Deployable knowledge–data hybrid models for day-ahead cooling load prediction under data scarcity: a case study and performance validation

Abstract

Traditional data-driven models for building cooling load prediction often fail during the initial operational stage due to insufficient training data, leading to unreliable forecasts. To address this challenge, a knowledge-data hybrid forecasting framework was proposed, it combines simplified heat-transfer-based load calculations with deep learning networks, where physics-based load estimates are embedded as auxiliary inputs to guide the data-driven predictor. Four models were evaluated, including three knowledge-embedded variants (using theoretical fresh-air load, envelope heat-transfer load, and their combination) and a purely data-driven CNN–LSTM baseline. Validated on an actual building under varying training-data richness, models based on the proposed framework reduce prediction errors by 39% to 69% and decrease error variance by nearly an order of magnitude compared with the baseline while effectively mitigating overfitting in small-sample scenarios. The computational simplicity and reliance only on easily accessible data underscore its practical value for early-stage engineering deployment.

Introduction

Accurate day-ahead cooling load forecasting is increasingly important as power systems integrate larger shares of variable renewable generation. At both grid and building scales, day-ahead predictions support demand response scheduling, optimal charging/discharging of thermal energy storage, and more efficient chiller dispatch, thereby contributing to energy-efficiency and carbon-reduction targets¹.

In practical deployments, forecasting models often face a critical constraint: insufficient historical data during the early operation stage of a building or after a building automation/energy management system is newly commissioned. Such data scarcity is common for newly constructed buildings, recently retrofitted systems or when building automation systems are first deployed. Under limited training data scenarios, data-driven methods can exhibit poor generalization, overfitting, and yielding large prediction uncertainty^2–4. This motivates forecasting frameworks that are both accurate and robust when only a small amount of site-specific data is available.

Cooling load modeling approaches are commonly grouped into physics-based, statistical, and data-driven paradigms. Physics-based tools derive load from thermodynamic and heat-transfer principles with high interpretability, but require comprehensive and detailed input parameters (building materials, occupancy schedules, weather conditions, and operational schedules) and are often impractical for real-time forecasting or deployment under uncertain future inputs^5,6. Statistical methods provide lightweight empirical relationships (e.g., temperature-frequency or degree-day style models), yet they often struggle to capture nonlinear dynamics and temporal dependencies in modern buildings⁷. Data-driven models, including neural networks, support vector machines, regression, and k-nearest neighbors, learn complex patterns directly from historical data and exogenous variables such as weather inputs⁸. But their performance is sensitive to data volume and quality and typically degrades under data scarcity^1,9,10.

Hybrid frameworks have therefore been proposed to integrate physical knowledge into data-driven forecasting. Representative directions include optimization-assisted learning^11,12, regression–ML hybrids^13–15, multi-network deep learning architectures (e.g., CNN–LSTM and Seq2Seq)^16–18, and physics-informed learning that injects thermodynamic/heat-transfer structure through constraints or surrogate signals¹⁹.

Despite those encouraging researches, several gaps remain for engineering deployment. First, many hybrid studies emphasize short-horizon prediction for real-time control, while day-ahead hourly forecasting is less systematically investigated^17,20,21. Second, data-scarce start-up conditions are often not evaluated explicitly, although they are common in practice and can expose severe overfitting in purely data-driven models^2,22. Third, some hybrid approaches rely on complex simulations or extensive building-specific calibration, which hinders scalability and deplorability^19,23. Finally, robustness is frequently reported qualitatively, and formal analyses of error dispersion or stability are still limited^24–26.

To address these gaps, this study develops and validates a deployable knowledge–data hybrid framework for day-ahead cooling load prediction under data-scarce conditions. The proposed framework computes readily obtainable physics-based load components using simplified heat-transfer formulations and embeds these signals as auxiliary inputs to a deep-learning-based predictor. Four models are evaluated, including three knowledge-embedded variants that incorporate theoretical fresh-air load, theoretical envelope heat-transfer load, and their combination, as well as a purely data-driven CNN–LSTM baseline. Model performance is benchmarked using an operational dataset from a real building under varying levels of data availability, ranging from 10% to 90% of the full dataset. Prediction accuracy, robustness, and overfitting mitigation are quantitatively assessed using RMSE-based metrics, error dispersion analysis, and effect-size evaluation, with the aim of providing a practical and low-cost solution for improving forecasting performance during the initial operational stages of modern building energy systems.

While recent studies have demonstrated the strong potential of ensemble learning methods and attention-based deep learning architectures for complex engineering prediction tasks, most existing works focus primarily on purely data-driven modeling paradigms^27–29. In the context of building cooling demand forecasting under data-scarce conditions, systematic integration of such advanced machine learning models with physics-informed hybrid frameworks remains relatively underexplored. Therefore, the present study focuses on establishing and validating a knowledge–data hybrid forecasting framework based on a CNN–LSTM backbone. The extension of this framework to incorporate alternative machine learning architectures, such as gradient boosting and transformer-based models, is left as an important direction for future work^30–32.

Unlike studies that primarily aim to develop a pre-trained model for direct cross-building generalization, this work focuses on a mechanism-oriented modelling framework that more effectively exploits information from a specific building to improve load forecasting performance for that target building. The significance of the study lies in the fact that the proposed framework itself is generalizable and can be transferred to other buildings and contexts.

Methodology

Building physics and data science offer complementary perspectives for interpreting and estimating building cooling load. From a building-physics perspective, cooling demand is explicitly quantified using analytical formulations derived from thermodynamic and heat-transfer principles. In contrast, data-science approaches infer cooling load through pattern recognition and time-series analysis, capturing nonlinear and temporal relationships directly from historical data. In this study, these two perspectives are integrated into a hybrid forecasting framework that leverages both physical consistency and data-driven predictive capability.

General description of the load demand prediction

From a data-science perspective, cooling load forecasting is inherently a time-series problem characterized by strong nonlinearity and temporal dependence, reflecting the dynamic interactions between weather conditions, building thermal inertia, and system operation. The cooling load of a building can be represented as a function of time-varying inputs, including ambient temperature, ambient humidity, solar radiation, occupancy, and appliance use, etc. In practice, these inputs are typically discretized into time series with past and present inputs as illustrated in Eq. (1) to simplify calculations.

