Abstract
Background
Influenza-like illness (ILI) serves as a representative climate dependent disease due to its climate sensitivity, widespread transmission, and significant impact on public health and societal functioning, highlighting the urgent need for reliable ILI prediction techniques that combine climate factors to support effective prevention and intervention. Existing compartmental model-based methods have limited ability to incorporate climate data and often struggle to capture the complex ILI transmission patterns. Many deep learning models generally lack interpretability and tend to overlook the interactions between climatic factors and ILI transmission. This study aims to develop a physics-guided framework that leverages the dynamical knowledge embedded in compartmental models and the multi-source data learning and representation capabilities of deep learning.
Methods
Our materials are the ILI case and climate data from Lanzhou, Gansu Province, China from 1st January 2023 to 26th October 2024 and Xi’an, Shaanxi Province, China from 1st January 2010 to 10th November 2016. Regarding the proposed model, first, we employ a physics-informed neural network (PINN) to learn the underlying dynamical knowledge under differential equation constraints and embed it into a long short-term memory network (LSTM) through a physics-guided representation alignment mechanism. Then, the transferred dynamical knowledge guides the LSTM to learn the mapping from climate and case data to ILI dynamics, enabling more accurate predictions jointly driven by domain knowledge and multi-source data.
Results
The results demonstrate that the proposed framework generally achieves better prediction accuracy and trend consistency than five advanced baselines. Specifically, compared with suboptimal baselines in Lanzhou, our model achieves reductions of at least 21.81%, 20.76%, 21.86%, and 19.44% in NRMSE, SMAPE, ND, and MAPE, respectively, while improving PCC by at least 14.96%. In addition, we derive explicit functional relationships between transmission rates and climate factors via symbolic regression, and verify the effectiveness and reliability of the model by statistical test and ablation study.
Conclusions
The study finds that physics-guided representation alignment mechanism not only offers the potential to enhance ILI prediction under varying climatic conditions but also provides new insights and methodological support for epidemic prediction.
Keywords: Climate dependent disease prediction, Climate factors, Compartmental model, Deep learning, Representation alignment
Background
Influenza is an acute respiratory infectious disease, and laboratory evidence has shown that climatic factors, particularly temperature and humidity, directly affect influenza virus activity [1, 2]. Moreover, influenza virus viability is enhanced under low-temperature and low-humidity conditions [3]. Meanwhile, influenza-like illness (ILI) exerts substantial impacts on multiple aspects of society, and the serious complications and deaths it may cause put a substantial disease burden on the public. Therefore, scholars have shown increasing interest in the relationship between climate change and ILI consequences. Predicting ILI-related health outcomes, such as infection cases and mortality, based on meteorological factors has emerged as a prominent research topic at the intersection of atmospheric science and biomathematics [4–10].
Compartmental models are built upon rational assumptions about transmission mechanisms and employ mathematical mechanisms to describe the dynamics of infectious diseases. Many studies have assumed that the transmission rate of ILI depends on climatic factors and have used compartmental models to investigate how climate change affects outcomes such as the severity, intervention intensity and duration of ILI. For example, in 2023, Servadio, Thai, Choisy et al. [11] linked climatic factors in tropical regions with influenza patterns and investigated whether influenza exhibits seasonality in tropical regions. In 2024, Shi, Wang, and Jin [12] examined the correlation between the periodic transmission rate of influenza and patterns of climatic indicators, while emphasizing the influence of climate trends on influenza prediction. In 2025, Ruan, Liang, Sun et al. [13] employed an SIRS model to investigate the regulatory effect of seasonal temperature increases on influenza patterns. Li, Liu and Qi [14] developed a metapopulation infectious disease model with seasonal fluctuations in climatic factors and applied it to study the seasonal outbreaks of influenza A viruses in temperate regions. These models describe and predict ILI dynamics based on mechanistic assumptions and capture certain characteristics of disease transmission to some extent. Meanwhile, climate variables have been incorporated into the above compartmental models to investigate their effects on ILI transmission dynamics. However, these models are primarily intended to improve the understanding of epidemiological mechanisms rather than to perform ILI prediction, which would limit their performance in prediction tasks. In addition, their construction and predictive performance rely heavily on model assumptions, making it difficult to capture the complex patterns of ILI and to generalize effectively to novel prediction tasks with heterogeneous external data.
Deep learning is a data-driven nonlinear mathematical tool with powerful learning capabilities and has demonstrated remarkable success in addressing a wide range of complex tasks. Considering that learning the characteristics of infectious diseases from data can help reveal the laws of transmission dynamics and the influence of real-world climatic factors, many scholars have developed climate data-driven deep learning frameworks and applied them to mathematical epidemiology. For example, in 2023, Amendolara, Sant, Rotstein [15] trained an LSTM model using ILI data and climatic factors, and it demonstrated excellent performance in short-term prediction tasks. In 2024, Zhu, Chen, Qin et al. [16] divided the ILI data into three periods(overall, non-epidemic, and epidemic) to analyze the effects of climatic factors on ILI across different periods, and employed an ensemble model integrating bi-directional LSTM and random forest for short-term prediction. In 2025, Li, Huang, Xie et al. [17] validated the relationship between climatic data and ILI incidence using Pearson correlation analysis, and introduced the Kernel Attention Network to achieve more accurate predictions of ILI incidence. Wang, Xiao and Song [7] employed deep neural networks to learn the incidence rate and, by integrating symbolic regression and sparse identification of nonlinear dynamics, explored optimal explicit form between incidence and climatic factors. Notably, they emphasized the significant nonlinear effects of climatic factors on ILI transmission dynamics, highlighting the need for caution when incorporating these factors into models. These models employ multi-source data to train general deep learning frameworks, thereby improving infectious disease prediction performance. However, they overlook the integration of climatic factors with mechanistic disease transmission modeling, resulting in inadequate characterization of infectious disease dynamics, limited interpretability, and challenges in providing reliable support for public health planning and response.
To balance accuracy, scalability, and interpretability in infectious disease prediction, integrating epidemiological knowledge with deep learning has emerged as a novel framework with transformative potential in epidemiology. The combination of deep learning and traditional compartmental models has advanced rapidly in recent years. For example, neural networks have been employed to estimate unknown components or parameters of compartmental models [18, 19], and Physics-Informed Neural Network (PINN) [20], which combines differential equations with neural networks, has been proposed and widely adopted [21, 22]. However, to our knowledge, despite their success in related tasks, these works remain limited in incorporating external multi-source data and capturing complex temporal dependencies. Thus, effectively integrating deep learning techniques with mechanistic models to achieve accurate and interpretable epidemic prediction remains a significant challenge.
