
==== Front
Patterns (N Y)
Patterns (N Y)
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2666-3899
Elsevier

S2666-3899(24)00162-4
10.1016/j.patter.2024.101029
101029
Article
Modularized neural network incorporating physical priors for future building energy modeling
Jiang Zixin 1
Dong Bing bidong@syr.edu
12∗
1 Department of Mechanical and Aerospace Engineering, Syracuse University, Syracuse, NY 13244, USA
∗ Corresponding author bidong@syr.edu
2 Lead contact

19 7 2024
09 8 2024
19 7 2024
5 8 10102929 2 2024
8 4 2024
27 6 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Summary

Building energy modeling (BEM) is fundamental for achieving optimized energy control, resilient retrofit designs, and sustainable urbanization to mitigate climate change. However, traditional BEM requires detailed building information, expert knowledge, substantial modeling efforts, and customized case-by-case calibrations. This process must be repeated for every building, thereby limiting its scalability. To address these limitations, we developed a modularized neural network incorporating physical priors (ModNN), which is improved by its model structure incorporating heat balance equations, physically consistent model constraints, and data-driven modular design that can allow for multiple-building applications through model sharing and inheritance. We demonstrated its scalability in four cases: load prediction, indoor environment modeling, building retrofitting, and energy optimization. This approach provides guidance for future BEM by incorporating physical priors into data-driven models without extensive modeling efforts, paving the way for large-scale BEM, energy management, retrofit designs, and buildings-to-grid integration.

Graphical abstract

Highlights

• A ModNN is developed

• ModNN is advanced by its physical structure, constraints, and modular design

• ModNN enables load prediction, dynamic modeling, retrofit, and optimal control

• ModNN demonstrates superior accuracy, scalability, efficiency, and consistency

The bigger picture

The building sector accounts for more than 30% of energy consumption and emissions globally. To understand its energy behavior, optimize its energy use, and conduct retrofit analysis, a well-calibrated building energy model is key. However, the scalability of such a physics-based model is a major barrier, and it is challenging to develop a data-driven model that guarantees physical consistency. To fulfill this research gap, this study proposes a modularized neural network incorporating physical priors for future building energy modeling. The authors demonstrate the advantages of incorporating physical knowledge into data-driven models, while also discussing structure of modular design. Such approach facilitates reusable calibrated modulars for different modeling tasks through model sharing and inheritance.

In this paper, the authors develop a ModNN for BEM. By integrating prior physical knowledge into a data-driven model, the proposed approach not only demonstrates superior prediction performance and scalability but also overcomes the generalization issues from typical purely data-driven models. This work provides a plug-and-play data-driven modeling solution for BEM, load prediction, retrofit design, control optimization, and building-to-grid integration.

Keywords

building energy modeling
data-driven
physics-inspired neural network
modularized neural network
model predictive control
load prediction
building retrofit
energy flexibility
Published: July 19, 2024
==== Body
pmcIntroduction

Currently, countries around the world are aiming to achieve climate neutrality by 2050 to slow global warming and address the energy crisis resulting from ever-increasing energy demands and carbon emissions.1 This highlights the critical role of the building and construction sector, which accounts for approximately 37% of greenhouse gas emissions and contributes to more than 34% of the global final energy consumption.2 To achieve this mission, it will be necessary to improve building energy efficiency. Building energy modeling (BEM) is one of the most important and advanced technologies for carbon mitigation in the building sector.3 BEM represents the complex building thermal dynamics based on weather conditions, building physical properties, desired indoor thermal conditions, and prior knowledge of occupants.4 BEM plays a key role in early stage building energy design,5 building energy assessments,6 building environment evaluations,7 building control operations,8 building-to-grid integration,9,10 energy retrofitting,11 and urban energy planning.12 It can facilitate the understanding of building owners, architects, and policy makers of how buildings can be optimally designed, constructed, retrofitted, maintained, and/or operated.13

