Post Constraint and Correction: A Plug-and-Play Module for Boosting the Performance of Deep Learning Based Weather Multivariate Time Series Forecasting

Abstract

Weather forecasting is essential for various applications such as agriculture and transportation, and relies heavily on meteorological sequential data such as multivariate time series collected from weather stations. Traditional numerical weather prediction (NWP) methods applied to multivariate time series forecasting are grounded in statistical principles such as Autoregressive Integrated Moving Average (ARIMA); however, they often struggle with capturing complex nonlinear patterns among meteorological variables and temporal variances. Currently, existing deep learning approaches such as Recurrent Neural Networks (RNNs) and transformers offer remarkable performance in handling complex patterns among meteorological multivariate time series, yet frequently fail to maintain weather-specific physical properties such as strict values constraints, while also incurring the significant computational costs of large parameter scales. In this paper, we present a novel deep learning plug-and-play framework named Post Constraint and Correction (PCC) to address these challenges by incorporating additional constraints and corrections based on weather-specific properties such as multivariant correlations and physical-based strict value constraints into the prediction process. Our method demonstrates notable computational efficiency, delivering significant improvements over existing deep learning time series models and helping to achieve better performance with far fewer parameters. Extensive experiments demonstrate the effectiveness, efficiency, and robustness of our method, highlighting its potential for real-world applications.

1. Introduction

Weather forecasting plays a crucial role in modern society, impacting various sectors from agriculture and transportation to daily life planning [1]. Meteorological multivariate time series collected from weather stations serve as one of the primary data sources for weather forecasting tasks, containing multiple variables such as temperature, humidity, pressure, wind velocity, and precipitation which can be used to describe the weather state at a specific location over a period of time.

Traditional numerical weather prediction (NWP) methods applied to multivariate time series forecasting tasks primarily consist of statistical and mathematical models, including Autoregressive Integrated Moving Average (ARIMA) [2], Kalman filtering [3], and Singular Spectrum Analysis (SSA) [4]. However, these which often struggle with nonlinear relationships and complex patterns among meteorological series [5]. Recently, the advancement of deep learning has transformed weather forecasting capabilities, driven by three key factors: algorithmic innovations, the unprecedented scale of available weather data, and advances in parallel computing hardware [6]. Deep learning time series forecasting models excel at automatically extracting complex temporal patterns and dependencies from time series data, achieving state-of-the-art accuracy [7,8].

Meteorological variables possess distinct physical characteristics that significantly influence their temporal dynamics and interrelationships. Temperature exhibits daily and seasonal cyclical patterns with gradual transitions, while precipitation demonstrates intermittent and often non-Gaussian distributions with sudden onsets and varying intensities [9]. Atmospheric pressure typically displays smoother temporal transitions, but can undergo abrupt changes during weather front passages. These variables are further governed by fundamental physical principles such as energy conservation and thermodynamic laws that impose constraints on their possible values and rates of change [10].

However, deep learning models often fail to properly account for meteorology-specific patterns of different meteorological variables such as the strict physical constraints and boundary conditions inherent in weather systems, potentially leading to imprecise or unreliable predictions. Meanwhile, due to the complexity and dynamic nature of these physical characteristics, which exhibit subtle or significant variations across different time periods and geographical regions [11], predefining these relationships in a static way would inevitably affect prediction accuracy and generalization capability. Furthermore, many advanced deep learning models employ complex architectures with massive parameter counts, resulting in high computational resource requirements. These issues limit their practical deployment in real-world timely weather forecasting systems where both accuracy and computational efficiency are critical [12].

To address these challenges, we propose a novel deep learning plug-and-play framework named Post Constraint and Correction (PCC). PCC explicitly incorporates data characteristics that emerge from complex physical processes into the normal prediction process, including numerical constraints and variable relationships, thereby contributing to prediction reliability. Our approach introduces two key sub-modules: (1) the Multi-variants Correlation Constraint (MCC) module, which captures and maintains the complex interdependencies between different meteorological variables, and (2) the State Correction (SC) module, which ensures the reasonability of predictions by incorporating additional correction terms into the final prediction. By introducing network modules that can adaptively learn the empirical relationships between meteorological variables, deep learning models can effectively capture the underlying dynamics without requiring precise definitions of basic laws.

Notably, our method adopts a linear architectural design that achieves significantly efficient computational complexity and parameter scale while delivering notable accuracy improvements, making it suitable for practical weather forecasting. The code implementation is available at a GitHub repository: https://github.com/Fubukipara/PCCforWeatherTS/ (accessed on 26 March 2025).

The main contributions of our work are threefold:

1.

We develop a plug-and-play deep learning module named PCC that incorporates variable relationships and maintains state reasonability for weather multivariate time series forecasting, enabling seamless integration with various forecasting models without requiring architectural modifications or additional preprocessing.

2.

We design a computationally efficient architecture that significantly improves backbone model performance on weather time series forecasting tasks with minimal additional computational overhead, allowing the enhanced model to achieve superior performance with a significantly reduced parameter count.

3.

We conduct comprehensive experiments, ablation studies, and visualizations to demonstrate and analyze the superiority of our method.