To address this challenge, on the one hand, compartmental models contain dynamic knowledge that guides epidemic prediction, while neural networks possess data-driven representational power. Their combination enables neural networks to follow infectious disease dynamics during the learning process. Accordingly, we employ PINN to capture the potential epidemic dynamics, since PINN can embed physical knowledge through unsupervised loss functions derived from differential equations. On the other hand, the multilayer perceptron architecture of PINN tends to describe independent and uniformly distributed data, making it less effective at capturing complex temporal dependencies. It is also limited in modeling climatic variables that are not involved in differential equations. Accordingly, we utilize LSTM to learn the temporal patterns of external data, as its memory cells and gating mechanisms make it particularly suitable for capturing long-term and cross-modal temporal dependencies.
This study develops a physics-guided framework for ILI prediction, which integrates the capability of learning underlying epidemic dynamics with the ability of multi-source data-driven modeling. First, PINN is employed to learn interpretable dynamic knowledge under the physical constraints of differential equations. Then, the learned knowledge is transferred to LSTM, enabling LSTM to perform data-driven epidemic modeling and prediction based on multi-source input data.
Methods
Compartmental model for mechanism discovery
In this study, to simulate the transmission dynamics of ILI, we used the susceptible-infectious-recovered-susceptible (SIRS) [23] epidemic model. The SIRS model divides total population (N) into three compartments: Susceptible (S), Infectious (I), and Recovered (R).
To accurately and effectively simulate the transmission dynamics of ILI, we assume that all parameters are functions of time t, and the SIRS model equations are as follows.
 |
1 |
where N = S(t) + I(t) + R(t), S(t), I(t), and R(t) represent the numbers of susceptible, infectious, and recovered individuals at time t, respectively, while β (t), σ (t), and γ (t) represent the transmission rate, the rate of immunity loss, and the recovery rate at time t, respectively.
In addition, let 
denote the state variables at time t, and let 
denote the set of parameters at time t. By solving system (1), the underlying epidemic dynamics characterized by 
and 
can be inferred within the observation period 
.
PINN module for dynamical knowledge
PINN, as a data-driven neural network algorithm, can not only accurately fit data using compartmental models but also effectively estimate time-varying parameters. The PINN module is parameterized by a multilayer perceptron that takes time t as input, i.e., the neural network NN(t) is a function of the time variable t. It learns the underlying dynamical knowledge from the observed data and predicts the SIRS model state 
at time t. The SIRS model describes the rate of change of certain functions with respect to t, that is, the gradient, denoted as 
. Considering that the gradient 
can be computed via automatic differentiation, this enables PINN to solve the SIRS model by jointly minimizing the residual between the two gradients and the observation error.
To ensure that the PINN produces predictions consistent with the dynamics described by system (1), we denote the output of NN(t) as the state variable 
, and require it to satisfy the constraint 
. Meanwhile, when fitting the observational data, both the observation error loss 
and the SIRS loss 
are minimized. Furthermore, since the employed SIRS model is time-varying, we introduce an additional time-varying parameter update loss 
.
Since the state variables are governed by the continuous-time SIRS system, the physics-informed loss is constructed using differential residuals. In contrast, the time-varying parameter vector 
is estimated at discrete observation times rather than being governed by an explicit differential equation. Therefore, a first-order difference regularization term is introduced to encourage smooth temporal evolution of the parameters and avoid unrealistic fluctuations. The three loss functions are formulated as follows.
 |
2 |
 |
3 |
 |
4 |
where 
represents the number of ILI infections predicted by PINN at time t and 
represents the observed data of the number of ILI infections at time t.
It is worth noting that the time-varying parameter update loss 
essentially serves as a regularizer. By constraining the magnitude of parameter updates between adjacent time steps, it prevents sharp fluctuations in the parameter sequence, encourages the model to capture the gradual evolution of the ILI transmission process, and mitigates overfitting under substantial data volatility, thereby enhancing the robustness and interpretability of the model.
LSTM module for multi-source knowledge
LSTM is an improved recurrent neural network designed to model long-term dependencies in time-series data [24–27]. The LSTM module takes multi-source time-series feature vector 
related to ILI (Including temperature, humidity, and ILI infections cases) as input; that is, the neural network 
is a function of the input feature 
that varies over time t. It learns the evolutionary trends of the given external data and predicts the SIRS model state 
at time t.
We denote the output of 
as the state variable 
and require that it satisfies the SIRS dynamical constraint 
. Let 
and 
denote the latent embeddings generated by the PINN and LSTM, respectively. Since the LSTM receives multi-dimensional feature 
rather than time alone, the temporal derivative 
cannot be directly computed via automatic differentiation.
To address this issue, we introduce an embedding alignment mechanism and minimize the embedding approximation loss 
to encourage 
. Here, the embedding approximation assumption is inspired by representation alignment techniques commonly used in transfer learning. Assuming that latent representations update smoothly and continuously over time, neighboring embeddings are expected to exhibit similar temporal evolution patterns. Therefore, when the embedding alignment is sufficiently accurate, it is reasonable to approximate 
.
Based on the chain rule, the temporal derivative of the LSTM state can then be expressed as 
, where both 
and 
can be directly computed via automatic differentiation.
Meanwhile, we further define the corresponding embedding approximation loss 
. Similar to PINN, when fitting the observational data, we consider the observation error 
and minimize the unsupervised SIRS loss 
. The three loss functions are formulated as follows.
 |
5 |
 |
6 |
 |
7 |
where 
represents the number of ILI infections predicted by LSTM at time t.
It is worth noting that 
is expected to hold when the embedding alignment is accurate and the latent dynamics are smooth. By minimizing 
, the LSTM-generated embedding 
is encouraged to remain close to 
, which is learned under the SIRS constraints. Meanwhile, this approximation provides a practical mechanism for transferring the dynamical information learned by the PINN to the LSTM under multivariate external feature inputs. Since 
is learned under explicit SIRS constraints, it implicitly encodes physically consistent epidemic evolution patterns and time-varying transmission dynamics. Through latent representation alignment, the LSTM is encouraged to preserve the dynamical information learned by the PINN, thereby maintaining physical interpretability while enhancing predictive flexibility.