However, developing a building energy model is non-trivial.8,14 In general, BEM can be classified into three categories15: white box (physical models), gray box (hybrid models), and black box (data-driven models). White box models such as EnergyPlus,16 TRNSYS,17 DeST,18 and Modelica,19 are fully based on solving physical equations.8 As a result, these models offer high-accuracy results but require expert knowledge, detailed building physical information, dedicated modeling efforts,20 and case-by-case calibrations.21 Gray box models fall between white and black box models; a physical based resistance-capacitance model (RC model) structure is established in a state-space form, and the model parameters are then identified using data.22 However, it is difficult for traditional RC models to capture the highly nonlinear behaviors of building systems.23,24 Furthermore, developing an RC model requires extensive experience to identify the correct model structure and involves repeated trial-and-error processes for model tuning and calibration,25,26 especially for multi-zone buildings.22,27,28 To improve the efficiency and effectiveness of BEM, black box models29,30 such as multiple linear regression,31 artificial neural networks,31,32,33 support vector machines,31,32 extreme gradient boosting,30,31 random forests,31 and variants of recurrent neural networks (e.g., long short-term memory [LSTM]30,31,34,35 and gated recurrent unit [GRU]30) can provide opportunities to learn building dynamics from extensive datasets. These models can provide lower engineering costs, less required domain knowledge, and greater adaptability.21 Nevertheless, data-driven models without physical guidance are subject to their limited interoperability and generalization ability,36 and they are highly sensitive to the data quality and quantity.37,38 If the training data do not include all possible operation conditions, it becomes difficult for these models to infer outputs for the out-of-pocket conditions.36 For example, a data-driven BEM might fail to respond correctly to varied climate conditions and set-point changes.

Therefore, to leverage the advantages of both data-driven models (scalability) and physical models (generalization ability), the state-of-the-art BEM combines physical and data-driven models.37,39 We summarize three typical combination methods as follows: (1) adjusting model structures,40,41,42,43 in which researchers adjust neural network structures based on prior knowledge to ground them in underlying physical laws, thereby improving the model generalization ability; (2) adding constraints to model parameters,40,40,42,43,44 in which researchers constrain model parameters to ensure correct model responses to inputs; and (3) customizing loss functions,44,45,46 in which researchers introduce a physical loss term calculated by a physical model to make the network adhere to the underlying physics. Compared with traditional, purely data-driven models, incorporating physical priors into neural networks demonstrates greater accuracy,37,42,43,44,4647 enhanced data effectiveness,37,48 and better generalization ability.37,40,4446,49 However, efficient incorporation of physical knowledge into data-driven models to fully exploit their generalizability remains a challenge. The aforementioned models were specifically designed to solve building control optimization problems only. Consequently, more generalized physics-inspired data-driven models that can simultaneously handle energy prediction, dynamic modeling, retrofitting, and control problems for buildings remain lacking.

Furthermore, with rapid urbanization, the spread of renewable energy sources, and innovative building technologies, BEM is becoming an increasingly complex and large-scale problem. For example, future BEM must not be only multi-scale (equipment, system, building, community, and city)50,51,52 and multi-component (e.g., building, energy storage, renewable generation, electric vehicles),53,54 but also multi-objective (e.g., indoor environment quality, energy efficiency, grid flexibility, carbon neutrality, and resiliency).10,55,56,57 These large-scale challenges present significant barriers for traditional BEM approaches, requiring expensive computational complexity, extensive modeling efforts, repeated model calibration, and difficult model communication owing to the lack of standardized model schemas.52,58,59,60,61 Thus, to develop such complex building models at a large scale, future BEM is expected to (1) learn from data efficiently, (2) respond properly to underlying physical laws, and (3) provide expandable interfaces for future multi-model connections.

Therefore, in this study, we developed a modularized neural network incorporating physical priors for future BEM. The proposed model provides (1) scalable modeling through data-driven methods, (2) reliable responses based on physical constraints, and (3) flexible extension through a modularized model hierarchy typology. A summary of this study is provided below. We first introduce a general concept of our modularized neural network incorporating physical priors (ModNN) model, including the model structure, physical incorporation, and module connections. We then evaluate its performance in four cases: (1) load prediction, (2) indoor environment modeling, (3) building retrofit design, and (4) energy optimization based on a real-world case study.

Results

MoDNN

The incorporation of physical knowledge can be summarized in four key points.