1

Where, denotes time; represents one time step before ; is a vector containing time-varying input variables at ; and is the time delay (i.e., the number of previous time steps related to the current time ); = The formula derived from the classical heat transfer equation and the thermal equilibrium equation.

To predict future loads, unknown future inputs are replaced with their predicted values, allowing forecasting to proceed step by step up to 24 h ahead, as shown in Eqs. (2)–(3).

2

3

Where, = predicted cooling load at time ; = predicted time-varying input variables at future moment ; and = 1,2, …, 24, steps ahead of moment .

General description of knowledge-based load calculation methods

From a building-physics perspective, cooling demand primarily originates from heat gains associated with outdoor air intake, heat transfer through the building envelope, and internal heat sources. These mechanisms can be described using simplified thermodynamic and heat-transfer principles. But a full calculation of total cooling load using physics-based methods requires comprehensive input data. However, the stochastic characteristics and measurement challenges associated with occupant behavior, air infiltration, lighting, and appliance loads make such all-inclusive knowledge-based modeling impractical. By contrast, envelope heat-transfer load and fresh air handling load can be reliably computed from routinely available data, making them the most valuable components for providing physics-informed signals to forecasting models. Therefore, this study does not attempt to construct a complete knowledge-based model; instead, it focuses on hybrid approaches that incorporate these computable yet incomplete load components into data-driven models to assess their contribution to forecasting performance.

The envelope heat transfer load , can be calculated via the Heat Balance Method (HBM)³³. And, theoretical fresh-air load can be similarly calculated according to the fresh air handler’s operation schedule data and measured ambient air enthalpy , and estimated supply-air enthalpy, as Eq. (4)⁶.

4

Where is the fresh air volumetric flow rate (m³/s); is the air density (≈ 1.2 kg/m³).

From the point of view of physics, building cooling load is formed by the envelope heat transfer processes resulting from the differences between internal and external conditions (e.g., solar radiation, outdoor temperature, outdoor humidity), internal heat sources (e.g., occupants, lighting, appliances), and air exchange between indoor and outdoor spaces. The total hourly cooling load is the sum of these three main components^5,34, as illustrated in Eq. (5).

5

Where, envelope heat transfer load (ENV); interior heat load; =air exchange load between indoor and outdoor spaces, mainly by fresh air handling (FA).

If the thermodynamic properties of the envelope and all relevant time-vary inputs including meteorological conditions, building operation and behavior parameters such as indoor temperature and humidity setpoints and the thermodynamic state of conditioned fresh air are all reachable, the building’s cooling load can be accurately determined through the process illustrated in Fig. 1.

Fig. 1.

Knowledge-based models for cooling load prediction.

General description of data-driven models

In general, ML-based forecasting algorithms perform an autoregressive mapping on historical observations. Let the current time be , then the one-step‐ahead prediction of cooling load at can be written as Eq. (6).

6

Where, is the predicted cooling load at time ; is the load demand predicting network training by historical demand observations and potentially other exogenous variables.

Likewise, the one-step-ahead forecast at time can be expressed as Eq. (7).

7

Where, is the predicted cooling load at ; and is the predicted inputs vector at .

Structure of the knowledge-data hybrid forecasting framework

This study develops and trains a novel knowledge–data hybrid model for cooling demand forecasting, intending to validate the complementarity between physics-based knowledge and data-driven learning. By embedding physically derived cooling load components into a data-driven predictive framework, the proposed model leverages the strengths of both paradigms while mitigating their individual weaknesses.

The cascade workflow is generally designed as illustrated in Fig. 2. As the first step, meteorological data and operational historical data are processed through dedicated prediction modules to generate future sequences of meteorological and operational conditions. These predicted sequences are then delivered to the building cooling load physics model which computes a deterministic estimate of cooling load based on fundamental heat balance principles. Finally, the predicted meteorological and operational sequences, together with the physics-based cooling load estimates , are jointly input into the main data-driven predictive model, where they serve as auxiliary inputs guiding the forecasting process. The main model integrates these heterogeneous signals to produce the day-ahead, hourly cooling load forecasts.

Fig. 2.

Framework of proposed knowledge-data hybrid model for cooling load prediction.

Through this data-physics-data cascaded integration, the hybrid mechanism is expected to benefit from the predictive continuity of exogenous drivers, the interpretability of physics-based calculations, and the flexibility of data-driven learning, resulting in a balanced and robust solution for day-ahead forecasting under data-scarce conditions.

Case building and dataset

Data for model training and validation were obtained from a typical office building located in Hangzhou, China, equipped with a centralized cooling system. Envelope thermal properties of the building were sourced from as-built documentation; meteorological and operational parameters were retrieved from the building automation system, and cooling load data from the building’s energy management system. The dataset spans one cooling season (00:00, May 4 to 23:00, October 31), totaling 4,344 h (181 days) of hourly records. As data pre-processing, outliers were removed and missing values interpolated using a Kalman filter. The variables include hourly outdoor dry‐bulb temperature, hourly solar irradiance, daily maximum and minimum temperature, and hourly cooling load.

The case study building is an office building located in Hangzhou, China. The building has a total gross floor area of 56,448.73 m², of which 36,067.5 m² are conditioned spaces served by the central air-conditioning system during summer operation. The building consists of 23 above-ground floors with a total height of 99.8 m. The ground floor footprint measures approximately 57.0 m × 32.5 m. Construction of the building began in 2020, and the project was completed in April 2022, with full operation starting in July 2022. The operational dataset used in this study was collected during the entire year of 2024, at which point the building had already reached stable operation. The building automation (BA) system continuously recorded environmental and operational parameters, enabling reliable and consistent data acquisition for model training and validation. The building envelope adopts double-glazed, triple-silver Low-E insulating glass. According to the manufacturer’s performance report, the glazing has a visible light transmittance of 51%, an outdoor reflectance of 9%, and a thermal transmittance (U-value) of 1.65 W/(m²·K). The glazing system complies with the Chinese standard JGJ 151, with a solar shading coefficient of 0.33. The building’s ventilation system is designed with a maximum fresh air supply rate of 137,200 m³/h, which constitutes a major component of the cooling load associated with fresh-air treatment in the HVAC system.