Physics-guided framework for final ILI prediction
Transfer learning aims to utilize the teacher model to extract knowledge from the source domain and transfer this knowledge to the target domain, thereby facilitating learning of the target task [28]. Inspired by transfer learning, PINN, LSTM, and ILI prediction are respectively regarded as the teacher model, student model, and target task within the proposed physics-guided framework in this study. Specifically, PINN is used to learn domain knowledge (i.e., potential ILI dynamics and time-varying parameters) from the source domain and transfer its learned representations to the target domain. LSTM is employed to accomplish the ILI prediction task in the target domain. By leveraging the knowledge transferred from PINN, it achieves both physical interpretability and high prediction accuracy while retaining a data-driven learning paradigm.
We choose LSTM as the student model because it is more suitable for handling multivariate time series data and can incorporate multi-source data that are not included in the SIRS model, thereby learning the mapping from 
to 
and realizing ILI prediction. Meanwhile, we perform knowledge transfer based on embedding approximate gradients and introduce the knowledge distillation loss 
[29] to enable LSTM to further learn the prediction curves and physical laws from PINN, as shown in the following formula:
 |
8 |
For the ILI prediction model proposed under the physics-guided framework, losses (2), (5), and (6) ensure that the model outputs remain close to the observed data, while losses (3) and (7) enforce consistency with differential equation constraints, loss (4) guarantees smoothness and rationality of the parameter sequence, and loss (8) further enhances model performance. Moreover, instead of simply concatenating the teacher and student models, we employ the pre-trained teacher model as the entry point of physics-guided representation alignment mechanism, that is, importing the PINN module and SIRS parameters as teacher knowledge into the new training stage. In this stage, knowledge transfer is accomplished by generating 
through forward propagation and computing 
using the chain rule. This allows the physical evolution laws learned by the PINN module to be transferred to the LSTM module, while the LSTM module only needs to learn how to adjust the dynamical parameters according to exogenous features based on these physical laws, and then make predictions on this basis.
Meanwhile, the training stage of the proposed model is as follows.
Stage 1: Freeze the LSTM module and train only the PINN module with SIRS constraints and the output layer, updating their parameters. In this stage, the model mainly relies on
and 
to ensure reliable capture of the temporal dynamics of ILI.Stage 2: Unfreeze the LSTM module to learn the impact of exogenous features on ILI transmission, while simultaneously training the LSTM module, the PINN module with SIRS constraints, and the output layer, updating their parameters. In this stage, the model depends on
, 
, 
, 
, 
, and 
to achieve effective integration of physical knowledge and exogenous features.Stage 3: Freeze the trained PINN and LSTM modules, and update the output layer as well as fine-tune the SIRS parameters based on
and 
. This step prevents overfitting of the model when handling complex data.
The overall illustration of the proposed model is visualized in Fig. 1. In Stage 1, the PINN receives time t as input and learns the epidemic states 
and latent representation 
under the guidance of observed ILI data and SIRS constraints. In Stage 2, the PINN continues to provide epidemic dynamic knowledge, while the LSTM receives the multi-source time-series feature 
(Including temperature, humidity, and ILI infections cases) and generates the latent representation 
and epidemic states 
. The physics-guided representation alignment mechanism transfers the learned epidemic knowledge from 
to 
. In Stage 3, the learned representations are fixed, and the final predicted number of ILI infections 
is obtained through output-layer refinement and SIRS parameter fine-tuning. All codes have been released at https://github.com/xhs66828C5C/Ours-ILI.
Fig. 1.
The overall framework of our ILI prediction model
Results
The datasets and indicators of case analysis
We use ILI infection data of Lanzhou dataset (from 1st January 2023 to 26th October 2024) and Xi’an dataset (from 1st January 2010 to 10th November 2016) [7], as well as temperature and humidity data for the same period collected from the China Meteorological Administration (https://weather.cma.cn/). Then, we group the data by week, took the mean within each group as the weekly data, and apply the Henderson moving average filter [30] to smooth the time series data. For visualization purposes, a Henderson moving average was applied only to the representative prediction window shown in this subsection. Since the Henderson filter is a symmetric smoothing method, it is not strictly causal. However, the smoothed series was used solely for trend visualization and was not involved in model training, parameter estimation, or forecasting. All quantitative forecasting evaluations reported were conducted using the original unsmoothed epidemic data. The final observed data used and its smoothed data are shown in Fig. 2.
Fig. 2.
The weekly observed data and its smoothed data (a-d) ILI cases and temperature with their smoothed data in Lanzhou and Xi'an
Meanwhile, for the infectious disease prediction task in this study, the Centers for Disease Control and Prevention usually require a 4-week-ahead forecast [31]. Therefore, we set the number of prediction weeks K = 4 for short-term prediction, and twice the common planning horizon K = 8 for long-term prediction. We further investigate whether the forecasts generated by different models in the long-term prediction task can accurately capture the actual ILI trends.
Then, following [32, 33], we adopt the Normalized root mean squared error (NRMSE), Symmetric mean absolute percentage error (SMAPE), Normal deviation (ND), and Mean absolute percentage error (MAPE) to evaluate the prediction performance, and the Pearson correlation coefficient (PCC) to assess the correlation of prediction trends. Among them, lower values of NRMSE, SMAPE, ND, and MAPE indicate better performance, whereas a higher PCC reflects stronger agreement with the actual trend. The calculation formulas of these indicators are given as follows.
 |
9 |
 |
10 |
 |
11 |
 |
12 |
 |
13 |
where K represents the number of prediction weeks, 
and 
represent the predicted and observed values of the number of ILI infections, respectively, 
, and 
.
Experimental results
Overall comparison results in Lanzhou and Xi’an
The Lanzhou dataset obtained a total of 95 weekly samples, corresponding to 202301-202443. The Xi’an dataset obtained a total of 358 weekly samples, corresponding to 201001-201645. To preserve the temporal structure of epidemic transmission and avoid information leakage, the dataset is divided sequentially rather than randomly shuffled. Specifically, for Lanzhou dataset, the first 59 weeks (202301-202407) were used as the training set, the subsequent 18 weeks (202408-202425) were used as the validation set, and the remaining 18 weeks (202426-202443) were used as the test set. For Xi’an dataset, the first 238 weeks (201001-201430) were used as the training set, the subsequent 60 weeks (201431-201538) were used as the validation set, and the remaining 60 weeks (201539-201645) were used as the test set.
To further assess the reliability and practical applicability of Ours, we additionally conduct rolling prediction over the entire test set. Specifically, after training the proposed model on the training and validation sets, rolling predictions are generated sequentially for each week in the test period. At each step, the model estimates the epidemic state at the corresponding future time point, and the prediction horizon is gradually extended until the entire test interval is covered.