2. Related Work

2.1. Weather Multivariate Time Series Forecasting

Multivariate time series data collected from weather stations serve as a fundamental cornerstone in modern weather forecasting systems. These data are characterized by their high dimensionality and complex temporal dependencies [13]. Traditional weather forecasting methods based on multivariate time series data primarily rely on various statistical approaches, including Autoregressive Integrated Moving Average (ARIMA) models, which assume linear relationships between variables, along with extensions such as SARIMA [14] for seasonal data. Other techniques have also been widely used, such as exponential smoothing [15] and Kalman filtering. Although these methods are interpretable and easy to implement, they often struggle to capture the complex and nonlinear relationships inherent in meteorological data, which may lead to unreliable predictions.

Driven by improvements in deep learning algorithms along with increases in the scale of available weather data and parallel computational hardware resources, recent advances in deep learning have revolutionized the field of weather forecasting and enabled the training of larger-scale models [16]. These new models are capable of effectively capturing complex and nonlinear patterns and dependencies within meteorological data in an end-to-end manner while eliminating the need for time-consuming manual feature engineering [17]. Thus, interest in deep learning models for NWP tasks there has been developing rapidly. For example, multiple deep learning methods such as Pangu-Weather [18] and FengWu [19] based on global-scale meteorological data in mesh space have shown the ability to outperform traditional physical models in terms of both prediction performance and efficiency.

For multivariate time series point data collected from weather stations, Recurrent Neural Networks (RNNs) and variants such as Gated Recurrent Unit (GRU) and Long Short-Term Memory (LSTM) networks have demonstrated remarkable success in prediction and analyzing tasks [20]. The stacked LSTM (stackLSTM) architecture stands out for its exceptional capacity to capture complex temporal dynamics at various time scales in weather time series by leveraging multiple LSTM layers [21,22,23]. Recently, several studies have explored the potential of transformers in the context of weather multivariate time series [24], leveraging the strength of transformers in capturing complex or long-term dependencies in sequential data to offer a promising alternative to traditional RNN-based approaches.

2.2. Deep Learning-Based Time Series Forecasting

Time series forecasting encompasses a broad spectrum of applications, ranging from financial markets [25] and traffic [26] to weather prediction. While traditional statistical methods such as ARIMA and VAR remain valuable for their interpretability, deep learning approaches have emerged as the dominant paradigm, offering superior performance in handling complex temporal and spatial patterns [27].

The evolution of deep learning models for time series analysis has been marked by significant architectural innovations. RNNs have been the primary choice for handling sequential data for the past several years, including NLP [28] and time series analysis [29]. Improved RNN architectures such as GRU [30] and LSTM [31] have addressed the fundamental limitations of vanilla RNNs in capturing long-range dependencies. These architectures have demonstrated remarkable successes [32], including in weather-related time series applications. However, there exists a significant computational time assumption due to the autoregressive structure of RNNs, and they also suffer from the gradient vanishing problem. To address these issues, SegRNN introduces parallel prediction mechanisms [33], enabling the model to predict multiple time steps in parallel. This approach improves both forecasting accuracy and computational efficiency.

Nowadays, the transformer architecture has gained significant attention in long sequence modeling [34] thanks to its multi-head self-attention mechanism, which has demonstrated remarkable ability to capture long-term dependencies and avoid the gradient vanishing problem that arises in RNNs. Consequently, transformers have shown great promise in sequential tasks such as time series forecasting. Although the attention mechanism demonstrates superior performance in long-term temporal modeling, it has drawbacks in terms of model complexity and computational cost [35]. Indeed, numerous methods have been proposed to address these issues, such as treating split patches or whole sequences as tokens in PatchTST [36] and iTransformer [37]. Another research line focuses on utilizing intricate temporal patterns within time series by leveraging techniques such as seasonal trend decomposition [38] and nonstationary compensation [39] to improve forecasting accuracy.

Recent developments in time series forecasting have also explored alternative architectures. MLP-based models such as DLinear [40] offer simple yet effective solutions for time series prediction, achieving comparable performance while requiring significantly fewer computational resources.

However, the existing deep learning-based time series forecasting models still lack the ability to capture the specific physical features of weather time series, including the strict values constraints of meteorological variables and their interdependencies, which may lead to suboptimal performance on weather time series prediction tasks. Therefore, in this paper we propose a generic deep learning plug-and-play module to help deep learning-based time series forecasting models to address these challenges effectively and efficiently.

3. Method

To deal with the specific features of weather time series which make deep learning-based time series forecasting models less effective, we propose a novel deep learning plug-and-play module called Post Constraint and Correction (PCC). The PCC module is designed to help the forecasting model capture more appropriate features of weather time series, resulting in more reliable predictions. As we demonstrate in Figure 1, the PCC module consists of two parts: a Multi-variants Correlation Constraint (MCC) module and a State Correction (SC) module. We introduce each part in detail in the following sections. We define the observed weather time series as $X\in {\mathbb{R}}^{O\times N}$, where O is the length of observed series and N is the number of variants. The ground truth and future prediction are denoted as $\overline{Y}\in {\mathbb{R}}^{P\times N}$ and $Y\in {\mathbb{R}}^{P\times N}$, respectively, where P is the prediction horizon (the number of future time steps predicted by the model).