Input data of the knowledge-data hybrid models

To establish a deployable hybrid model, the first step is to screen available input variables and extract those most strongly correlated with the predicted load demand. These variables are listed in Table 1, including observed meteorological data, along with theoretical load components and . Pearson’s product-moment and distance correlation coefficients are both adopted³⁵. The results, as shown in Table 1, revealed there are strong positive associations (> 0.6) between building cooling load records and outdoor solar irradiance, together with theoretical fresh‐air load and theoretical envelope load . The summed value of showed the highest correlation with total cooling load. These results indicate that including theoretical fresh‐air treatment load, envelope heat transfer load, or their sum as inputs to the predicting neural network is prospective to enhance prediction performance.

Table 1.

Correlation analyses of building cooling load and its related parameters.

Characteristics	Parameter id.	Parameter	Pearson product-moment correlation coefficient	Distance correlation coefficient		Time lags (hour)
Meteorological parameters	1	Hourly outdoor temperature	0.58	0.52		0
	2	Hourly outdoor solar irradiance	0.89	0.91		2
	3	Daily maximum temperature†	0.20	0.21		0
	4	Daily minimum temperature†	0.22	0.23		0
Knowledge-based information	5	Theoretical fresh air load	0.76	0.74		0
	6	Theoretical envelope load	0.78	0.81		2
	7		0.88	0.88		1

Parameters marked with † source from local weather forecast service.

Table 2 summarizes the results of Principal Component Analysis (PCA) on variables associated with cooling load. The weekday_id encodes non-working days (weekends and holidays) as 0 and working days as 1 to capture different load patterns. The PCA was performed to identify a reduced and informative subset of inputs for the hybrid model, aiming to simplify the model structure and enhance its practical deployability. The result showed that the first three principal components (PC1-PC3) collectively explain 87% of the total variance of all optional inputs.

Table 2.

Variance contribution of building cooling load related parameters.

Characteristics	Parameter	PC1	PC2	PC3	PC4	PC5	PC6	PC7	PC8
Meteorological parameters	Hourly outdoor temperature	0.414	−0.313	0.011	0.024	0.068	−0.834	−0.169	0
	Hourly outdoor solar irradiance	0.343	0.49	−0.006	0.058	−0.269	0.116	−0.743	0
	Daily maximum temperature†	0.316	−0.487	0.002	−0.121	−0.754	0.249	0.128	0
	Daily minimum temperature†	0.341	−0.435	0.016	−0.318	0.568	0.429	−0.292	0
Tag	Weekday_id	−0.007	0.02	1	−0.007	−0.01	0	0.005	0
Knowledge-based information	Theoretical fresh air load	0.378	−0.04	0.01	0.825	0.15	0.201	0.19	−0.276
	Theoretical envelope load	0.384	0.382	−0.009	−0.443	0.036	−0.039	0.376	−0.606
		0.451	0.295	−0.004	−0.055	0.085	0.042	0.376	0.746
PC Explained Variance		0.5009	0.2425	0.1249	0.0752	0.0257	0.0213	0.0094	~ 0

Parameters marked with † source from local weather forecast service.

Based on PCA, Table 3 further aggregates the contributions of each variable according to the weights of PC1-PC3. The summed variance contribution indicated that hourly outdoor solar irradiance was the most significant meteorological input.

Table 3.

Total variance contribution of building cooling load related parameters.

Characteristics	Parameter	Variance contribution
Meteorological parameters	Hourly outdoor temperature	0.285
	Hourly solar irradiance	0.291
	Daily highest temperature	0.277
	Daily lowest temperature	0.278
Tag	weekday	0.133
Knowledge-based information	Theoretical fresh air load	0.200
	Theoretical envelope load	0.286
		0.298

Therefore, together with weekday_id, which captures weekly load trends, these features were selected as key inputs for the data-driven sub-model. In parallel, three independent physics-based sub-models incorporating , and + were constructed respectively to provide auxiliary information, thus forming the three types of hybrid model.

Details of the knowledge-data hybrid models

Figure 3 illustrates the detailed structure and data flow of the proposed hybrid forecasting model based on the general framework introduced in Sect. 2.4. The prediction process follows a cascaded flow across input, data-driven layers, and knowledge-based layers. First, historical records are pretreated and passed to data-driven models to generate future meteorological sequences (LSTM-CNN) and the operation time series of fresh air units (WMA). Then these predicted data are then transferred to the knowledge-based layer, which computes deterministic estimates of the theoretical cooling load components, including and . As the third step, physics-informed outputs and are fused with the predicted meteorological and operational variables and jointly fed into the LSTM-based prediction network as part of the input vectors to execute predicting task.

Fig. 3.

Flow chart of the proposed knowledge-data hybrid model.

As summarized in Table 4, the model variants differed in whether the predictive core incorporated , , their combination, or relied solely on data-driven inputs. In all cases, the physics-based load estimates were not used as final predictions but were embedded as auxiliary inputs to constrain the feature space of the main CNN–LSTM predictor. By combining the predicted meteorological and operational sequences with these physics-informed load components, the main predictor was guided by both empirical data patterns and physical priors, thereby generating the day-ahead hourly cooling load forecasts.

Table 4.

The input data structure of models.

Model id	Meteorological parameters	Tag	Extra information from physics model
Inputs	Hourly outdoor solar irradiance	Weekday_id	Theoretical fresh air load		Theoretical envelope load
Model 1	Adopted	Adopted	Adopted
Model 2	Adopted	Adopted		Adopted
Model 3	Adopted	Adopted			Adopted
Model 4 (baseline)	Adopted	Adopted

Model 4 (the pure data-driven CNN–LSTM model) is treated as the baseline; Models 1–3 are knowledge-embedded variants of this baseline.