We comprehensively compare the performance of Ours with the following five advanced baselines and appropriately select the corresponding basic parameters.
PINN-Joint [34]: A PINN based on an SIRS module with a time-varying transmission rate. PINN-Joint consists of three independent neural networks used to learn the susceptible state S(t), infectious state I(t), and the time-varying transmission rate β (t), respectively. All networks are optimized using the Adam optimizer with a learning rate of
. The neural networks for S(t) and I(t) each contain four hidden layers with 32 neurons per layer and use the tanh activation function. The neural network for β (t) also consists of four hidden layers, but with 64 neurons per layer, enabling stronger representation capability for complex time-varying transmission dynamics.PINN-Split [34]: PINN-Split adopts the same network architecture and hyperparameter settings as PINN-Joint, but differs in its training strategy. Specifically, PINN-Split employs a reduced-split training approach, where the epidemiological data loss and the residuals of the system equations are optimized separately, resulting in more reliable training and faster convergence.
PINN-TFC [35]: A PINN based on theory of functional connections. We select the PINN-TFC based on an SEIRS module with a fixed transmission rate. The hyperparameter settings we used follow the standard PINN-TFC, with the same values as in the original configuration.
PINN-SAIRD [36]: A PINN based on an SAIRD module with a fixed transmission rate. PINN-SAIRD employs a fully connected neural network to approximate the hidden epidemic states. The network consists of four hidden layers, each containing 32 neurons with the tanh activation function. The output layer contains five neurons corresponding to the five epidemic compartments in the SAIRD model. To ensure non-negativity and conservation of the population proportions, a softmax activation function is applied to the output layer.
DNNLEF [7]: A fully connected neural network to approximate the time-varying transmission rate. DNNLEF predicts the spread of epidemics based on the SIRS model with learned transmission rate sequence. The hyperparameter settings we used follow the standard DNNLEF, with the same values as in the original configuration.
The comparison results in Lanzhou dataset are shown in Table 1 and Fig. 3. The comparison results in Xi’an dataset are shown in Table 2 and Fig. 4. It can be observed from Table 1 that Ours achieves the best performance across all evaluation metrics, including NRMSE, SMAPE, ND, MAPE, and PCC in Lanzhou dataset. Furthermore, Figs. 3 and 4 shows the following phenomena:
The prediction curves generated by Ours, PINN-Joint, and PINN-Split are the closest to the ground-truth epidemic trajectory. Notably, these three models all incorporate time-varying transmission rates within the PINN framework, suggesting that dynamically modeling transmission processes is essential for accurately characterizing the temporal evolution and fluctuation patterns of ILI dynamics.
In contrast, PINN-TFC and PINN-SAIRD employ compartmental dynamics with fixed parameters, which limits their ability to adapt to temporal variations in transmission intensity. As a result, their prediction curves deviate significantly from the observed epidemic trends, especially during periods of rapid fluctuation.
Although DNNLEF also models time-varying transmission rates using a deep neural network framework, it lacks explicit epidemiological dynamic constraints, leading to weaker temporal consistency and reduced forecasting accuracy compared with PINN-based approaches.
In Xi’an dataset, PINN-TFS and PINN-SAIRD generate overly smooth prediction curves with limited temporal fluctuations, resulting in relatively low PCC values. Nevertheless, their conservative predictions remain close to the overall average level of the observations, leading to lower point-wise errors and superior performance in NRMSE, SMAPE, ND, and MAPE. In contrast, Ours and other time-varying parameter baselines better capture temporal variations in epidemic transmission and external factors, producing prediction trajectories that more closely follow the observed epidemic trends and thereby achieve higher PCC values. Although this flexibility may slightly increase point-wise deviations, Ours still ranks third in NRMSE, SMAPE, ND, and MAPE, outperforming all remaining baselines while achieving the best trend consistency.
Table 1.
Comparison between Ours and advanced baselines in Lanzhou dataset
| Model | Metric (%) | ||||
|---|---|---|---|---|---|
| NRMSE | SMAPE | ND | MAPE | PCC | |
| PINN-Joint | 28.83 | 25.91 | 24.71 | 25.23 | 43.86 |
| PINN-Split | 27.70 | 25.48 | 23.88 | 24.85 | 48.65 |
| PINN-TFC | 38.60 | 35.31 | 30.13 | 28.16 | 28.61 |
| PINN-SAIRD | 30.18 | 26.37 | 25.39 | 27.28 | 24.27 |
| DNNLEF | 33.11 | 27.34 | 26.65 | 25.87 | 47.08 |
| Ours | 21.66 | 20.19 | 18.66 | 20.02 | 55.93 |
The bold and underlined values denote the optimal and the suboptimal performance, respectively
Fig. 3.
ILI case prediction curves on entire test interval in Lanzhou
Table 2.
Comparison between Ours and advanced baselines in Xi’an dataset
| Model | Metric (%) | ||||
|---|---|---|---|---|---|
| NRMSE | SMAPE | ND | MAPE | PCC | |
| PINN-Joint | 48.54 | 51.47 | 42.94 | 40.00 | 88.10 |
| PINN-Split | 40.66 | 39.11 | 34.91 | 31.76 | 86.34 |
| PINN-TFC | 35.75 | 28.84 | 28.35 | 28.57 | 67.32 |
| PINN-SAIRD | 35.30 | 26.63 | 28.13 | 28.69 | 66.85 |
| DNNLEF | 48.48 | 42.25 | 44.88 | 43.37 | 69.43 |
| Ours | 35.98 | 35.48 | 31.17 | 29.19 | 88.42 |
The bold and underlined values denote the optimal and the suboptimal performance, respectively
Fig. 4.
ILI case prediction curves on entire test interval in Xi’an
Results and analysis on specific time windows for Lanzhou and Xi’an
Here, we compare the proposed model (denoted as Ours) with five baselines to evaluate their performance, and provide the following experimental setups.
Ours: The weights of
, 
, 
, 
, 
, 
, and 
are 1, 5, 0.0001, 0.5, 1, 1, and 1, respectively. All modules use Adam optimizer with learning rate 
, and training batch of size 64. The LSTM module uses “tanh” for the candidate cell state, and “sigmoid” for all gates.SIRS [23]: We use the “odeint” and “pymc” in Python to compute the numerical solution of the SIRS system (1) for ILI case prediction, with initial values S(0) = 254.857, I(0) = 2.429, and R(0) = 1, and with the time-varying parameters β (t), σ (t), and γ (t) estimated by Markov Chain Monte Carlo method, where one time step corresponds to one epidemiological week.