3.1. Initial Prediction

We first feed the observed weather time series X to the backbone model (the underlying time series forecasting model, such as LSTM, that serves as the foundation for forecasting). Then, the backbone model generates the initial prediction $Y_i$:

$$Y_i=Backbone\left(X\right)$$

(1)

where $Y_i\in {\mathbb{R}}^{P\times N}$ is the initial prediction of the future series. The backbone model can be any general time series forecasting model, such as an RNN, transformer, or MLP model. Our PCC module is designed to be plug-and-play, which means that there is no need to change backbone model’s architecture, data normalization, training trajectory, etc. It is only necessary to deploy the PCC module on the forecasting series’ channel dimension after the backbone model.

3.2. Multi-Variants Correlation Constraint

There is a distinct correlation among different meteorological variables (e.g., temperature, precipitation) that play important roles in climate change [41]. Thus, we consider that offering additional insights on these relationships may help the model to make more accurate predictions.

However, unlike some other multivariate time series data, such as traffic flow and electricity demand, the correlation patterns among meteorological variables are primarily or directly based on physical properties [42] which are strict, specific, stable, and hard to modify or change across the time. However, some methods focus on whole sequences or patches of different channels. For instance, iTransformer considers every channel’s sequence as a token, which may introduce irrelevant temporal variances and lead to less effective capture of these temporally independent physical correlation patterns.

To address the above issues, we design the Multi-variants Correlation Constraint (MCC) module, which is deployed after the backbone model on the channel dimension of the forecasting series. The detailed architecture of the MCC’s network is shown in Figure 2.

After the backbone generates the initial prediction $Y_i$, we first define the last observed time step $X_O$ as the reference point and representation of the historical states, then calculate the bias between the initial prediction $Y_i$ and $X_O$:

$$B=Y_i-X_O$$

(2)

where $B\in {\mathbb{R}}^{P\times N}$ is the bias matrix. The MCC module leverages the bias matrix B as a physically meaningful representation of how each variable evolves from the reference state. By using the change relative to a reference point ($X_O$) rather than absolute values, the module is able to effectively capture the temporal dynamics of weather evolution.

Then, we deploy a fully-connected layer $f_{mp}$ and a nonlinear activation function $\sigma$ on the channel dimension of B to project the bias of every time steps into the hidden space:

$$h_{mc}=\sigma \left(f_{mp}\left(B\right)\right)$$

(3)

where $h_{mc}\in {\mathbb{R}}^{P\times H}$ is the hidden representation of the variable correlation and H is the predefined dimension of the hidden space. We deploy a dropout layer after the hidden representation $h_{mc}$ to avoid overfitting [43]:

$$h_{mc}=Dropout\left(h_{mc}\right).$$

(4)

The fully connected layer $f_{mp}$ learns the complex interdependencies between changes in different meteorological variables through its learnable weights. For example, it can model how changes in temperature relate to corresponding changes in humidity, air pressure, and other variables. The nonlinear activation function $\sigma$ enables modeling of the nonlinear patterns inherent in complex and dynamic meteorological systems.

Then, another fully connected layer $f_{mc}$ is introduced to generate the constraint term $C_{mc}$:

$$C_{mc}=f_{mc}\left(h_{mc}\right).$$

(5)

Next, we add the constraint term $C_{mc}$ to the initial prediction’s bias B to generate the constrained bias $B_{mc}$:

$$B_{mc}=B+C_{mc}.$$

(6)

Finally, we convert the constrained bias $B_{mc}$ back to a time series by adding the reference point $X_O$:

$$Y_{mc}=B_{mc}+X_O.$$

(7)

During the process above, the variable correlation constraint term $C_{mc}$ is inferred by the MLP through the bias matrix B separately and independently at every time step, thereby avoiding the irrelevant temporal variances and leading to more precise capture of the variable correlation patterns. Unlike fixed correlation rules, the MCC module can dynamically adjust the relationships between meteorological variables through the learnable weights of the linear layers, thereby enhancing the accuracy and reliability of the model’s predictions.

During the training stage, the MCC module can effectively learn the mapping between the bias matrix and the constraint term through a loss-based gradient propagation mechanism:

$${\theta }_{MCC}\leftarrow {\theta }_{MCC}-\eta {\displaystyle \frac{\partial \mathcal{L}}{\partial {\theta }_{MCC}}}$$

(8)

where $\eta$ is the learning rate, $\mathcal{L}$ is the loss function, and ${\theta }_{MCC}$ represents the parameters of the MCC module. The gradient calculation follows the chain rule:

$${\displaystyle \frac{\partial \mathcal{L}}{\partial {\theta }_{MCC}}}={\displaystyle \frac{\partial \mathcal{L}}{\partial Y}}·{\displaystyle \frac{\partial Y}{\partial Y_{mc}}}·{\displaystyle \frac{\partial Y_{mc}}{\partial C_{mc}}}·{\displaystyle \frac{\partial C_{mc}}{\partial {\theta }_{MCC}}}.$$

(9)

This update forces the MCC module to learn parameters that generate a $C_{mc}$ term, which minimizes the overall prediction error when applied to the initial prediction and bias matrix. Because meteorological variables exhibit intrinsic physical relationships in the training data, the module naturally learns these relationships through the optimization process without requiring explicit formulation of physics equations.