In this study, the hybrid forecasting architecture was implemented by integrating Convolutional Neural Networks (CNNs), Long Short-Term Memory (LSTM) units, and a Weighted Moving Average (WMA) algorithm. The choice of a CNN–LSTM architecture is motivated by two complementary considerations. First, CNN layers are effective at extracting local patterns and reducing redundancy from multivariate time series, which is particularly beneficial when combining meteorological variables, weekday tags, and physics-informed load components¹⁶. Second, LSTM units have been shown to capture both seasonal behaviors (long-range temporal dependencies) and short-term (24-hour) patterns in building energy demand, offering superior performance and stability compared with shallow recurrent or feed-forward networks^21,36,37. By stacking CNN and LSTM layers, a high-performance baseline model is designed to benchmark against which the added value of physics-based knowledge embedding can be rigorously assessed.

In parallel, the operation sequence of fresh air handling units (FAUs), which is relatively simple and repetitive, was regressed using a WMA algorithm, with the weekday feature incorporated following the ASHRAE Handbook guidelines³⁴.

The overall architecture, illustrated in Fig. 4, integrates meteorological inputs, temporal tags, and physics-informed input features with CNN-LSTM based main prediction network to generate day-ahead hourly cooling load forecasts.

Fig. 4.

Structure of CNN-LSTM based main prediction model.

The main prediction model is a data riven deep neural network consisted of a stack of multi-dimensional convolutional layers followed by more LSTM layers and a fully connected output layer that maps the final hidden state to the 24-hour-ahead load sequence. For each data-richness level, early stopping with a patience of several epochs was applied to halt training when the validation loss no longer improved, thereby reducing the risk of overfitting.

Partitioning of training datasets to represent different levels of data sufficiency

This study focuses on evaluating the influence of physics-informed knowledge embedding on forecasting performance under varying levels of data availability. Particular attention is given to the initial deployment phase, where limited training data often causes underfitting in conventional data-driven approaches. The objective is to assess whether embedding theoretical load components can enhance model generalization and accuracy in data-scarce scenarios.

To investigate the effect of knowledge embedding under different degrees of data sufficiency, a resampling technique was employed. Training subsets were extracted from a full-season cooling dataset, ranging from 10% (14 days) to 90% of the original data. This procedure simulated data availability conditions from extreme scarcity to near completeness. For each subset, the initial segment was used to train the model, while the subsequent five days were reserved for testing. All features were normalized using min–max scaling before training.

To ensure robustness and reflect realistic deployment scenarios, nine independent subsets with different starting points were generated for each data proportion. This design emulated alternative model initialization points within the cooling season. As an illustration, Table 5 presents the nine subsets generated under the 10% data-richness condition, serving as a representative case for evaluating model performance in data-scarce scenarios.

Table 5.

Dataset segmentation scheme for simulating different data sufficiency.

Dataset id	Training subset			Test subset
	Start date	End date	Date span	Start date	End date	Date span
1	1	14	14	15	19	5
2	17	30	14	31	35	5
3	33	46	14	47	54	5
4	49	62	14	63	67	5
5	65	78	14	79	83	5
6	81	94	14	95	99	5
7	97	110	14	111	115	5
8	113	126	14	127	131	5
9	129	142	14	143	147	5

Model performance evaluation metrics

Four types of models were trained under varying levels of data richness. For each model, prediction errors were recorded along with their training data richness, multiple independent training and prediction were conducted to mitigate the stochastics come from model training and data set dividing.

To quantitatively evaluate the predictive accuracy and stability of the models, two statistical indicators were employed: the Root Mean Squared Error () and Cohen’s .

Prediction accuracy was measured by³⁸. It measures the average magnitude of the predicting errors of time independent model. It penalizes larger deviations more heavily. It is defined as Eq. 8.

8

Where, is the number of trainings deployed with the same model type and same data richness. Lower RMSE indicates higher prediction accuracy.

Prediction stability was evaluated from the dispersion of forecasting errors and was visualized through boxplots. To further assess potential overfitting, each trained model was used to predict hourly cooling load for five consecutive days, covering both weekdays and weekends to capture variations in occupant behavior.

To compare the performance difference between two models Cohen’s effect size was calculated. In this study, each model’s prediction errors from 10% to 90% data richness were collected in the following sets Eq. (9):

9

Where, , denote two different models; and each sample in represents the obtained under a given data richness level (n = 9). Cohen’s d is then defined as Eq. (10):

10

Where, and are the sample means of sets and ; and is their pooled standard deviation of and .

According to Cohen (2009), indicates negligible performance difference between model and , small effect, medium effect, and large effect³⁹. A positive indicates that model has higher (worse performance), while a negative indicates that model has lower RMSE (better performance).

Results

This section presents the performance of the four forecasting models under different levels of data richness. The focus is on day-ahead hourly prediction under extreme data scarcity; Performance trends when data richness increases; Robustness analysis based on the distribution of prediction errors; A comparison of model training costs.

Day-ahead hourly prediction under extreme data scarcity

To rigorously evaluate the models under extreme data scarcity, all four models summarized in Table 4 were first trained using only 10% of the available data, following the segmentation scheme described in Sect. 2.9. Five consecutive days were selected as an illustrative test window, nine independent training subsets were generated by random sampling to reflect different possible early-stage deployment scenarios. The evaluation is based on hourly prediction error with the day-head full 24-hour trajectory.

Figure 5 presents a representative 5-day forecasting example under 10% data richness, together with the distribution of RMSE for the nine training subsets. The dashed line denotes the measured hourly cooling load, while the colored solid lines indicate the predictions of Models 1–4. Under this challenging setting, the pure data-driven baseline (Model 4) systematically underestimates the morning start-up ramp and afternoon peak loads, and exhibits a noticeable phase lag around abrupt load transitions. In contrast, the hybrid models track both the timing and magnitude of the major load variations more faithfully: Model 1, which incorporates physics-informed fresh-air load components, closely follows the rapid load increase at the start of occupancy periods; Models 2 and 3 maintain better alignment during low-load night-time intervals and shoulder hours.