PINN [20]: We use all weeks prior to the prediction start week as the training set and the subsequent eight weeks as the test set. The model is trained for 5,000 steps, with loss function weights and optimizer settings kept consistent with those of the PINN module of Ours.
LSTM [24]: We construct the training set using a sliding window of length 90, which spans all available historical data from the beginning to the end of the training period. Each window is paired with the subsequent 56 days of ILI cases as its output. The final input window at the end of the training period is then used to predict the ILI cases for the next 56 days. The model is trained for 5,000 steps with a batch size of 32, and the optimizer settings are kept consistent with those of the LSTM module of Ours.
Temporal convolutional network (TCN) [37]: We keep the original TCN architecture unchanged, maintaining the same sliding window length, prediction horizon, and training strategy as those used in the LSTM module of Ours.
Deep neural network learning-explaining framework (DNNLEF) [7]: We train DNNLEF on a truncated training interval and then use the learned transmission rate to forward-simulate the epidemic system for predicting the number of cases over the next 8 weeks.
For fair comparison, all data-driven models (LSTM, TCN, DNNLEF, and Ours) use the same input features. The remaining models (SIRS and PINN) follow their original formulations and therefore do not use climatic variables as external inputs.
Then, we predict the weekly ILI cases in Lanzhou dataset over three time periods: from 2nd July 2023 to 26th August 2023 (202326-202333), from 12th November 2023 to 6th January 2024 (202345-202352), and from 1st September 2024 to 26th October 2024 (202435-202442), and also predict the weekly ILI cases in Xi’an dataset over three time periods: from 22nd December 2012 to 14th February 2013 (201252-201307), from 22th August, 2014 to 16th October, 2014 (201435-201442), and from 5th August, 2016 to 29th September, 2016 (201633-201640).
Meanwhile, we compute the evaluation indicators based on the obtained prediction results and summarize them in Tables 3 and 4. In addition, Figs. 5, 6, 7, and 8 provide a visual illustration of the performance of different models in terms of predictive ability and trend consistency.
Table 3.
Performance comparison of the prediction task in three windows of Lanzhou dataset
| 202326-202333 | Short-term prediction (forecast the first four weeks) | Long-term prediction (forecast the last four weeks) | Trend matching | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | NRMSE | SMAPE | ND | MAPE | NRMSE | SMAPE | ND | MAPE | PCC |
| SIRS | 0.194 | 0.176 | 0.171 | 0.164 | 0.592 | 0.442 | 0.559 | 0.586 | -0.904 |
| PINN | 0.139 | 0.118 | 0.117 | 0.110 | 0.158 | 0.127 | 0.128 | 0.139 | 0.926 |
| LSTM | 0.911 | 0.555 | 0.797 | 0.866 | 1.048 | 0.665 | 1.005 | 1.020 | 0.016 |
| TCN | 1.297 | 0.762 | 1.240 | 1.310 | 1.195 | 0.694 | 1.097 | 1.150 | 0.234 |
| DNNLEF | 0.564 | 0.760 | 0.550 | 0.544 | 0.784 | 1.357 | 0.782 | 0.790 | 0.824 |
| Ours | 0.171 | 0.147 | 0.149 | 0.162 | 0.389 | 0.331 | 0.386 | 0.396 | 0.957 |
| 202345-202352 | Short-term prediction (forecast the first four weeks) | Long-term prediction (forecast the last four weeks) | Trend matching | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | NRMSE | SMAPE | ND | MAPE | NRMSE | SMAPE | ND | MAPE | PCC |
| SIRS | 0.848 | 1.245 | 0.788 | 0.761 | 0.755 | 1.162 | 0.739 | 0.732 | 0.060 |
| PINN | 0.602 | 0.682 | 0.539 | 0.496 | 0.349 | 0.322 | 0.285 | 0.261 | 0.047 |
| LSTM | 0.908 | 1.438 | 0.851 | 0.833 | 0.867 | 1.501 | 0.857 | 0.856 | 0.359 |
| TCN | 0.951 | 1.529 | 0.886 | 0.860 | 0.865 | 1.432 | 0.839 | 0.825 | -0.639 |
| DNNLEF | 0.507 | 0.504 | 0.436 | 0.389 | 0.416 | 0.481 | 0.395 | 0.383 | 0.758 |
| Ours | 0.198 | 0.223 | 0.192 | 0.200 | 0.491 | 0.658 | 0.485 | 0.491 | 0.850 |
| 202435-202442 | Short-term prediction (forecast the first four weeks) | Long-term prediction (forecast the last four weeks) | Trend matching | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | NRMSE | SMAPE | ND | MAPE | NRMSE | SMAPE | ND | MAPE | PCC |
| SIRS | 2.834 | 1.181 | 2.827 | 2.982 | 2.949 | 1.207 | 2.932 | 3.526 | 0.281 |
| PINN | 0.677 | 0.905 | 0.632 | 0.612 | 0.788 | 1.048 | 0.715 | 0.664 | -0.284 |
| LSTM | 0.383 | 0.239 | 0.293 | 0.298 | 2.088 | 1.051 | 2.078 | 2.363 | 0.288 |
| TCN | 0.372 | 0.221 | 0.253 | 0.287 | 1.168 | 0.751 | 1.120 | 1.440 | 0.072 |
| DNNLEF | 0.846 | 0.589 | 0.828 | 0.836 | 1.015 | 1.416 | 1.001 | 1.225 | -0.135 |
| Ours | 0.204 | 0.195 | 0.199 | 0.211 | 0.340 | 0.293 | 0.273 | 0.398 | 0.452 |
The bold and underlined values denote the optimal and the suboptimal performance, respectively
Table 4.