3.3. State Correction

The range of each variable is strictly limited based on the meteorological variables’ physical characteristics; for instance, the humidity should be within [0,100]. These ranges are also influenced by the location of the observation station; for example, the temperature at the North Pole should be lower than at the equator. Furthermore, certain variables are interdependent; when precipitation occurs, the humidity is typically high, and high wind velocities often occur together with lower temperatures.

These characteristics lead to strict constraints on the ranges of variable values, and result in a high risk of unreasonable states occurring during training and forecasting due to the randomness of model parameter optimizations, such as high precipitation and low humidity at the same time. Therefore, we propose that providing extra monitoring of the predicted states and correcting the unreasonable ones can lead to more reliable prediction results and more robust models.

After obtaining the constrained prediction $Y_{mc}$, we project $Y_{mc}$ from the temporal domain to the hidden space by a fully connected layer $f_{sp}$ and a nonlinear activation function $\sigma$:

$$h_s=\sigma \left(f_{sp}\left(Y_{mc}\right)\right)$$

(10)

where $h_s\in {\mathbb{R}}^H$ is the representation of each state and H is the predefined dimension of the hidden space. To avoid overfitting and make the model more robust, we introduce a dropout layer after the hidden representation $h_s$, as previously deployed in the MCC module network:

$$h_s=Dropout\left(h_s\right).$$

(11)

The SC module focuses on ensuring that each predicted state adheres to the physical constraints of the meteorological variables. Unlike the MCC module, which operates based on changes in the variables, the SC module directly examines each predicted state at each time step independently. This direct projection of each time step’s state into the hidden space allows the model to assess whether that state is reasonable irrespective of how it was reached.

To enhance the MLP’s ability to capture the state’s subtle features among the large scale of meteorological states, we introduce a differential operation on representation $h_s$ inspired by the differential transformer [44], as demonstrated in Figure 3. This differential operation $diff$ can be calculated as follows:

$$diff\left(h_s\right)=exp\left({\lambda }_1\right)\odot h_s-exp\left({\lambda }_2\right)\odot h_s$$

(12)

where ${\lambda }_1,{\lambda }_2\in {\mathbb{R}}^{P\times H}$ are learnable matrices and ⊙ denotes element-wise multiplication.

Similar to a differential amplifier [45], this differential operation acts as a feature enhancement mechanism, helping to distinguish important patterns from noise in the hidden representation. By applying different learnable weights ($exp\left({\lambda }_1\right)$ and $exp\left({\lambda }_2\right)$) to the same hidden representation $h_s$, the operation can selectively amplify subtle but important features while suppressing common noise patterns.

Then, we deploy another fully connected layer $f_{sc}$ to judge the state and generate a correction term $C_{sc}$:

$$C_{sc}=f_{sc}\left(h_s\right).$$

(13)

Finally, we add the correction term $C_{sc}$ to the constrained prediction $Y_{mc}$ in order to adjust the states at each time step and generate the final prediction Y:

$$Y=Y_{mc}+C_{sc}.$$

(14)

Similar to the MCC module, the SC module learns through the gradient propagation mechanism during training. The gradient flow for the SC module is as follows:

$${\theta }_{SC}\leftarrow {\theta }_{SC}-\eta {\displaystyle \frac{\partial \mathcal{L}}{\partial {\theta }_{SC}}}$$

(15)

where ${\theta }_{SC}$ represents the learnable parameters in the SC module, including the differential operation parameters ${\lambda }_1$ and ${\lambda }_2$. The gradient of the loss with respect to these parameters is

$${\displaystyle \frac{\partial \mathcal{L}}{\partial {\theta }_{SC}}}={\displaystyle \frac{\partial \mathcal{L}}{\partial Y}}·{\displaystyle \frac{\partial Y}{\partial C_{sc}}}·{\displaystyle \frac{\partial C_{sc}}{\partial {\theta }_{SC}}}.$$

(16)

Through our Equation (14), we have ${\displaystyle \frac{\partial Y}{\partial C_{sc}}}=1$, which provides a direct gradient path from the loss to the SC module parameters. This direct gradient flow allows the SC module to efficiently learn which aspects of individual meteorological states require correction.

Therefore, the PCC module’s dual-component design addresses different challenges caused by physical characteristics in weather time series forecasting. The MCC component ensures that the relationships between variables stay reasonable during the prediction process. It operates on the correlation patterns, ensuring that when one variable changes, the related variables change in correct ways. Meanwhile, the SC component ensures that the individual states at each time step remain physically plausible regardless of how they were evolved, acting as a correction mechanism that identifies and adjusts unrealistic states.

Together, these components complement each other, enabling the model to learn the variable constraints from the data without requiring explicit physical equations or laws. This makes the resulting model more adaptable and robust across varying scenarios, such as different climate regimes.

4. Experiments

4.1. Experiment Materials and Setup

4.1.1. Dataset

We conducted experiments on the weather time series dataset collected from the Max Planck Institute for Biogeochemistry, Jena, Germany in tabular format. The datasets contained 21 different meteorological variables, including temperature, humidity, and wind speed, as listed in the Appendix A. The variables were recorded every 10 min over the whole year of 2020. Using a ratio of 7:1:2, we split the dataset into a training set for training the model, a validation set for tuning the hyperparameters, and a test set for evaluating the model’s performance after training.