Physics-inspired modularization

(Equation 1) mair·cair·dTairdt=∑j=1NAjqconvection,in,j,θ″+q˙infiltration,θ+q˙system,θ+q˙internal,conv,θ.

Here, mair is the mass of space air (kg), cair is the specific heat of space air (J/ (kg·°C)), Aj is the area of the jth surface (m2), qconvection,in,j,θ″ is the convective heat flux (W2/m), q˙infiltration,θ is the heat transfer due to infiltration (W), q˙system,θ is the heat transfer due to the heating/cooling system (W), and q˙internal,conv,θ is the convective portion of the internal heat gains due to people, lights, or equipment (W).

Inspired by the heat balance equation (Equation 1), we incorporated physical knowledge by modularizing the model structures to construct a heat balance framework. Specifically, we developed different neural network modules to estimate each distinct heat transfer term of the dynamic building system, as shown at the bottom of Figure 1A. For instance, the internal heat gain in buildings typically originates from three sources: (1) metabolic heat generated by occupants, (2) heat from electrical equipment and appliances, and (3) heat from lighting. Thus, we designed an internal heat gain module that uses occupancy, time, and space air temperature as inputs to predict the internal heat gain. Similarly, the heating, ventilation, and air conditioning (HVAC) heating/cooling load module is constructed using parameters such as HVAC power and space air temperature. Unlike traditional neural network structures that process all inputs collectively—feeding all related features (examples are listed at the top of Figure 1A) into one model directly without any guidance, thus losing physical interpretability—our approach refines the model based on heat balance principles. This enables each module within the model to be physically meaningful. The heat balance decomposition details can be found in Supplemental experimental procedures 6. Depending on the different prediction tasks, the selection of input features and the diagram of the module connections will vary; more detailed information can be found in Supplemental experimental procedures 2 and Tables S3 and S4.Figure 1 Diagram of the developed ModNN model structure

A cell is defined as the input vectors, as shown at the top of Figure 1A. They form the basis of neural network modules; module is defined as different neural network modules, as shown at the bottom of (A). These are developed based on the underlying heat transfer laws, which can be used to estimate the heat transfer terms of a dynamic building system; the model is a combination of different modules and cells. It is ready-to-use for different modeling tasks.

(A) Neural network module library; each module is designed to estimate one heat transfer term.

(B) Outline of the encoder-decoder GRU module, where σ represents the sigmoid activate function; tanh represents the hyperbolic tangent activate function; FC represents the fully connected neural network; xt,wt,yt represent the state variable, disturbance, and output, respectively; and ht represents the hidden state.

(C) Neural network module embedding.

Physics-inspired model structure

Inspired by the state-space formulation, we adopted a sequence-to-sequence encoder-decoder model structure, as shown in Figure 1B, to configure the modules mentioned above. In this structure, an encoder is designed to extract historical information, a current cell is designed to measure data from the current time step, and a decoder is designed to predict the system responses based on future inputs. Specifically, the encoder is used to transform the heat stored in building envelopes and furniture owing to thermal inertia into a high-dimensional hidden vector. Subsequently, the current cell updates the measured real-time space air temperature in the current time step to eliminate the accumulated modeling errors. The encoder and current GRU cells act as feature extractors to capture past and current information, thereby providing a more accurate initial state for the decoder. The decoder represents the system dynamics; using the initial state from the encoder and current cell as an initial point, the decoder can generate a sequential system response according to future disturbances and system inputs.

Physics-inspired model constraints

To improve the generalization ability of the proposed ModNN model, we introduce physical consistency constraints to ensure that the model responds appropriately (following the underlying physical laws) to a given model input. For example, the magnitude of the conduction heat flux passing through a wall should decrease as the R value increases, and the indoor air temperature should decrease with an increasing HVAC cooling load, and vice versa for heating. This consistency can be ensured by forcing the partial derivative of the model output to be positive with respect to its input. Detailed information regarding the physical consistency constraints can be found in Supplemental experimental procedures 2.2.