Performance comparison of the prediction task in three windows of Xi’an dataset (The bold and underlined values denote the optimal and the suboptimal performance, respectively)
| 201252-201307 | Short-term prediction (forecast the first four weeks) | Long-term prediction (forecast the last four weeks) | Trend matching | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | NRMSE | SMAPE | ND | MAPE | NRMSE | SMAPE | ND | MAPE | PCC |
| SIRS | 0.671 | 0.923 | 0.642 | 0.627 | 0.735 | 0.880 | 0.647 | 0.595 | 0.222 |
| PINN | 0.515 | 0.617 | 0.483 | 0.467 | 0.566 | 0.501 | 0.448 | 0.371 | 0.004 |
| LSTM | 0.465 | 0.487 | 0.405 | 0.378 | 0.735 | 1.041 | 0.690 | 0.683 | 0.213 |
| TCN | 0.283 | 0.269 | 0.249 | 0.233 | 0.355 | 0.294 | 0.282 | 0.261 | 0.684 |
| DNNLEF | 0.341 | 0.358 | 0.288 | 0.281 | 0.469 | 0.442 | 0.398 | 0.351 | 0.587 |
| Ours | 0.175 | 0.146 | 0.146 | 0.134 | 0.201 | 0.169 | 0.172 | 0.158 | 0.982 |
| 201435-201442 | Short-term prediction (forecast the first four weeks) | Long-term prediction (forecast the last four weeks) | Trend matching | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | NRMSE | SMAPE | ND | MAPE | NRMSE | SMAPE | ND | MAPE | PCC |
| SIRS | 0.233 | 0.209 | 0.233 | 0.233 | 0.256 | 0.239 | 0.225 | 0.209 | -0.917 |
| PINN | 0.471 | 0.382 | 0.472 | 0.471 | 0.170 | 0.151 | 0.147 | 0.153 | 0.879 |
| LSTM | 0.123 | 0.111 | 0.118 | 0.117 | 0.233 | 0.208 | 0.200 | 0.182 | 0.824 |
| TCN | 0.179 | 0.197 | 0.179 | 0.179 | 0.446 | 0.498 | 0.410 | 0.388 | 0.543 |
| DNNLEF | 0.458 | 0.291 | 0.371 | 0.368 | 0.388 | 0.395 | 0.341 | 0.316 | -0.577 |
| Ours | 0.325 | 0.276 | 0.321 | 0.321 | 0.191 | 0.165 | 0.157 | 0.163 | 0.467 |
| 201633-201640 | Short-term prediction (forecast the first four weeks) | Long-term prediction (forecast the last four weeks) | Trend matching | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Model | NRMSE | SMAPE | ND | MAPE | NRMSE | SMAPE | ND | MAPE | PCC |
| SIRS | 0.085 | 0.073 | 0.072 | 0.069 | 0.037 | 0.032 | 0.032 | 0.031 | 0.609 |
| PINN | 0.986 | 0.662 | 0.987 | 0.989 | 1.117 | 0.718 | 1.119 | 1.119 | -0.685 |
| LSTM | 0.397 | 0.323 | 0.387 | 0.388 | 0.863 | 0.584 | 0.838 | 0.840 | -0.765 |
| TCN | 0.261 | 0.225 | 0.253 | 0.255 | 0.537 | 0.421 | 0.533 | 0.534 | -0.572 |
| DNNLEF | 0.309 | 0.300 | 0.280 | 0.280 | 0.563 | 0.532 | 0.479 | 0.472 | -0.413 |
| Ours | 0.268 | 0.227 | 0.259 | 0.256 | 0.270 | 0.217 | 0.247 | 0.248 | 0.652 |
The bold and underlined values denote the optimal and the suboptimal performance, respectively
Fig. 5.
The predicted ILI cases obtained by SIRS, PINN, LSTM, TCN, DNNLEF and Ours (a–c) Predicted curves for the Lanzhou dataset in three different periods (d–f) predicted curves for the Xi’an dataset in three different periods
Fig. 6.
The trend consistency comparison results for two datasets (a) Lanzhou dataset (b) Xi'an dataset
Fig. 7.
Performance comparison results in Lanzhou dataset (a, b) Short-term and long-term performance comparisons during 202326-202333 (c, d) during 202345-202352 (e, f) during 202435-202442
Fig. 8.
Performance comparison results in Xi’an dataset (a, b) Short-term and long-term performance comparisons during 201252-201307 (c, d) during 201435-201442 (e, f) during 201633-201640
For Lanzhou dataset, as shown in Table 3 and Figs. 5, 6, and 7, it is not difficult to find that, compared with SIRS, PINN, LSTM, TCN, and DNNLEF, the proposed model demonstrates lower prediction errors than most baseline methods under multiple prediction periods. Specifically:
In the short-term prediction task, the proposed model achieves the optimal performance for both 202345-202352 and 202435-202442. Compared with the suboptimal results, our predictions are significantly improved, where NRMSE, SMAPE, ND, and MAPE improving by at least 45.16%, 11.76%, 21.34%, and 26.48%, respectively. Meanwhile, the proposed model attains the suboptimal performance during 202326-202333.
In the long-term prediction task, although the proposed model only achieves the optimal performance during 202435-202442, it still attains the suboptimal performance in both 202326-202333 and 202345-202352, showing excellent prediction performance.
In terms of trend consistency, the proposed model performs the best across all three periods. Compared with the suboptimal results in each period, PCC improves by at least 3.35%, 12.14%, and 56.94%, respectively. This indicates that the proposed model can effectively capture the dynamic variation trends in the ILI transmission process, demonstrating its advantages and reliability in practical applications.
For Xi’an dataset, as shown in Table 4 and Figs. 5, 6, and 8, it is not difficult to find that, compared with SIRS, PINN, LSTM, TCN, and DNNLEF, the proposed model can provide reliable long-term predictions and has good trend consistency. Specifically:
In the short-term prediction task, the proposed model only achieves the optimal performance for 201252-201307 and the suboptimal for 201633-201640. It should be noted that, for the Xi’an dataset with a long observation period, the SIRS model converges to a steady-state equilibrium during the prediction interval. In this regime, although it achieves the best performance across evaluation metrics, it struggles to capture time-varying transmission dynamics and non-stationary epidemic patterns.
In the long-term prediction task, the proposed model achieves the optimal performance for 201252-201307 and the suboptimal performance for both 201435-201442 and 201633-201640, showing excellent prediction performance.
In terms of trend consistency, the proposed model performs the best across 201252-201307 and 201633-201640. Compared with the suboptimal baselines, PCC improves by at least 43.57% and 7.06%, respectively.
It is worth mentioning that we can observe the following phenomena:
For the Lanzhou dataset, the prediction performance of SIRS gradually declines across 202326-202333, 202345-202352, and 202435-202442. This may be because the predictive accuracy of traditional compartmental model is highly dependent on the time span, and the cumulative effect of errors increases over time, resulting in poor performance during 202435-202442. For the Xi’an dataset, the SIRS model exhibits gradually improved performance, which can be attributed to the convergence of the epidemic dynamics to a steady-state equilibrium. As a result, the structural assumptions of the SIRS become well aligned with the underlying dynamics, leading to improved metric performance despite its limited ability to capture time-varying transmission mechanisms.