Fig. 5.

Comparison of model performance for day-ahead hourly cooling load prediction.

(The dashed line denotes the actual observed hourly cooling load, while the solid line represents the average forecast across all model runs. Prediction errors are visualized using error boxplots.)

The boxplots in Fig. 5 summarize the RMSE distributions across the nine random training subsets at 10% data richness, where each RMSE value is calculated from the corresponding hourly prediction errors over the 5-day test window. Specifically, the pure CNN–LSTM baseline (Model 4) exhibits an average RMSE of 0.058, whereas the three hybrid models achieve substantially lower average RMSE values of 0.027, 0.035, and 0.029 for Models 1, 2, and 3 (refer to Table 6). All three hybrid models exhibit lower medians and narrower interquartile ranges than the pure CNN–LSTM baseline, indicating that knowledge embedding improves both the accuracy and the consistency of hourly forecasts across different training segments. Within the group of hybrid models, the differences are much smaller: in this example, Model 1 shows the lowest median RMSE and a relatively compact spread, while Models 2 and 3 follow closely with overlapping interquartile ranges. This suggests that, although all hybrid variants substantially outperform the purely data-driven approach in reconstructing the hour-by-hour load evolution under extreme data scarcity, Model 1 has a slight but consistent edge in terms of both hourly error level and variability.

Table 6.

Comparison of daily and RMSE values for the four models under extreme data scarcity (10% training data).

	Day1	Day2	Day3	Day4	Day5	Averaged RMSE
Model 1 with FA info	0.043	0.041	0.043	0.035	0.035	0.027
Model 2 with FA&ENV info	0.099	0.040	0.053	0.039	0.048	0.035
Model 3 with ENV info	0.039	0.032	0.061	0.033	0.033	0.029
Model 4 pure data-drive	0.150	0.121	0.071	0.041	0.118	0.058

The performance of the models under different levels of data richness

To investigate how model performance evolves with increasing data richness, all four models were trained and evaluated under nine levels of training data availability, from 10% to 90%. For each richness level, nine independent datasets were generated according to the segmentation scheme illustrated in Sect. 2.9, and ten independent training runs were performed per dataset. In all cases, the forecasting task remained day-ahead hourly prediction of the cooling load profile.

The numerical results of this analysis are summarized in Table 7, which reports the mean RMSE of each model at each data-richness level. Across almost all data-richness levels, the three knowledge–data hybrid models (Models 1– Models 3) outperform the pure data-driven baseline (Model 4) in terms of mean RMSE. When averaged over the entire range of data-richness levels, the hybrid models that incorporate fresh air handling (FA) and/or enclosure heat transfer (ENV) load information achieve significantly lower range-averaged RMSE values (RMSE = 0.0192–0.0202) than the purely data-driven baseline Model 4 (averaged RMSE = 0.0461), corresponding to overall improvements of approximately 54.7–58.4%. Table 7 also shows that the least improvement (39.3%) occurs for Model 2 at 10% data richness, whereas the most pronounced improvement (68.7%) is observed for Model 3 at 50% data richness compared with Model 4.

Table 7.

Comparison of prediction errors between the hybrid models and the pure data-driven model.

Model types	Prediction error under different levels of data richness (RMSE)
	10%	20%	30%	40%	50%	60%	70%	80%	90%	mean
Model 1 with FA info	0.0242	0.0239	0.0194	0.0212	0.0179	0.0213	0.0174	0.0200	0.0165	0.0202
Model 2 with FA & ENV info	0.0293	0.0262	0.0228	0.0250	0.0190	0.0176	0.0173	0.0178	0.0133	0.0209
Model 3 with ENV info	0.0286	0.0250	0.0194	0.0223	0.0153	0.0166	0.0155	0.0160	0.0138	0.0192
Model 4 pure data driven (baseline)	0.0483	0.0467	0.0388	0.0541	0.0489	0.0496	0.0476	0.0428	0.0382	0.0461

Under low data richness (10–30%), the improvements are particularly notable, reflecting the fact that the FA and ENV load calculator provide additional structure and guidance when the amount of training data is insufficient to fully capture the variability in the load time series. As the richness level increases, the performance gap generally narrows, yet the proposed hybrid models retain a clear advantage in most cases, indicating that the embedded physical information is beneficial beyond the extreme data-scarce regime.

To further quantify effect size of calculated FA and ENV load hybrid, Table 8 presents the Cohen’s d statistics comparing the prediction errors between pairs of models at each richness level. Negative values indicate that the model in the row outperforms the model in the column (i.e., has lower RMSE). Under low data richness (10–20%), large negative effect sizes are observed for comparisons between the hybrid models and the baseline, confirming that the hybrid approaches achieve substantially better accuracy in the data-scarce regime. At higher richness levels, the effect sizes tend to shrink in magnitude, yet remain mostly favorable to Models 1–3, supporting the conclusion that physics-informed features provide a consistent advantage.

Table 8.

Cohen’s d effect size analysis of prediction errors between the four models.

Model comparison	Cohen’s d	Interpretation
Model 1 vs. Model 2	−0.175	Small effect (d < 0.2); negligible difference
Model 1 vs. Model 3	0.253	Small effect (0.2 < d < 0.5); minor difference
Model 2 vs. Model 3	0.342	Medium effect (0.2 < d < 0.5); minor difference
Model 4 vs. Model 1	6.22 ↑	Very large effect (d > 0.8); highly significant difference
Model 4 vs. Model 2	4.85 ↑	Very large effect (d > 0.8); highly significant difference
Model 4vs. Model 4	5.23 ↑	Very large effect (d > 0.8); highly significant difference

Figure 6 offers a graphical representation of the trends reported in Table 7, by plotting the mean RMSE of each model against data richness. The three hybrid models exhibit relatively smooth and gradually decreasing error curves, indicating that their performance improves in a stable manner as more training data become available. By contrast, the purple curve in Fig. 6, corresponding to Model 4 (the pure CNN–LSTM baseline), displays pronounced non-monotonic fluctuations and reaches a local maximum around a richness level of 0.4. This pattern reveals that the purely data-driven model is highly sensitive to the specific composition of the training subsets: when the selected training windows include abrupt cooling-load spikes caused by sudden weather or operational changes, the baseline network tends to overfit these local patterns and generalizes poorly to the subsequent 5-day test horizon, resulting in higher RMSE.

