing the Informer model to be adaptively calibrated in response to the fluctuating vegetation drought data, therefore nearing the global ideal solution. Nevertheless, in the context of regional multi-objective prediction, conventional optimization algorithms often do not attain the performance of the Informer model. Integrating WOA with the Informer model enhances prediction accuracy, diminishes time complexity, optimizes resource use, and markedly elevates overall efficiency and performance. In contrast to conventional long-time series models like Transformer and LSTM, Informer is specifically optimized for managing dynamic long-time series multi-objective data, thereby minimizing computational demands while enhancing the extraction of critical information through multi-scale information selection. The WOA method effectively decreases the computational load while preserving prediction accuracy by optimizing the network architecture in conjunction with the ProbSparse attention mechanism for handling extensive time-series data. The Informer model employs ProbSparse attention for the dynamic prediction of vegetation drought, enabling it to manage extensive time-series data. Additionally, WOA aids the Informer model in identifying intricate non-stationary data characteristics, hence improving the regional model’s generalizability. Furthermore, the WOA–Informer model exhibits enhanced robustness while managing non-smooth and variable data. The primary reason is that although the Informer model has resilience in handling non-stationary data, it remains susceptible to local optimum solutions in long-term time-series forecasting, while WOA offers superior global optimization capabilities. Despite Informer’s superior predictive capability with intricate dynamic time-series data, a degree of uncertainty persists. WOA may enhance the performance of Informer by delivering more consistent prediction intervals and improved uncertainty estimates. We use WOA with the Informer model, applicable for migration models in multi-task learning, to enhance the model’s generalizability. The WOA-optimized Informer architecture enhances the model’s capacity for transfer learning and mitigates the generalization deficiencies seen in previous models across other domains. The WOA–Informer model demonstrates a substantial increase in prediction accuracy, with an average MSE of just 0.011 ± 0.012 over many predictions (refer to Figure 7). The integration of WOA and Informer effectively optimizes model parameters, enhances prediction accuracy, manages intricate multi-objective time-series data, and bolsters model robustness while also augmenting the model’s generalization capacity for large-scale time-series challenges in complex settings, including climate forecasting, water-quality assessment, and vegetation-health evaluation.

3.2. Effect of WOA on Informer Model Efficacy

The WOA algorithm markedly enhanced the dynamic prediction ability of the vegetative-drought TVDI index over an extensive time series, exhibiting substantial changes (the trend changes of NDVI, LST, TVDI, TVPDI, DSI, and ESI index from 2010 to 2023 are shown in Figure 8). Nevertheless, in the training of regional multi-objective data, attaining good long-term time-series predictions with modest parameter tweaks to the Informer deep-learning model becomes challenging. The mean MSE value for the 16 locations was 0.174 ± 0.031. Nonetheless, the Informer model optimized using the WOA method consistently surpassed the conventional Transformer model (p < 0.001). In contrast to Transformer, which decodes sequences incrementally, Informer uses generative decoding to forecast the full sequence in a single forward propagation. This significantly enhances the inference speed of lengthy sequence predictions and mitigates the prevalent cumulative error issue in the stepwise decoding process. In the vegetation-drought-prediction research, the WOA–Informer model showed superior predictive performance in the study region, with an average MSE of 0.0012 over 16 monitoring stations, whereas the Transformer exhibited the worst performance, with an average MAE of 0.16 (Figure 9). The WOA–Informer model has shown many significant benefits for multivariate predictions: (1) In comparison to the conventional Transformer, the Informer markedly diminishes time complexity and memory consumption, thereby enhancing its efficiency in managing long series prediction tasks; (2) the model is adept at processing ultra-long input sequences, which is essential for the long-term prediction of vegetation drought; (3) Informer substantially enhances predictive capability by proficiently capturing long-range dependencies, demonstrating exceptional performance; (4) the WOA algorithm exhibits robust global search capabilities, preventing the model from becoming trapped in local optima and facilitating the discovery of superior solutions within an expansive search space; (5) the WOA algorithm also possesses rapid convergence during the optimization process, enabling the model to swiftly identify optimal or near-optimal hyperparameters, thereby reducing training duration.

The research revealed that the Transformer model struggles to accurately capture the intricate relationships among multivariate variables, exhibiting an initial loss of 0.35, but the Informer and WOA–Informer models demonstrate a more rapid reduction in loss throughout training. The WOA algorithm and the sparse-attention mechanism effectively manage non-stationary time-series data, proving particularly useful for multi-scale dynamic information processing in extended time series. The WOA–Informer model requires just 13 epochs to achieve an acceptable loss level, demonstrating rapid convergence during training and proficiency in handling the intricacies of non-smooth time-series data.