Physics-inspired model assembly

Finally, we developed another fully connected neural network module to connect different modules based on physical typology, as shown in Figure 1C. For example, for dynamic modeling of the space temperature, which is a function of the internal heat gain, heat transfer through the building envelope, HVAC load, and heat transfer from adjacent zones, we can feed the outputs from these modules to a fully connected module to learn their relations. Following the same logic, the modules can be easily changed and flexibly connected for different applications. For example, once an HVAC load module is developed, it can be easily assembled with others to (1) predict the HVAC load with the current module, (2) predict the HVAC energy consumption by combining it with a coefficient of performance (COP) output module, (3) predict the total building load by combining it with an other load output module that estimates the power consumed by lighting or other equipment, and (4) predict the HVAC load for multi-zone buildings by combining single-zone load prediction modules through a graph neural network module (Supplemental experimental procedures 2.4 and Figure S7) that estimates the heat transfer from adjacent zones. Compared with traditional data-driven modeling approaches, which require the development and training of a new model for each prediction task, such modularization offers the potential for model sharing and inheritance. As a result, it becomes significantly easier to connect a new module or fine-tune individual modules, substantially reducing the modeling effort and time. More details about the module selection and connection can be found in Supplemental experimental procedures 2 and Table S4.

Predicting the building HVAC load and energy consumption

To understand the energy prediction performance and modularization efficiency of the proposed ModNN model, we evaluated it using a synthetic dataset generated by EnergyPlus. We selected two Department of Energy prototype buildings located in Denver as case studies: a single-family house and a small five-zone office building; detailed data can be found in Tables S1 and S3 and Figures S1 and S2, . To demonstrate the model performance, we evaluated it in four scenarios: (1) basic performance: the model is trained and tested on the baseline dataset using an 80/20 data split; (2) generalization: the model is trained on the baseline dataset but tested on different datasets generated through (a) adjusting temperature set-points and (b) future weather conditions; (3) modularization: a well-trained HVAC load prediction model is connected to (a) a COP module and (b) a total building load module for HVAC energy consumption and total building load prediction; and (4) scalability: five single-zone modules are connected by a graph neural network to achieve multi-zone HVAC load prediction. The detailed model training and evaluation methods can be found in Supplemental experimental procedures 3. The evaluation matrix used in this case study comprises the coefficient of determination (R2), mean absolute value (MAE) and mean bias error (MBE), which are described in Supplemental experimental procedures 4.6 and Equations S8–S10.

Figures 2A–2F show the overall performance of the model. The X axis represents the EnergyPlus baseline, while the Y axis represents the results predicted by ModNN. Red and blue colors denote the heating and cooling loads, respectively. The overall R2 values range from 0.79 to 0.94, indicating a strong correlation between the predicted and actual values, while the MAE varies from 0.11 kW to 0.73 kW, demonstrating the accuracy of the model for predicting energy loads. Specifically, when applying ModNN for HVAC load prediction in scenario (1), as illustrated in Figure 2A, the well-trained model achieves R2 values of 0.91 for heating and 0.89 for cooling, with MAEs of 0.16 kW and 0.35 kW, respectively. The MBE is −4.02% for heating and −3.5% for cooling.Figure 2 Performance of ModNN for energy prediction tasks

(A) Single-zone HVAC load prediction on the raw testing set.

(B) Single-zone HVAC load prediction with adjusted temperature set-points.

(C) Single-zone HVAC load prediction with the future weather dataset.

(D) Single-zone HVAC energy consumption prediction embedded with COP module.

(E) Total building load prediction embedded with the total building load module.

(F) Multi-zone HVAC load prediction.

The x and y axes represent groundtruth and prediction results from ModNN, repsectively. The violin plot presents the prediction error distribution. The dashed lines from bottom to top indicate the first quartile (Q1), median, and third quartile (Q3), respectively. The error bars describe the corresponding standard deviation.

A similar accuracy level, as shown in Figure 2B, is observed for predictions of the HVAC load under varied temperature set-points (we increased the set-point by 2°C in this scenario). This highlights the potential for exploring the influence of temperature set-points on HVAC loads and evaluating different building control strategies. In addition, the model can be easily converted into a control-oriented model, where the temperature set-point serves as the control input and the HVAC load serves as the control output, which can be used for energy optimization and management. For example, as shown in Figure S14B, we increase the control input (set-point) by 2°C, and the control output (HVAC load) decreases accordingly.