Fig. 6.

Comparison of prediction errors between the hybrid models and the pure data-driven model.

The local peak of the purple curve at 40% richness coincides with the shortest training time reported for Model 4 in Table 9, suggesting that the baseline CNN–LSTM often converges very quickly to a suboptimal local minimum under these conditions. Such premature convergence is a typical manifestation of training instability and overfitting in deep learning models. In contrast, knowledge embedding via physics-based load components effectively constrains the hypothesis space and regularizes the learning process, leading to more monotonic error reduction and a substantially smaller gap between the best and worst cases across the considered richness levels. This behavior corroborates the claim that the proposed hybrid framework improves both accuracy and stability of day-ahead load forecasting under data scarcity.

Table 9.

Main prediction model training cost (unit: seconds).

Data richness	Model 1 with FA info	Model 2 with façade info	Model 3 with façade info	Model 4 data driven only
0.1	11.2	11.0	10.8	10.9
0.2	11.7	11.9	11.7	11.9
0.3	13.6	12.9	12.6	12.8
0.4	8.9	8.7	8.7	8.7
0.5	14.9	14.2	14.6	14.5
0.6	16.0	15.4	15.9	15.8
0.7	16.9	16.7	16.5	16.6
Average	13.3	13.0	13.0	13.0

Training deployed on a computer with13th Gen Intel Core i5-13500 H CPU, 2.60 GHz, with 32 GB RAM with 7 times of training.

In summary, compared with the purely data-driven Model 4, all three hybrid models deliver lower average RMSE across most data-richness levels and exhibit more regular, predictable performance trends as additional data are provided. This supports the use of physics-informed features for improving the generalization capability of CNN–LSTM-based forecasting models, particularly when training data are limited.

Robustness of hybrid models

Beyond minimizing average RMSE, a practical forecasting model must also exhibit robust performance, characterized by low error variability and resistance to outliers. Figure 7 presents boxplots of prediction errors across the four models and data richness levels, each based on ten independent training runs. Compared with the purely data-driven model, Models 1–3 exhibited narrower error distributions and lower variance, particularly under higher data richness. Their shorter boxes with fewer outliers indicated reduced stochastic errors, while lower medians reflected smaller overall prediction errors. Together, these results demonstrated the greater robustness of the hybrid approaches.

Fig. 7.

Distribution of prediction errors under different data abundances.

The statistical indicators of RMSE distributions further substantiated these findings. Here, Q1 (first quartile) and Q3 (third quartile) represent the 25th and 75th percentiles of the error distribution, and their difference (IQR, interquartile range) measures the spread of the middle 50% of the data. Model 1 with fresh-air information achieved the lowest variance (8.6 × 10⁻⁵) and the narrowest IQR (0.0104, Q1 = 0.0186, Q3 = 0.0289), confirming its stable performance across repeated runs. Model 2 with both fresh-air and envelope information achieved competitive accuracy but showed greater variability (variance = 2.1 × 10⁻⁴; IQR = 0.0254, Q1 = 0.0202, Q3 = 0.0456). Model 3 with envelope information performed at an intermediate level (variance = 1.5 × 10⁻⁴; IQR = 0.0176, Q1 = 0.0221, Q3 = 0.0397). In contrast, the purely data-driven Model 4 displayed the largest variance (9.3 × 10⁻⁴) and the widest IQR (0.0478, Q1 = 0.0295, Q3 = 0.0772), undersco Despite those encouraging researches ring its instability and sensitivity to training randomness.

Compared to purely data-driven model, Models 1–3 exhibited narrower prediction error distributions and lower error variance, particularly under higher data richness. Meanwhile, their shorter boxes with fewer outliers in Fig. 7 indicate the output from these hybrid models have lower stochastic errors. While their lower medians reflect smaller overall prediction errors, the narrow error distribution further indicates greater robustness. Model 3 performed moderately well, whereas Model 4, purely data-driven, displayed the widest error dispersion and the largest number of outliers, reflecting poor stability and greater predictive risk.

Figure 7 further shows that Models 1–3 maintained stable performance even under extreme data scarcity (only 16 days of training data, 10% of the dataset). As data richness increased from 10% to 30%, their RMSE values declined steadily, whereas Model 4 exhibited a non-monotonic trend, with initial error reduction followed by an increase at 40%. This highlighted the superior generalization capability of hybrid models in data-scarce scenarios.

Among the hybrid models, Model 1 (embedding only fresh-air treatment load) showed the best performance under limited data, whereas Model 3 (embedding both fresh-air and envelope loads) outperformed the others as the dataset size increased. Statistical analysis across all data richness levels using Dunn’s test with Bonferroni correction revealed no significant differences among the three hybrid configurations (p > 0.05), suggesting that all of them provided comparable stability.

Model training cost comparison

Training cost is a key concern for the practical deployment of forecasting models, particularly when implemented in on-site micro-processing units where computational resources are restricted. Table 9 summarizes the training time under different levels of data richness, excluding CPU time for physics-based load calculations.

Training time reached a minimum at 40% data richness across all model types. At this level, training time was the shortest, but the prediction accuracy of the purely data-driven model (Model 4) declined significantly. This counterintuitive result was likely caused by premature convergence of the neural network, leading to rapid but incomplete learning, which was reflected in poor predictive performance and irregular error trends.

The meteorological forecasting module, which had a similar CNN–LSTM structure to the main predictive model, incurred comparable CPU cost, and its training could be executed asynchronously with the main predictor. Importantly, the runtime comparison in Table 9 shows that the hybrid models (Models 1–3) require CPU times that are very close to the purely data-driven baseline (Model 4) across data-richness levels. The CPU time results further indicated that the inclusion of prior knowledge does not significantly increase computational cost compared with the purely data-driven model in real deployment, while providing substantially improved accuracy and robustness.