3.3. Benefits of the WOA–Informer Model in Forecasting Extended Time Series of Dry Vegetation Indicators

We examine the efficacy of the WOA–Informer model in forecasting the vegetation-drought index within the framework of multi-objective optimization for extensive time-series data. Various variables may directly or indirectly influence the performance, training efficiency, and final generalization capability of a neural network model throughout the training process. The criteria include learning rate, network weights, loss function, overfitting and underfitting, batch size, and activation function (

Table 1. WOA–Informer model hyperparameter.

	Learning Rate	Overfitting	Batch Size	Activation Function
Transformer	0.0001	may	128	Relu
Informer	0.001	no	128	Gelu
WOA–Informer	0.001	no	128	Gelu

). The learning rate dictates the magnitude of weight updates in each gradient-descent iteration, the weight-initialization method influences convergence speed and model performance, the batch size impacts training efficiency and generalization capability, and the activation function establishes the level of nonlinearity in the neuron output. Consequently, effective regulation of these elements is essential to enhance the convergence rate, performance, and generalization capacity of the model. To avert the emergence of overfitting or underfitting, we include the Early Stopping module in the WOA–Informer model to oversee the validation set’s performance. Nevertheless, in the context of multi-objective data, the WOA–Informer model demonstrates an enhanced capacity for information acquisition, hence augmenting its predictive efficacy. The fundamental mathematical concepts include the following: (1) Constricting Enclosure and Spiral Method: By reducing the proximity to the present ideal solution and meticulously refining the location using the spiral method, it ultimately conducts a global search by randomly deviating from the current optimal solution. Sparse-Attention Mechanism: Enhance model efficiency by reducing computational requirements. Sequence-distillation operation: Incrementally decrease the sequence length to minimize memory and computational burden. Generative decoder: Enhance the inference velocity and mitigate error accumulation. As the volume of source-domain data used to train the WOA–Informer model escalates, the model’s predictive accuracy is markedly enhanced. Nonetheless, when the quantity of regional targets escalates to 16, the model’s predictive accuracy seems to peak, encountering its growth limitation. This indicates that the efficacy of the WOA–Informer model may be constrained by the computing capacity of the sparse-attention mechanism. Consequently, considering computational efficiency and model performance, we find it fair and suitable to establish the number of targets in the research area at 16.

3.4. Prolonged Observation and Examination of Vegetation Dynamics and Their Response to Drought

This work analyzes the weight correlations of multi-objective vegetation-drought indicators (TVDI, TVPDI, DSI, and ESI) over an extended time period from 2010 to 2024 to improve the precision of regional forecasts prior to regional forecasting. The findings indicate that the WOA–Informer model, using a regionalized methodology, has considerable superiority compared to locally trained deep-learning models. In the WOA–Informer model, we delineate the weight connection among the goals by formulating a loss function for multi-objective optimization and assessing their interactions (Figure 10). In comparison to the locally trained combinatorial model, the WOA–Informer model excels at capturing multi-objective synergistic interactions. Moreover, the WOA–Informer model is better suited for predicting analogous areas, owing to the simplicity of its parameters, which significantly enhances the model’s generalizability. This methodology examines the interconnections among several goals from a broader viewpoint, offers a more appropriate resolution for source domains, and aids in addressing the difficulties of regional forecasting and real-time assessment of vegetation drought.

3.5. WOA–Informer: Precise Long-Term Dynamic Sequence Forecasting

We picked the vegetation-drought index data spanning from 1 January 2010, to 18 April 2024, including 174 months, for multi-objective regional prediction. The training set comprises data from 1 January 2010, to 18 April 2023, while the test set includes data from 19 April 2023, to 22 April 2024. To evaluate the WOA–Informer model for the multi-objective parameter-optimization issue, minimal hyper-parameter adjustment was conducted, resulting in final hyper-parameter settings of a learning rate of 0.001, 20 iterations, and a batch size of 128. The WOA–Informer model’s prediction results for the test set indicate the following error metrics: MAE ≤ 0.05, MSE ≤ 0.001, MSPE ≤ 0.01, and MAPE < 5% (Figure 11). The research demonstrates that the WOA–Informer model has great accuracy in regional long-term time-series forecasting, with errors remaining constant and exhibiting little fluctuation, making it particularly appropriate for extensive dry regions. A reduced learning rate with fewer iterations successfully mitigates overfitting and strengthens the model’s generalization capability, while an increased batch size expedites model training and further bolsters model stability. The multi-objective prediction outcomes for extended time series indicate that regional multi-objective prediction may attain high accuracy and demonstrate significant applicability and universality.