Figure 2C illustrates the performance of the model under future weather conditions using a typical meteorological year from 2085 to 2095 in Denver.62 (Detailed data information can be found in Supplemental experimental procedures 1.3, Table S2, and Figure S5.) Owing to global warming, a significant increase in the cooling demand is observed, primarily concentrated in the lower left quadrant of the graph, whereas the heating load demand decreases. These results offer valuable insights for resilience-related studies on future building energy consumption.

Figures 2D and 2E illustrate the performance of the model for HVAC energy and total building load predictions, respectively. First, we trained an HVAC load prediction model on the baseline dataset and then fixed all of the trained model parameters. Subsequently, we connected this well-trained HVAC load prediction module to a COP module for HVAC energy prediction and a total load prediction module for whole building load prediction. The detailed model architecture is elaborated in Supplemental experimental procedures 2 (Figure S6), and the training process is outlined in Supplemental experimental procedures 3 (Figures S8–S10 and Table S5). Compared with the HVAC load prediction, the R2 values of the energy and total load predictions are similar, ranging from 0.85 to 0.91. This indicates a high model interpretability, in which more than 85% of the variation in the output variables can be explained by the input variables. However, a slight increase in the MAE and a uniform error distribution are observed. This may be caused by the uncertainty introduced by the COP module and total building load module. For instance, unlike HVAC load prediction, which primarily focuses on establishing relationships between weather conditions, building dynamics, and space set-points, predicting the total building load is significantly more complex. This complexity arises from the incorporation of occupancy behavior, a factor characterized by its stochastic nature, and is thus more challenging to predict accurately. As a result, the MAE increases, but the error is uniformly distributed, which means that the model does not systematically overestimate or underestimate the predicted values; it maintains a robust performance over varying conditions. This hierarchical modularization model structure not only simplifies the process of BEM development, but also significantly reduces the effort required for model calibration. Compared with traditional data-driven models that behave as a so-called black box, making it difficult to understand the underlying reason why a certain prediction is made,63 our approach allows for a detailed examination of the internal structure of the model and offers clear physical insights into its performance. For example, when the total building load prediction fails, it is extremely difficult to diagnose the model using a traditional data-driven approach.64 However, owing to the modular structure of our approach, it is easier to investigate the root causes of errors in the HVAC load prediction module or other aspects of the load prediction modules.

Figure 2F shows the results of the multi-zone load prediction, with MAE of 0.23 kW and 0.26 kW for cooling and heating loads, respectively. Considering the greater internal heat gains in office buildings, the cooling demand in such buildings may be higher than the heating demand, which is why the prediction results are concentrated at the bottom left corner of Figure 2F. In this model, each single-zone module is treated as a node, which are interconnected through an adjacency matrix to describe their topological relationships. This modular approach also provides future application potential for multi-scale, multi-component building modeling. By leveraging graph neural networks, the model can effectively scale from single buildings to a larger scale by incorporating the appropriate connection adjacency matrices. Furthermore, a one-month evaluation is presented in detail in Supplemental experimental procedures 4.4 and Figures S1A–S1F. The prediction results from the ModNN model are consistent with the baseline across various scenarios, underscoring its superior predictive performance.

Predicting building indoor environmental dynamics

Another important application of BEM is the modeling of building indoor environmental dynamics. In this case study, we use the temperature and humidity as examples to evaluate the performance of our model. The inputs and model structure are detailed in Supplemental experimental procedures 2 and Table S4. In contrast to the load prediction where we use the current input, xt, to estimate the load/energy consumption, yt, within the same time step, building dynamic modeling is more focused on forecasting the future output, yt, based on the future disturbance, wt+T, and system input, ut+T. The time resolution is 15 min in this study, and we evaluate the model performance with prediction horizons from 15 min to 24 h on a 1-month dataset. The model consists of four neural network modules: an external heat transfer module, internal heat gain module, HVAC load module, and adjacent heat transfer module (the latter is only for multi-zone problems).