Discussion

Improvement of overfitting by knowledge embedding

The purely data-driven model in this research exhibited an unstable and non-monotonic prediction error trend with the increase of data richness (Fig. 6). This fluctuation suggested a considerable risk of overfitting, particularly evident as the model’s performance deteriorated despite additional training data. Such instability can be partly explained by abrupt cooling load spikes triggered by outdoor temperature increases, which exceeded indoor thermal comfort thresholds and sharply elevated cooling demand in the case studied. This observation is consistent with findings reported in related research⁴⁰. More importantly, it demonstrated the inherent limitations of purely data-driven approaches in reliably capturing abrupt demand transitions under varied operating conditions.

This deficiency may also be attributed to intrinsic characteristics of deep learning architectures, such as LSTM networks, which are known to be sensitive to limited or volatile datasets and prone to overfitting under such conditions⁴¹. These findings further highlighted the importance of embedding physics-based knowledge to enhance prediction stability and generalization performance under data-scarce conditions.

It should be noted that this improvement is not primarily achieved through specialized or aggressively tuned regularization techniques within the data-driven component. In this study, only commonly adopted regularization practices, such as basic weight regularization and early stopping, were employed, without dedicated optimization of regularization hyperparameters. Therefore, the observed reduction in overfitting can be mainly attributed to the knowledge-embedding strategy itself, which introduces additional physically consistent structure into the learning process. This highlights the role of physics-based auxiliary inputs as an implicit and effective form of regularization, particularly valuable during the early operational stages when training data are scarce.

Data accessibility of the knowledge-data hybrid models in practical application

In addition to prediction performance, the implementation cost of knowledge-embedded models, both in terms of data collection and training, must also be considered. Unlike purely data-driven approaches, knowledge–data hybrid models that incorporate physics-based features require additional building and operational information. This increases setup complexity and may influence practical deployment. Therefore, it is necessary to compare the computational burden associated with different types of physics-informed inputs.

Models 1–3 incorporated real-time calculations of fresh-air handling and envelope heat transfer. The fresh-air treatment load, primarily determined by the temperature and humidity difference between indoor and outdoor air together with ventilation rates, is relatively simple to compute. The required parameters are normally available from weather forecast stations and building automation systems, making these inputs readily accessible.

In contrast, calculating envelope heat transfer involves more variables and parameters than computing the fresh-air load. As summarized in Table 10, these often require site-specific measurements or detailed construction documentation, increasing the complexity and limiting scalability.

Table 10.

Complexity for fresh air and envelope heat-transfer load calculations.

Factor in consideration	fresh air treatment load calculation	Envelope heat-transfer load calculation
Related characteristics	Indoor/outdoor environmental parameters (temperature, humidity), ventilation rate	Heat transfer modes (conduction, convection, radiation), material thermal properties, weather, wall/window/roof structure
Data acquisition difficulty	Low (Weather and indoor climate data are readily available)	High (Requires architectural design details, material thermal parameters, solar radiation, etc.)
Computational complexity	Low (Based on simple formulas with transient response characteristics)	High (Involves nonlinear transient heat transfer equations with thermal storage and lag effects)
Adaptability to parameter variation	High (Ventilation load responds approximately linearly to variable air volume control)	Low (Thermal response of the envelope is delayed, and indoor temperature fluctuations introduce calculation errors)

Consequently, Model 1, which relied solely on predictive calculations of fresh-air treatment load, offered a favorable balance between simplicity and predictive accuracy, making it particularly suitable for engineering applications.

Comparison with other models

Table 11 positions the proposed knowledge–data hybrid framework alongside several representative hybrid approaches recently reported in the literature, including physics-informed neural models and decomposition-based machine-learning methods. These works collectively demonstrate that introducing prior structure into data-driven models is a promising way to improve building energy forecasts. Because the underlying datasets, case buildings, climatic conditions and task definitions differ across studies, however, the numerical performance indicators in Table 11 are not directly cross-comparable and should be interpreted as indicative rather than as a strict ranking.

Table 11.

Comparative analysis of this study against recent literature.

Study	Task/data context	Methodology	Key metrics (per original study)	Comparison with this work
This work	Day-ahead hourly prediction; training from performance evaluation coverage as little as 16 days	Physics-informed hybrid: load component priors (fresh air/envelope) + CNN–LSTM	RMSE: 0.0192–0.0202 vs. 0.0461 baseline (↓ 56–58%); Cohen’s d: 4.85–5.23; reduced error dispersion (boxplots)	Demonstrates robust performance under extreme data scarcity; explicitly evaluates deployment feasibility and prior-data acquisition cost
Ma et al., 2025	Small-sample cooling/heating load; hourly & sub-hourly	SN-NODE + RC physics	CVRMSE ≈ 0.30 (hourly)/0.20 (sub-hourly)	Similar small-sample goal; different modeling paradigm (neural ODE)
Huang et al.25	Office building cooling load; short-term prediction	EMD + LSTM + Markov hybrid	RMSE = 4.53; MAPE = 0.84%; RMSE reduction of 40–94% vs. baselines	Strong signal-processing hybrid; no physical prior embedding; no systematic small-sample validation
Kondath et al.17	Day-ahead hourly in tropical commercial buildings	DNN, CNN, RNN, LSTM, Seq2Seq	CVRMSE ≈ 6–10%	No physics embedding; no systematic small-sample evaluation
Di Natale et al.23	Multi-zone thermal dynamics	Physically Consistent Neural Networks (PCNN)	Up to ≈ 40% improvement over RC models; lower overfitting risk	“Structural consistency” physics integration; different task/metric scope
Chaganti et al.38	Model Design-stage simulated dataset	Ensemble ML	R² ≈ 0.997	Not operational-period hourly prediction; no scarce-data scenario
Drgoňa et al.22	Multi-zone thermal dynamics	Physics-constrained deep learning	Emphasis on data efficiency and consistency	Not focused on day-ahead hourly forecasting; conceptually complementary
Salman & Liu41.	General deep learning	Overfitting mechanism analysis	Theoretical and empirical basis for overfitting under small/volatile data	Supports the rationale for physics priors to mitigate overfitting

*The performance metrics (e.g., RMSE, CVRMSE, MAPE) listed in this table are sourced from different original studies, each based on its own case building(s), dataset, climatic context, and forecasting settings. As a result, these values are not directly cross-comparable and are provided only for indicative, qualitative positioning rather than for strict numerical ranking of model performance.