4. Discussion

4.1. The Significance of Integrating WOA and Informer Models in Long-Term Time-Series Forecasting

This study integrates WOA with the Informer model, which incorporates a sparse self-attention mechanism, to introduce a novel modeling approach for regional multi-objective forecasting of the vegetation-drought index. We employ the Bionic Population Intelligent Optimization technique to enhance the hyperparameters of the Informer model. WOA effectively identifies the ideal hyperparameter combination, and when integrated with the ProbSparse attention mechanism, it accurately predicts lengthy time series in multi-objective forecasting [47]. The findings indicate that, in contrast to deep-learning model-optimization methods utilizing modal-decomposition techniques, WOA demonstrates superior adaptability in managing nonlinear, irregular, and complex data, and it is proficient in addressing single-objective, multi-objective, and constrained optimization challenges [48]. Furthermore, WOA possesses robust global search capabilities, successfully circumventing local optimization pitfalls, whereas modal-decomposition techniques frequently succumb to local optimization challenges when addressing complex non-convex or multi-peak structures [49]. When the training data are enough, WOA can replicate the “bubble net feeding” behavior of whales, together with appropriate configurations of population size and iteration count, yielding more optimal optimization outcomes [50]. Brodzicki, A. et al. demonstrated that the WOA-optimization algorithm can expedite the coupling process in high-dimensional search spaces by employing distributed or parallel computation via the population search mechanism [51]. Additionally, Ji, C. et al. discovered that WOA can dynamically modify its search strategy through the adaptive hunting behavior of whales, allowing it to respond to variations in the objective function over extended time-series data, thereby enhancing optimization efficacy [52]. The WOA–Informer model introduced in this paper exhibits robust regional predictive proficiency in the long-term multi-objective forecasting job. The model offers a more precise instrument for predicting the vegetation-aridity index in arid and semi-arid regions, addressing the information deficit in global vegetation predictions for these areas [53]. The model’s training duration escalates with an increase in the number of regional multi-targets. The WOA–Informer model can enhance prediction accuracy by fine-tuning the pre-trained model.

4.2. Influence of Prolonged Regional Drought-Index Forecasts

Thus far, limited research has been undertaken on the selection of deep-learning models for the implementation of long time series to forecast the dynamic information of the vegetation-drought index. In 2021, Sanjoy Chakraborty et al. integrated the SHADE and WOA algorithms to address global optimization challenges within evolutionary optimization techniques, assessing the efficacy of the new algorithms in resolving real-world issues through two unconstrained and four constrained engineering design problems [54]. In 2020, Eslam M. Hassib et al. evaluated the efficacy of novel algorithms for addressing real-world challenges by employing BRNN-based models alongside the WOA-optimization algorithm to analyze large datasets [55]. They compared GWO-MLP with Particle Swarm Optimization (PSO-MLP), Genetic Algorithm (GA-MLP), Ant Colony Optimization (ACO-MLP), Evolutionary Strategies (ES-MLP), and Population-Based Incremental Learning (PBIL-MLP), demonstrating that the combination of WOA and BRNN achieves commendable accuracy and a high rate of local optimal avoidance [56]. The conventional Transformer model exhibits a computational complexity of O(L²) per layer and may provide predictions from extensive time-series datasets. However, this typically entails substantial computational expense and memory usage [57]. The Transformer model incorporates a self-attentive mechanism. Nonetheless, its performance deteriorates due to sequence overfields [58]. Consequently, for regional forecasting utilizing extensive datasets, it is imperative to assess the model’s generalizability during pre-training and furnish a precise instrument for predicting the drought dynamics of vegetation in the designated arid and semi-arid domains by amalgamating regional predictions and establishing the weighting relationships among the targets within this area.