Both LSTM and TCN take relatively sparse and noisy humidity features as inputs, which increases the risk of overfitting. Moreover, these models rely on autoregressive extrapolation beyond the training distribution and lack the incorporation of domain knowledge. Although TCN offers improved architectural flexibility, both TCN and LSTM remain purely data-driven approaches without explicit epidemiological mechanisms.
PINN relies on SIRS constraints to learn the underlying dynamics, which makes it less prone to overfitting and enable relatively strong predictive performance. However, this reliance also leads to the neglect of exogenous factors such as climate, limiting its flexibility in accurately capturing the actual trends of ILI cases.
DNNLEF primarily relies on input ILI case counts and meteorological variables to fit epidemic curves and infer epidemiological parameters, employing customized activation functions to accommodate seasonal patterns. This design leads to limited robustness when handling noises and data fluctuations. Although DNNLEF demonstrates reliable predictive performance on two datasets, meteorological factors are incorporated as direct inputs in a relatively coarse manner, without effectively modeling their interactions with the intrinsic epidemic transmission dynamics, thereby constraining its ability to explain climate effects.
The observations above indicate that different baseline models excel in different aspects but also exhibit corresponding limitations. By applying a physics-guided representation alignment mechanism to combine PINN and LSTM, the proposed model consistently achieves competitive performance across experimental settings. This suggests that the proposed physics-guided framework provides an effective mechanism for integrating epidemiological knowledge and multi-source data, thereby improving both prediction accuracy and robustness.
Statistical test
To assess whether the performance improvements of the proposed method are statistically reliable rather than resulting from random variation, we further conduct the Nemenyi nonparametric statistical rank order test based on the average ranks of all methods. The Nemenyi test is particularly suitable for comparing multiple prediction models because it does not require the performance measures to follow a specific distribution and evaluates differences based on model ranks rather than raw metric values. Moreover, it provides a post-hoc multiple-comparison procedure that can identify whether the rank differences between models are statistically significant. The statistical test results for Lanzhou dataset are reported in Table 5 and Fig. 9.
Table 5.
Ranking results for different models on five metrics in Lanzhou
| Metric | Classifier ranking | |||||
|---|---|---|---|---|---|---|
| PINN-Joint | PINN-Split | PINN-TFC | PINN-SAIRD | DNNLEF | Ours | |
| NRMSE | 3 | 2 | 6 | 4 | 5 | 1 |
| SMAPE | 3 | 2 | 6 | 4 | 5 | 1 |
| ND | 3 | 2 | 6 | 4 | 5 | 1 |
| MAPE | 3 | 2 | 6 | 5 | 4 | 1 |
| PCC | 4 | 2 | 5 | 6 | 3 | 1 |
| Average rank | 3.2 | 2.0 | 5.8 | 4.6 | 4.4 | 1.0 |
| Critical difference (CD) | 3.3718 | |||||
The bold value denotes the optimal performance
Fig. 9.
CD diagram of five advanced baselines and Ours in Lanzhou
In Table 5, Ours significantly outperforms all baselines on five metrics and average rank. And the critical difference (CD) value is 3.3718, which means that the two models are not significantly different at the p < 0.05 level when their rank difference is less than 3.3718. In Fig. 9, horizontal lines are used to join the classifiers that are not statistically significant, Ours is significantly superior to PINN-TFC, PINN-SAIRD, and DNNLEF at the p < 0.05 level. Therefore, although Ours is numerically superior in five metrics and average rank than PINN-Joint and PINN-Split, it is not statistically significantly superior to them.
Ablation study
To investigate whether several key loss terms are beneficial to the ILI prediction performance, we study the ablation of Ours by selectively disabling them on the Lanzhou dataset. Specifically, we introduce three variants of Ours: w/o 
, w/o 
, and w/o 
. Here, the weights of 
, 
, 
, 
, 
, 
, and 
in Ours are 1, 5, 0.0001, 0.5, 1, 1, and 1, respectively. For each variant, we remove the corresponding component by setting its loss weight to zero while keeping all other architectures, hyperparameters, and training settings unchanged.
From the ablation results in Table 6, we have the following observations:
Overall, Ours consistently outperforms all three variants. This result not only demonstrates that each loss component can significantly enhance the final prediction performance, but also verifies that using the embedding alignment to retain epidemic dynamics knowledge is a simple and effective strategy.
The removal of
leads to a sharp decline in performance. This confirms that the latent representation transfer is the key mechanism enabling the LSTM to inherit epidemic dynamics learned by the PINN.Although the w/o
and w/o 
achieve higher performance than the w/o 
, the contribution of these two layers of physical constraints to the overall framework cannot be underestimated. Specifically, 
enables the teacher PINN to learn physically consistent epidemic dynamics and reliable time-varying parameters, while 
further constrains the student LSTM to preserve epidemic evolution patterns during prediction.
Table 6.
Performance of Ours and its variants in Lanzhou
| Model | Metric (%) | ||||
|---|---|---|---|---|---|
| NRMSE | SMAPE | ND | MAPE | PCC | |
w/o
|
28.04 | 24.32 | 22.83 | 22.76 | 43.82 |
w/o
|
34.50 | 32.81 | 30.36 | 29.65 | 25.58 |
w/o
|
29.16 | 26.36 | 24.98 | 25.79 | 41.24 |
| Ours | 21.66 | 20.19 | 18.66 | 20.02 | 55.93 |
The bold value denotes the optimal performance
Symbolic regression and interpretability analysis of the transmission rate function
The proposed model does not yield explicit mathematical expressions. To better understand the relationships among transmission rate, temperature, and humidity, we employ symbolic regression as a mechanism discovery tool to derive optimal symbolic expressions. While symbolic regression can fit the data effectively and produce novel formulations of the contact transmission rate, the resulting expressions are often overly complex and difficult to interpret [7].
Symbolic regression is configured with temperature sequence 
and humidity sequence 
as input variables. The initial set of operators includes binary operators (+, −, × ) and unary operators (\sin , \exp ). The number of iterations is set to 200, the population size to 100, and the maximum complexity to 16, with MSE as the loss function. The best-performing expression is then selected as the final output. The resulting formulation of the transmission rate in Lanzhou is given as follows:
 |
14 |
Then, to quantify the temporal evolution of the transmission rate under different intervention strategies in Lanzhou and to provide guidance for more effective control policies, we adopt three rate functions introduced in [2, 38, 39], with forms given as follows.
 |
15 |
 |
16 |
 |
17 |
Using the inferred transmission rate sequence in 202301–202443 of Lanzhou as observations, we estimate the unknown parameters in Eqs. (15) - (17) via least squares. The optimal parameter estimates are reported in Table 7.