As demonstrated in Figure 4, some of the meteorological variables exhibit unstable and nonstationary properties with noise and fluctuations, making it challenging for general forecasting models to capture useful features precisely and resulting in less effective performance on forecasting tasks.

4.1.2. Backbone Models

For our experimental evaluation, we selected three different types of time series models as the backbone models for our PCC method to serve as the baselines for the comparison. The selected backbone models have shown the state-of-the-art performance among their categories in multivariate time series forecasting tasks. The chosen models included an RNN-based model (SegRNN), two transformer-based models (iTransformer and PatchTST), and an MLP-based model (DLinear). In addition to these advanced general models, we also selected stackLSTM as a backbone model, as it has been extensively utilized in numerous weather time series research studies to model temporal variations and has demonstrated versatile performance in this regard. The detailed settings of the backbone models are listed in Appendix D.

4.1.3. Training and Evaluation Setup

During the training process, we set the default length of the observed series O to 96 (equivalent to 16 h) and set the length of the future series P as $\{96,192,336,720\}$, which means predicting the weather for the next 16, 32, 56, and 120 h.

Although our PCC module is simple and lightweight compared to the backbone models, it introduces additional complexity as a supplementary component, which may introduce additional risk of unstable training. We carefully selected the hyperparameters of the PCC module as follows:

1.

The hidden dimension for the two submodule networks was set as 128, balancing performance and computational complexity.

2.

Tanh was chosen as the activation function $\sigma$ due to its symmetry around zero and smooth gradient properties. These characteristics ensure training stability, which is particularly important because weather variables often contain extreme values and rapid fluctuations.

3.

The dropout rate was set to 0.3, representing an optimal tradeoff between regularization to prevent overfitting and maintaining sufficient network capacity.

4.

The learnable matrices ${\lambda }_1,{\lambda }_2$ were both initialized to 1, ensuring that the differential terms start with equivalent contribution weights. This helps to prevent initial bias that may cause unstable parameter updates during the early training stages.

We evaluated PCC’s sensitivity to the settings of the hidden dimension, dropout rate, and initialization of the differential matrix in Appendix C.

Our PCC module functions as a plug-and-play component that can be integrated into backbone models by simply connecting it at the feature dimension. This integration requires no modifications to the backbone architecture itself nor additional data preprocessing steps such as normalization. Therefore, in our experiments we followed the same settings for each model as in their original publications. In addition, we used the official code implementations, including data processing methods, normalization methods, hyperparameters, learning rate, etc., ensuring fairness of the comparison and reproducibility of the results. The detailed training configurations of all the backbone models are provided in Appendix D.

We evaluated the models’ performance on the test set using the metrics of mean absolute error (MAE) and mean squared error (MSE). To ensure reliability, each numerical result was the average of five runs with different random seeds.

The experiments were conducted on a server with an Intel Xeon Platinum 8474C CPU, 80 GB memory, and an NVIDIA RTX 4090D GPU with 24 GB memory. The PCC module and backbone models were implemented with PyTorch 2.0.0 and CUDA 11.8 with Python 3.8 on Ubuntu 20.04.

4.2. Main Results and Discussion

4.2.1. Main Forecasting Results

As shown in

Table 1. MSE and MAE of forecasting results for different backbones with and without PCC across different forecasting horizons ({96,192,336,720}) with a consistent look-back length of 96. Lower value of MAE and MSE indicate better forecasting performance, and are indicated in bold.

Model	Design	96		192		336		720		Average
		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
stackLSTM	Original	0.298	0.318	0.387	0.373	0.501	0.459	0.548	0.495	0.434	0.411
	+PCC	0.221	0.266	0.242	0.289	0.285	0.314	0.369	0.376	0.279	0.311
DLinear	Original	0.207	0.233	0.244	0.269	0.286	0.306	0.345	0.354	0.271	0.290
	+PCC	0.152	0.197	0.196	0.239	0.243	0.276	0.304	0.318	0.224	0.258
iTransformer	Original	0.175	0.215	0.225	0.258	0.281	0.299	0.358	0.350	0.260	0.281
	+PCC	0.159	0.204	0.208	0.248	0.267	0.291	0.347	0.344	0.245	0.272
PatchTST	Original	0.175	0.216	0.221	0.257	0.280	0.298	0.352	0.347	0.257	0.280
	+PCC	0.157	0.205	0.207	0.249	0.270	0.293	0.345	0.344	0.245	0.273
SegRNN	Original	0.162	0.200	0.208	0.243	0.264	0.285	0.347	0.340	0.245	0.267
	+PCC	0.144	0.189	0.189	0.233	0.238	0.273	0.299	0.317	0.218	0.253

, the PCC module can effectively improve the performance of all backbone models on weather time series forecasting tasks across different prediction horizons. The models augmented with the PCC module all achieve significantly lower MAE and MSE than the original models. Notably, our PCC respectively reduces the MSE of the DLinear and SegRNN models by 17.34% and 11.02% on average. The improvement is particularly pronounced for shorter prediction horizons, with a 26.57% reduction in MSE for DLinear and 11.11% reduction for SegRNN with a prediction horizon of 96. The detailed error bars, which include the standard deviation and the confidence interval, are listed in Appendix B.