4.3. Influence of Multi-Objective Optimization on WOA–Informer Model Efficacy

This study demonstrates that enhancing the performance of the WOA–Informer model is achievable by augmenting the number of regional multi-targets. Our analysis revealed that the performance of the WOA–Informer model significantly improved while training due to the use of extensive time series of multi-target data, particularly when the number of monitoring points surpassed 10. Nonetheless, when the quantity of monitoring points attains 16, the model’s performance appears to approach its maximum development potential. This indicates the existence of a multi-objective threshold for WOA–Informer to execute a global search throughout the modeling process. This may result from the model’s susceptibility to local optima due to data complexity and high dimensionality [59]. Moreover, we discovered that WOA is susceptible to parameter configurations, and excessive tuning of these parameters may impact its performance and convergence rate. Consequently, determining whether excessive data input or a moderate amount suffices while utilizing the WOA–Informer model is a critical consideration for modeling efficiency [60]. Upon comparing the performance of the Transformer, Informer, and WOA–Informer models in the context of multi-objective optimization, it is evident that the prediction accuracy of WOA–Informer models surpasses that of the other models significantly [61]. The comprehensive acquisition of extensive time-series data allows the WOA–Informer model to more effectively manage the equilibrium among several objectives [62]. Furthermore, the WOA–Informer model enhances computing performance, reduces the number of parameters, and decreases model complexity, hence streamlining the parameter-tweaking process of the Informer model [63]. The study revealed that a reduction in monitoring points leads to a 30% to 42% decline in the performance of the WOA–Informer model. Nonetheless, it still outperforms the Transformer and Informer models despite this degradation [64]. Nonetheless, when sufficient monitoring points are present, the performance disparity between the WOA–Informer and Informer models is markedly diminished, mostly due to the ProbSparse attention mechanism and Hierarchical Time Embedding utilized in the Informer model [65]. This demonstrates that WOA is proficient in enhancing the Informer model, particularly for regional monitoring purposes. The integration of WOA and Informer can proficiently address the prediction issues associated with drought-prone vegetation areas and efficiently tackle the challenge of forecasting lengthy time-series data.

4.4. Limitations and Prospects

The primary objective of this study is to tackle the critical challenges in multi-objective region prediction, particularly the generalizability and parameter complexity of deep-learning models managing extensive time-series data. While our suggested WOA–Informer model exhibits a marginally reduced training speed relative to the Informer model, its parameter-optimization procedure is more straightforward and can more rapidly identify the best global solution [66]. Subsequent studies may enhance this training velocity. For instance, migration learning can be employed to explore the spatial and physical determinants of human activities, climate change, and regional forecasting to identify the principal elements influencing the vegetation-drought index [67]. The incorporation of reinforcement learning can improve the generalization capacity of deep-learning models in extended time series, hence addressing continuous decision-making challenges. WOA necessitates numerous iterations throughout its search process, and in the context of intricate high-dimensional, non-convex problems, it may encounter early convergence, resulting in the failure to identify the global optimum [68]. The WOA–Informer model necessitates great data quality and is particularly susceptible to outliers, which can compromise the accuracy of its predictions. The model has inadequate sensitivity in managing extremely non-stationary data, hindering its ability to accurately capture the intricate variations within the data. Another problem is the optimization of hyperparameters, such as population size, maximum iterations, number of attention heads, window size, and learning rate, which generally necessitates considerable time and domain expertise. While the WOA–Informer model has considerable benefits in areas like water resource management and environmental time-series forecasting, its utilization in real-time or online prediction contexts is constrained by substantial computational resource requirements. The implementation of the differential evolution mechanism can effectively circumvent local optima by directing the search trajectory based on the disparities between solutions, hence offering a more resilient modeling strategy for regional forecasting. These enhancements and expansions will considerably augment the efficacy and applicability of our system in predicting the regional vegetation-drought index.

5. Conclusions

This paper presents a high-performance, regional, multi-target prediction model that integrates WOA with the Informer model to accurately forecast the vegetation-aridity index over an extended time series in arid and semi-arid locations. We assessed the model’s performance at 16 aerial photography test locations in the hinterland of the Mu Us Sandy Land region in the Inner Mongolia Autonomous Region. Following the pre-training of the research area, we implemented the proposed WOA–Informer model in the target region, primarily utilizing the meta-heuristic optimization technique to augment its global search efficacy and significantly diminish the computational cost associated with the conventional self-attention mechanism. Experimental results indicate that the model’s prediction accuracy markedly surpasses that of the conventional Transformer model, while the complexity of the parameter adjustment is much diminished in comparison to the Informer model. The research demonstrates that the WOA-optimization technique significantly enhances the performance of the Informer model. In comparison to the Informer model alone, the coupled model demonstrates superior performance in addressing the intricate nonlinear, regional, multi-objective prediction of the vegetation-drought index, exhibiting effective solution capabilities. This indicates that our suggested model can effectively address the challenging issue of vegetation monitoring and prediction in arid and semi-arid regions while also offering an optimum approach for attaining precise regional predictions globally in the future.