Table 7.
Parameter estimation for three rate functions in Lanzhou
| Variables | Values | Variables | Values | Variables | Values | Sources |
|---|---|---|---|---|---|---|
 |
0.3851 |  |
0.3833 |  |
21.3659 | Estimated |
 |
-0.0055 |  |
-0.0160 |  |
-3.2453 | Estimated |
 |
0.0004 |  |
0.0012 |  |
0.0001 | Estimated |
 |
1.8667 | Estimated |
Figure 10 presents the corresponding fitting results and RMSE. It can be observed that, for Lanzhou, either 
or 
is more suitable for extrapolating the transmission rate during this period. This finding suggests that epidemic control strategies should be tailored to local conditions rather than adopting fixed and uniform measures. In practice, interventions should be determined based on the specific context and dynamically adjusted to better meet evolving needs.
Fig. 10.
Transmission rate inference, rate function fitting and RMSE in Lanzhou
Discussion and conclusion
In this study, to develop an infectious disease prediction model that integrates dynamics knowledge-driven modeling with multi-source data, we design a novel physics-guided representation alignment mechanism that transfers the dynamics learned by PINN into an LSTM model capable of incorporating climatic factors. Our results demonstrate that the proposed framework effectively combines the strengths of SIRS, PINN, and LSTM while overcoming their respective limitations. Specifically, the lack of flexibility and generalizability of SIRS in modeling complex ILI patterns, the exclusion of exogenous climate data in PINN, and the limited predictive interpretability in LSTM can all be reasonably explained and mitigated from the perspective of knowledge transfer.
The results demonstrate that the proposed model achieves competitive prediction performance across different tasks, particularly in capturing epidemic trends and maintaining relatively reliable prediction accuracy. This success arises from two key factors: first, the effective integration of transmission mechanisms derived from the compartmental model into the neural network; and second, the seamless incorporation of multimodal data through a physics-guided representation alignment mechanism, which enhances both learning and representation capabilities. As a result, the proposed model not only enables gradient alignment and knowledge transfer through embedding approximation and the chain rule, but also leverages physical knowledge from the source domain to guide the LSTM in mapping input data to the underlying ILI dynamics. Furthermore, it is worth noting that this study employs a specific compartmental model (i.e., the SIRS model) to generate dynamic knowledge within the physics-guided framework. This design allows decision-makers to extend the approach by directly modifying the compartmental model, thereby adapting it to the transmission and prediction of various emerging infectious diseases.
Note that in classical compartmental models, the impact of climatic factors on ILI is usually modeled by assuming specific functional forms in advance-for example, a mixture of sinusoidal and linear functions [38], a purely sinusoidal function with time variables [39–41], an exponential function derived from laboratory experiments [42], or a quadratic function in tropical and subtropical regions [43]. In other words, for ILI under different scenarios, it is often necessary to search for relevant prior knowledge to predefine the functional form of climatic factors, which significantly constrains the performance of compartmental models. The proposed model effectively overcomes this limitation by eliminating the need for compartmental models to provide domain knowledge about climatic factors. Instead, it leverages climate data in the target domain to drive the student model LSTM, which can automatically capture the influence of climatic factors on ILI. This general framework can be flexibly applied across different ILI scenarios, thereby improving the adaptability and reliability of ILI prediction and supporting public health surveillance and intervention planning.
On the one hand, based on the analysis of the prediction results for Lanzhou during 202326-202333, we observe that LSTM performs poorly in both ILI prediction and trend matching tasks, whereas the proposed model achieves strong performance in terms of both accuracy and correlation. This suggests that relying on a small amount of data for prediction with LSTM may result in unsatisfactory outcomes, while physics-guided representation alignment can provide an effective mechanism for epidemic prediction under limited-data scenarios. Its results should be regarded as one of the multiple sources of evidence for public health decision-making, which aligns with the conclusions of existing studies [44, 45]. On the other hand, in the case studies across three periods, we find that the performances of SIRS and PINN (domain knowledge-driven) as well as LSTM and TCN (multi-source data-driven) are unstable. The former indicates that fixed sources of domain knowledge may not perfectly adapt to epidemic prediction tasks under varying scenarios, while the latter suggests that generating reliable infectious disease prediction results may require not only observational data, but also additional epidemiological constraints and dynamical prior knowledge. These two findings are consistent with existing studies [33, 46]. Therefore, a coupled framework that jointly integrates multi-source data and dynamical knowledge can help reduce prediction errors and trend fluctuations in ILI prediction.
Although we believe this study opens new avenues for incorporating epidemic domain knowledge and exogenous climatic factors into deep learning models for more reliable ILI trend prediction, several limitations remain. On the one hand, the compartmental model we employed does not consider the effects of non-pharmaceutical interventions or vaccination, which somewhat constrains its ability to capture the actual epidemic transmission process. Incorporating these factors into the proposed model is necessary to more comprehensively capture the dynamics of epidemic spread. On the other hand, we have not conducted a systematic assessment of transferability before applying source-domain knowledge to the target domain, which may introduce the risk of “Negative Transfer”. Selecting epidemiological models and prior assumptions that are better suited to specific real-world transmission scenarios is crucial for mitigate such risks and enhance predictive performance. Furthermore, although the model achieves satisfactory prediction accuracy, it does not generate predictive distributions or confidence intervals that could characterize forecast uncertainty. In practical epidemic surveillance and public health decision-making, uncertainty information is often valuable for risk assessment and resource allocation. Addressing these issues will be the focus of our future research.
Acknowledgements
Not applicable.
Abbreviations
- ILI
Influenza-like illness
- PINN
Physics-informed neural network
- LSTM
Long short-term memory network
- SIRS
Susceptible-infectious-recovered-susceptiblk
- NRMSE
Normalized root mean squared error
- SMAPE
Symmetric mean absolute percentage error
- ND
Normal deviation
- MAPE
Mean absolute percentage error
- PCC
Pearson correlation coefficient
- TCN
Temporal convolutional network
- DNNLEF
Deep neural network learning-explaining framework
- CD
Critical difference
Authors’ contributions
Li, T participated in data analysis and manuscript writing. Li, T and Li, W proposed the main structure of this study. Li, T, Li, W and Hu, S provided useful suggestions and revised the manuscript. All authors reviewed the manuscript.
Funding
Major Project of Guangzhou National Laboratory (GZNL2024A01004)
National Science and Technology Major Project (2025ZD01900407)
R&D Program of Guangzhou Laboratory (SRPG22-007).
Data availability
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.