Within this broader context, the present study contributes a complementary hybrid strategy that explicitly embeds physics-based fresh-air and envelope load components into a CNN–LSTM architecture. The results show that such knowledge embedding can deliver simultaneous gains in accuracy and robustness, particularly when only 10–20% of the historical data are available for model training. Across the examined data-richness levels, the proposed hybrid models consistently reduce range-averaged RMSE by roughly 55%−60% relative to the purely data-driven baseline, with improvements reaching nearly 70% at certain intermediate richness levels. Effect size analysis (Cohen’s d = 4.85 to 5.23) confirmed that these gains far exceeded the conventional “large effect” threshold, underscoring their practical significance. At the same time, the error distributions become noticeably narrower and less heavy-tailed, indicating fewer extreme hourly prediction errors and more stable performance under different training subsets.

In this sense, the proposed framework offers a practically deployable option for early-stage, data-scarce deployment in real buildings compared other research. By leveraging readily obtainable physical information about fresh-air and envelope loads, the method provides a structured way to reduce day-ahead cooling-load prediction errors and to stabilize forecast behaviors when measured data are still limited, thereby supporting more reliable operation and control of building energy systems in the initial phases of monitoring and optimization.

Robustness to missing or noisy sensor data in deployment

From a deployment perspective, missing or noisy sensor data are common challenges in real-world building energy systems. Importantly, the proposed framework does not rely on any single indoor or outdoor sensor variable to determine cooling demand. Instead, the physics-based component represents the building’s overall average operational state, rather than conditions in individual rooms, and thus does not depend on measurements from specific indoor temperature or humidity sensors at the zone level.

In practice, short-term data missing in meteorological or operational inputs can be handled using simple and widely adopted preprocessing approaches, such as temporal interpolation or carry-forward schemes, weather forecast data, while high-frequency measurement noise can be mitigated through basic smoothing as it has been ad in this paper. More fundamentally, by embedding physics-based load estimates derived from aggregated thermodynamic relationships, the hybrid model becomes less sensitive to localized sensor noise or occasional data loss. The recurrent structure of the data-driven component further exploits temporal context, which helps stabilize predictions under imperfect sensing conditions. These characteristics support robust deployment without requiring dense or highly reliable room-level sensing infrastructure.

Conclusion

This study proposed and validated an easy-to-deploy forecasting framework that integrates physics-based load calculators into a deep learning model to improve day-ahead building cooling load prediction, particularly under data-scarce conditions.

• Consistent high performance across varying levels of data availability. The case study demonstrates that knowledge–data hybrid models embedding fresh-air and/or envelope heat-transfer load calculators maintain robust predictive performance under different data availability conditions, with particularly pronounced advantages under extreme data scarcity. Compared with purely data-driven models, the proposed hybrid approaches significantly reduce prediction errors—achieving up to a 58.4% improvement over the baseline—while enhancing prediction stability, mitigating overfitting risks, and improving model generalization.

• Recommendation for fresh air-load hybrid in data-scarce cases. Among the three hybrid variants, the configuration embedding the fresh air treatment load predictive calculator is recommended for data-scarce deployment. In the extreme data scarcity condition, this model achieved RMSE = 0.0202 and maintained stable performance across all data richness levels, with minimal computational cost due to the simplicity of fresh air treatment load calculations.

• Support continuous online performance updating. The proposed hybrid framework enable usable cooling load predictions at the early stage of building operation, when historical data are scarce, while maintaining the ability to update or retrain the model as additional data become available.

• Framework extensibility beyond CNN–LSTM. Although the discussed knowledge-data hybrid model leverages a CNN-LSTM, the proposed framework is not limited to this configuration and can be extended into other deep learning frameworks to further improve forecasting performance.

• Limitations and future research prospects. This study is based on detailed data from a case office building with well-documented physical and operational characteristics. As such, the results are best viewed as an initial validation of the proposed knowledge–data hybrid framework. Future work will extend the same modelling workflow to a broader portfolio of buildings with diverse typologies and climatic conditions, in order to further examine and demonstrate the robustness and transferability of the approach.

Beyond extending validation across multiple buildings or campus-scale, future research should also explore the application of the proposed knowledge–data hybrid modelling in buildings with on-site renewable energy systems and active demand-side response or other transient load interventions, thereby broadening its applicability to intelligent building energy management and grid-responsive operational strategies.

Abbreviations

ANN

Artificial neural network

ASHRAE

American society of heating, refrigerating and air-conditioning engineers

CNN

Convolutional neural network

CPU

Central processing unit

CVRMSE

Coefficient of variation of the root mean square error

ENV

Envelope heat-transfer load

FA

Fresh-air load

FAU

Fresh air unit

GMDH

Group method of data handling

HBM

Heat-balance model

HVAC

Heating, ventilation, and air conditioning

IQR

Interquartile range

LSTM

Long short-term memory

MAPE

Mean absolute percentage error

ML

Machine learning

PCA

Principal component analysis

RAM

Random access memory

RMSE

Root mean square error

Seq2Seq

Sequence-to-sequence

SVM

Support vector machine

WMA

Weighted moving average

Author contributions

All authors contributed to the study conception and design. Jian Chen: funding acquisition, project administration, writing—review and editing. Tao Sun: conceptualization, methodology, data analyses, investigation, writing—original draft, drawing of charts, editing. Yuyan Zhang: data process and analyses, Lu Chen and Wenjuan Zhang: data collection, Tongyu Xu and Weijun Gao: supervision, writing—review. Weishang Liu: drawing of charts. All authors have read and agreed to the published this version of the manuscript. All authors read and approved the final manuscript.

Data availability

Data are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.