4.2.2. Different Length of Observed Time Series

For a more comprehensive evaluation,

Table 2. MSE of forecasting results for different backbones with and without PCC across different lengths of observed time series ({48,96,192,336,720}) with a consistent prediction horizon of 96.

Backbone	Design	Observation Length
		48	96	192	336	720
stackLSTM	Original	0.326	0.298	0.326	0.322	0.349
	+PCC	0.236	0.221	0.213	0.195	0.228
DLinear	Original	0.230	0.207	0.190	0.177	0.168
	+PCC	0.188	0.152	0.145	0.143	0.142
iTransformer	Original	0.202	0.175	0.170	0.162	0.175
	+PCC	0.181	0.159	0.154	0.151	0.159
PatchTST	Original	0.211	0.175	0.159	0.150	0.147
	+PCC	0.188	0.157	0.150	0.146	0.145
SegRNN	Original	0.203	0.162	0.150	0.146	0.142
	+PCC	0.169	0.144	0.139	0.138	0.138

Bold values indicate better results.

lists the performance of the PCC module with different lengths of the observed time series. As the table shows, the PCC module improves the performance of the different backbone models across varying observation lengths. The improvements are more significant in shorter observed time series. This indicates that PCC can help to reduce the model’s dependence on the length of the observed series. For instance, the DLinear model with PCC achieves better performance based on 8 h of observed time series than the original model based on 32 h of observation. Thus, our PCC module can help the model to capture the underlying patterns more effectively in the data scarcity scenario, leading to more robust and reliable results on the weather forecasting task with less data storage.

4.2.3. Key Variables Analysis

We also conducted a key variables analysis to demonstrate the effectiveness of the PCC module with different crucial meteorological variables, including air temperature, specific humidity, wind velocity, precipitation, etc. These variables are widely utilized in weather forecasting tasks, have shown significant impacts on climate change, and correlate closely with real-world applications such as agriculture, transportation, etc. Importantly, they also exhibit significant correlation patterns between each other and have specific ranges of values. A

Table 3. Prediction results of key variables using DLinear as the backbone. Results are reported as MSE and MAE, including predictions with and without PCC. The forecasting range and look-back length are both 96.

Variable	Air Temperature		Specific Humidity		Wind Velocity		Precipitation
	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Original	0.099	0.223	0.097	0.212	0.001	0.021	0.067	0.055
+PCC	0.082	0.218	0.062	0.173	0.001	0.018	0.052	0.035

Bold values indicate better results.

demonstrates, the PCC module can improve the performance of different backbone models with respect to these different key meteorological variables.

4.2.4. Impact of Strongly Correlated Variables

One potential concern with the dataset used in our experiments is the presence of strongly correlated variables, particularly those related to humidity measurements (e.g., Tdew, rh, VPmax) and some temperature-related variables (e.g., T, Tlog). These variables describe similar physical properties but do so in different mathematical representations, which might introduce unwanted biases or redundancies in the model.

To investigate this concern, we conduct additional experiments using a reduced variable set that excluded these potentially redundant features. Specifically, we removed the following variables: Tdew (dew point temperature), VPmax (saturation water vapor pressure), VPact (actual water vapor pressure), VPdef (water vapor pressure deficit), sh (specific humidity), H2OC (water vapor concentration), Tpot (potential temperature), and Tlog (temperature in log).

Table 4. Performance comparison (MSE) of models with the full variable set versus the reduced variable set. The forecasting and look-back length are both 96.

Backbone	Design	Reduced Variable Set		Full Variable Set
		MSE	MAE	MSE	MAE
SegRNN	Original	0.219	0.208	0.162	0.200
	+PCC	0.196	0.195	0.144	0.189
DLinear	Original	0.284	0.253	0.207	0.233
	+PCC	0.207	0.207	0.152	0.197

Bold values indicate the best results.

presents a comparison of model performance with the full variable set and the reduced variable set. Contrary to what might be expected, the models trained on the full variable set consistently outperformed those trained on the reduced set regardless of whether PCC was applied. Moreover, PCC maintained its performance improvement ability even with the reduced variable set.

These results suggest that while these particular variables are mathematically related or describe similar weather conditions, they also contribute unique and valuable information that is relevant to the forecasting task. The relationships between these variables appear to provide useful signals rather than misleading correlations. Furthermore, our PCC module effectively leverages these relationships rather than being hindered by them, as evidenced by its consistent performance improvement across both variable sets.

These findings highlight the robustness and generalization capability of our approach even when processing highly correlated input features, which is a common characteristic in meteorological datasets.

4.3. Method Analysis Results and Discussion

4.3.1. Ablation Study

To verify the effectiveness of each design of the PCC module, we conducted an ablation study. We removed or replaced each part of the PCC module, including removal of the two submodules, differential representation inside the state correction network, and inverting the order of the two submodules. The experiments used SegRNN as the backbone model with a look-back window of 96 and prediction horizon of 96. As demonstrated in

Table 5. Ablation results (MSE) of different PCC designs. The ablation experiments used SegRNN as the backbone model, with a look-back window of 96 and prediction horizon of 96. Here, ✓ denotes that the design is used and unchanged, w/o means that it is removed, and ‘Invert’ means that the order of the two modules is inverted. ‘FC-only’ means that the design was replaced with one fully connected layer, while ‘Diff-Rep’ denotes differential representation.