First, the results of the one-week humidity and temperature predictions in a single zone and one-week multi-zone temperature predictions are presented in Figures 3A–3C, respectively. The red lines represent the space temperature/humidity predictions from the ModNN model, with each line describing the temperature/humidity trajectory 1 day ahead (96 time steps), updated every 15 min. The green line indicates the ground truth results from EnergyPlus. It is evident that our predictions align closely with the ground truth. The discrepancies may be due to two factors: (1) model training errors and (2) the use of a synthetic dataset generated by EnergyPlus as the baseline. The EnergyPlus model assumes that space temperatures can maintain the set-point perfectly, given sufficient system capacity. This assumption may lead to some mismatches between the predicted results (more realistic) and the baseline. The monthly humidity prediction error across various prediction horizons is shown in Figure 3D, where the average MAE ranges from 0.0002 to 0.0005 kg water/kg air.Figure 3 Performance of ModNN for building indoor environmental dynamics prediction tasks

(A) One-week single-zone humidity prediction.

(B) One-week single-zone temperature prediction.

(C) One-week multi-zone temperature prediction.

(D) Error distribution of the one-month single-zone humidity prediction.

(E) Error distribution of the 1-month single-zone temperature prediction.

(F) Error distribution of the 1-month single-zone temperature prediction under future weather conditions.

(G) Error distribution of the 1-month multi-zone temperature prediction.

The red lines represent the space temperature/humidity predictions from ModNN, and the green line indicates the ground truth results. The violin plot presents the prediction error distribution. The central line represents the median, the box edges denote the first quartile (Q1) and third quartile (Q3), and the error bars describe the corresponding standard deviation.

The monthly temperature prediction errors are illustrated in Figure 3E. The average MAE increases slightly from 0.39°C to 0.41°C, and we do not observe any significant increase in the accumulated prediction error as the prediction horizon65,66 increases. This can be explained by three factors: (1) the incorporated physical laws enhance the model robustness; (2) it is assumed that future disturbances, such as ambient temperature and solar radiation, can be perfectly predicted, although real-world implementations may introduce additional uncertainties; and (3) disturbance such as the internal heat gain in EnergyPlus are schedule based, and thus they can be learned by the time feature. However, real-world datasets may contain more stochastic disturbances, potentially increasing the accumulated prediction errors, as discussed later in the “A real-world case study” section. The overall results indicate a good and robust performance of the ModNN model.

To further assess its generalization capability, the trained model was frozen and evaluated using the future weather dataset mentioned in the previous section. The overall error distribution, shown in Figure 3F, reveals a slight increase in the MAE from 0.51°C to 0.61°C compared with Figure 3E; however, the model still demonstrates robust performance. Additionally, the temperature prediction accuracy of the model for a multi-zone office building is shown in Figure 3G. Here, each zone precisely predicts the space temperature over a one-month evaluation period, with overall MAEs ranging from 0.27°C to 0.39°C. Zone 1 exhibits a slightly higher prediction error, possibly because of two factors: (1) its position as the central zone adjacent to all other zones, which could introduce more uncertainties, and (2) its different schedule settings leading to different temperature set-points. Zone 2 also presents a slightly higher prediction error because it faces south and is thus significantly affected by solar radiation.

We also demonstrated the long-term prediction capacity of ModNN, as shown in Figure S15, which depicts the results of the 7-day-ahead prediction evaluated on a 2-week basis. At each time step, ModNN predicts the indoor temperature for the next 7 × 24 h (2,520 time steps). Notably, the prediction demonstrates high accuracy and can be used for long-term dynamic building modeling. For example, the model could be used to evaluate the indoor environment of a building under long-term power outage events.

Building retrofitting using ModNN

In this section, we apply the proposed ModNN model to assess building energy retrofits. First, we decompose the entire building load prediction model into several heat balance modules, with each module responsible for predicting the heat flux resulting from convection, conduction, and radiation. Second, we incorporate physical consistency constraints into the model parameters. For instance, during the heating season, the derivative of the building heating load with respect to the window solar heat gain coefficient (SHGC) should be negative. This is because a higher SHGC indicates that more solar heat is transmitted through the window into space, leading to an increase in the zone temperature and a consequent reduction in the heating demand. Third, after completing the model training, we freeze all of the model parameters and adjust only the weight of the embedding layer based on the retrofitting measures to predict the energy performance for a post-retrofit case. For instance, in the case of a wall retrofit in which we focus exclusively on adding insulation, the conductive heat flux is expected to decrease in proportion to the increase in the insulation R value. Accordingly, we adjust the weight of the corresponding neural network layer to reflect this R value proportion, thereby predicting the energy consumption after the retrofit.