Design	Operation	Forecasting Length
		96	192	336	720
MCC	w/o	0.149	0.193	0.241	0.302
	FC-only	0.145	0.190	0.239	0.301
SC	w/o	0.146	0.196	0.255	0.332
	FC-only	0.148	0.192	0.245	0.312
Diff-Rep	w/o	0.146	0.192	0.243	0.307
PCC	w/o	0.162	0.208	0.264	0.340
	Invert	0.147	0.192	0.244	0.309
	✓	0.144	0.189	0.238	0.299

Bold values indicate the best results.

, the PCC module with default settings achieves the best performance. Removing the State Correction module leads to the most significant performance degradation, particularly for longer prediction horizons. This indicates that the SC module plays a crucial role in stabilizing predictions and achieving improvement, given the distinct unstable fluctuations and strict physical value constraints among the weather time series.

4.3.2. Scalability Analysis

To evaluate the scalability of our approach, we conducted experiments comparing the parameter efficiency between the original backbone model and our PCC-enhanced version. We used SegRNN as the backbone model and varied its parameter scale through different hidden dimensions: {64, 128, 256, 512, 1024} for the original design and {16, 32, 64} for the PCC-enhanced design, both based on the same observation and prediction length of 96. According to the fitting curves of the results in Figure 5, the PCC-enhanced SegRNN has better accuracy with learnable parameters of less than about $1\%$ of the original SegRNN’s parameters. This indicates that our PCC helps the original forecasting models to more effectively capture crucial meteorological patterns and achieve higher accuracy with fewer parameters. This reduction in model size directly translates to lower computational demands and deployment costs on hardware devices.

4.3.3. Training Cost and Stability

To more practically and comprehensively validate our method’s efficiency and resource utilization, we present the training time and memory consumption of our method in

Table 6. Comparison of training time and maximum memory consumption of the SegRNN backbone with and without PCC. The look-back length is 96, while the prediction horizons are 96 and 720. We set the number of training epochs as 30 and the patience as 10.

Design	MSE		Time (s)		Memory (MB)
	96	720	96	720	96	720
Original	0.162	0.347	176	227	204.55	865.39
+PCC	0.144	0.299	214	219	204.47	868.00

, using SegRNN as backbone model with the same look-back and forecasting horizon of 96. The results show that the performance improvement requires only modest and acceptable increases in computational overhead. The shorter training time with the 720 horizon indicates that despite PCC introducing additional computational complexity, it can help the model to achieve more stable training and faster convergence when the forecasting horizon is longer and more likely to be unstable.

Notably, the memory consumption remains nearly identical (less than 0.5% difference) when the prediction horizon is 720. Counter-intuitively, the maximum memory consumption with PCC is slightly lower than the original SegRNN when the prediction horizon is 96. We attribute this to the additional influence of the PCC module on the GPU memory allocation optimization strategies of PyTorch, as the additional memory consumption introduced by PCC is too small to be a significant factor in the total memory consumption.

To evaluate the training process, Figure 6 illustrates the temporal evolution of the training and validation losses for the SegRNN backbone with and without PCC integration. The experiments were conducted with a fixed look-back window of 96 and forecasting lengths of 96 and 720. The training loss trajectories (depicted by blue lines) for both model variants exhibit stability and convergence in the latter stages of training, demonstrating that incorporation of the PCC module maintains training stability while enhancing predictive performance.

4.3.4. Comparison with Complex Mechanisms

The computational complexity of our PCC method primarily stems from the simple fully connected layers and element-wise tensor manipulations in the differential operations. Both of these can be accelerated by parallel computing on GPUs, resulting in lower computational overhead compared to more complex structures such as self-attention mechanisms. We conducted a comparison of performance and computational cost between the proposed PCC module and some complex mechanisms, again using SegRNN as the backbone model.

Table 7. Performance comparison between PCC and more complex mechanisms in terms of MSE, inference time, and number of parameters. The backbone is SegRNN with a look-back length of 96 and prediction horizons of 96 and 720.

Design	MSE		Time (ms)		Parameters (k)
	96	720	96	720	96	720
SAMP + SC	0.152	0.304	0.325	0.401	86.5	246.3
LIFT + SC	0.149	0.303	0.348	0.455	1694.7	2738.2
saPCC	0.147	0.301	0.311	0.411	112.5	3173.9
PCC	0.144	0.299	0.297	0.375	35.6	195.4

Bold values indicate the best performance.

shows the results. The methods for comparison included the following:

• SAMP + SC: Replaces the MCC module with SAMP [46], a self-attention based method for capturing variable correlation in time series.
• LIFT + SC: Replaces the MCC module with LIFT [47], a channel dependence correction method for time series forecasting using linear structures with complex calculations such as Fourier transforms.
• saPCC: Replaces the MLP structure in both PCC submodules with a multihead self-attention mechanism.

The implementation details of these three methods are described in Appendix E.

The results show that PCC achieves superior performance with its simple linear structure while maintaining significantly lower computational complexity, making it more suitable for resource-sensitive practical scenarios. We attribute this to the temporal independence in the design of the MCC module, which avoids the negative effects of temporal variance, making it more effective in capturing physical correlation patterns.

4.3.5. Robustness Analysis

As deep learning models are data-driven methods, it is well known that they are sensitive to data quality. Noisy data may introduce negative impacts on model training and lead to poor performance [48]. Unfortunately, time series data collected from weather stations are sometimes noisy [49]. In this section, we evaluate the robustness of our PCC module under various noise conditions to validate its effectiveness in practical forecasting scenarios.

First, we introduce Additive Gaussian Noise (AGN), then scale the AGN by a predefined weight and add it to the dataset. We evaluated two training cases, first adding noise only to the input sequence and then adding it to the complete sequence. We used the SegRNN model as the backbone, with the same look-back window and prediction horizon of 96. The scale weights of the AGN were 0.1 and 0.3. The results listed in

Table 8. Results of the PCC module under different noise conditions. The backbone is SegRNN with a look-back length of 96 and prediction horizons of 96 and 720.

Noisy Input	0.1		0.3
	MSE	MAE	MSE	MAE
SegRNN	0.167	0.211	0.176	0.228
SegRNN + PCC	0.149	0.198	0.159	0.210
Noisy Sequence	0.1		0.3
	MSE	MAE	MSE	MAE
SegRNN	0.178	0.240	0.267	0.357
SegRNN + PCC	0.161	0.228	0.250	0.346

Bold values (except the column headers “0.1” and “0.3”) indicate the better results.

show the robustness of our PCC module against noise and demonstrate its ability to effectively capture the underlying patterns in noisy scenarios while improving the backbone model’s performance.

4.3.6. Visualization of the Two Submodules in PCC

Figure 7 visualizes the weights and behavior of the Multi-variants Correlation Constraint (MCC) and State Correction (SC) submodules using heatmaps. We used the DLinear model as the backbone, with the same look-back and prediction length of 96.

To visualize the submodules’ operational behavior while mitigating the negative impact of extreme or zero values on the visualization, we calculated the mean squared roots of the ratios between the submodules’ absolute output and input values instead of using the ratios directly.

The visualizations of the weights and ratio metrics demonstrate that the two submodules have different preferences for the time steps and variables. The MCC module is more active in the early prediction stages, which are closer to the reference point, as well as for those variables with strong correlation patterns, such as humidity, wind velocity, and precipitation. The SC module places more emphasis on the later stages, following the regular pattern that the more distant points are more likely to be unstable during model prediction. In addition, it focuses more on variables which are more likely to be unstable or have more complex and significant variance patterns, such as temperature, water vapor pressure, wind velocity and direction, and photosynthetically active radiation. After the initial period, the activity of both modules becomes stable and time-insensitive.

4.3.7. Visualization of Differential Operation

To provide more intuitive insight into the SC module’s effectiveness, we directly visualize the differential operation’s learnable matrices ${\lambda }_1$ and ${\lambda }_2$ in the state correction module. First, we reparameterize these two matrices as

$${\lambda }_{diff}=exp\left({\lambda }_1\right)-exp\left({\lambda }_2\right),$$

(17)

where ${\lambda }_{diff}$ is the reparameterized differential operation matrix. Then, we visualize ${\lambda }_{diff}$ in Figure 8, demonstrating that the differential operation prefers to enhance those features that are farther away from the observation series.

5. Conclusions and Future Work

In this paper, we have presented a generic plug-and-play approach named Post Constraint and Correction (PCC) for improving deep learning-based multivariate weather time series forecasting by explicitly incorporating domain knowledge such as the physical characteristics of meteorological variables. Our method addresses two key challenges in weather time series prediction: (1) missing or improper alignment of the variables’ physical interdependencies, and (2) imprecise or unreasonable predictions caused by neural network randomness and the strict physical constraints of meteorological variables.

Extensive experimental results across different cases clearly demonstrate the effectiveness of our PCC approach in terms of improved accuracy and cost-efficiency. For instance, our method reduces the prediction MSE by up to 11% on average with the SegRNN backbone while requiring less than 0.5% additional GPU memory consumption. Our analysis of PCC’s scalability demonstrates that the proposed module enables backbone models to achieve better performance with a smaller number of parameters, making it particularly suitable for resource-constrained environments where computational efficiency is as critical as forecasting accuracy.

Additionally, our ablation studies validate the importance of each component in the proposed method along with the reasonableness of our design. Visualization analysis further reveals the interpretability of the module’s behavior, showing how it effectively learns and leverages the specific properties of meteorological time series to achieve improved model performance.

While our PCC module demonstrates significant performance improvements, we acknowledge interpretability limitations common to deep learning approaches in physical systems. Unlike traditional physics-based models, neural networks often function as ‘black boxes’, making it difficult to fully explain how they incorporate domain knowledge [50]. In our opinion, there is a promising approach to address this challenge by developing hybrid architectures that more explicitly incorporate physical relationships. For example, implementing gating mechanisms that selectively apply predefined physical constraints to the model outputs may offer an optimal balance between purely data-driven approaches and physics-constrained models. Such hybrid approaches can not only enhance interpretability but might further reduce model complexity while maintaining or improving prediction accuracy. We consider this a promising direction for our ongoing research, and plan to thoroughly investigate such mechanisms in subsequent work.