We assess the model performance based on three common retrofitting measures: window SHGC retrofit, window U value retrofit, and wall R value retrofit, as illustrated in Figures 4A–4C. In the results, the red bar indicates the outcomes from a neural network of the same structure (BaseNN) but lacking physical consistency (traditional data-driven model), the green bar represents the ModNN model (our model), and the blue bar shows the reference results from EnergyPlus (baseline). It is observed that increasing the window SHGC and wall insulation thickness leads to a decrease in the heating demand by up to 22%. Conversely, an increase in the window U value results in a heating demand increase of up to 53%. The ModNN model, which incorporates physical consistency, consistently follows the trend given by the EnergyPlus model. In contrast, the model without physical consistency frequently produces trends that are opposite to the expected outcomes. For instance, increasing the SHGC values from 0.4 to 0.5 should multiply the heat flux through the window by a factor of 1.25 during the retrofit prediction process. By constraining the derivative of the HVAC load relative to the heat flux through the window term, a higher SHGC will lead to predictions of reduced HVAC load consumption. Without this constraint, the model output fails to adhere to this logical trend. We also observe a trade-off between model interpretability and performance,49,64,67 as shown in Figure 4D. With the incorporation of physical consistency constraints, a minor performance reduction is observed when the BaseNN model is tested on its baseline testing dataset (SHGC = 0.35); it exhibits slightly better performance compared with ModNN. This is because the physical constraint narrows the solution space, leading to a marginally lower accuracy compared with models without such constraints. However, models with greater prediction accuracies may result in less trustworthy predictions.49 When tested outside the baseline dataset (SHGC ≠ 0.35), ModNN maintains reasonable predictions; however, BaseNN provides opposite predictions.Figure 4 Performance of ModNN for building energy retrofitting prediction tasks

(A) HVAC load difference under a window SHGC retrofit.

(B) HVAC load difference under a window U-value retrofit.

(C) HVAC load difference under a wall R value retrofit.

(D) Monthly HVAC load under a window SHGC retrofit.

(E) Comparison of the normalized conductive heat flux among the three models.

The red bar indicates the results from BaseNN, the green bar represents ModNN, and the blue bar shows the baseline results. The boxplot displays the distribution of normalized heat conduction flux. The central box represents the interquartile range (from the first quartile to the third quartile), the median is indicated by a line within the box, and the error bars denote the corresponding standard deviation.

Another finding is that the mismatch between the ModNN predictions and the EnergyPlus results increases when the retrofit measures increasingly diverge from the baseline. It is understandable that, as the testing data move further outside the training pocket, they will become more difficult to achieve accurate predictions. Figure 4E shows a comparison between the predicted conductive heat flux and the conductive heat flux calculated using EnergyPlus. With the implementation of physical consistency, the blue line closely aligns with the green line, and both maintain the same trend. This alignment indicates that the module can accurately predict the conductive heat flux as intended. In contrast, the red line represents the model without physical consistency, whose outputs fail to closely follow the EnergyPlus results. The most recent data-driven retrofit study relied on synthetic data generated by surrogate models.49 These models use detailed building characteristics, such as envelope material properties, stories, and orientation, to discover their relationship with the building energy consumption (i.e., they establish mapping from the building meta-data to building energy consumption). However, collecting such a comprehensive dataset for model training is challenging, and to date, there has not been a reported instance of a data-driven building retrofit model based on operational data.49 This study (which establishes mapping from building operation data to building energy consumption) addresses this gap by decomposing the heat balance calculations into individual neural network modules. By incorporating physical constraints, each module can accurately predict the trends in its respective heat transfer terms. Furthermore, post-retrofit energy performance is predicted by adjusting the weights of the neural network layers. This approach offers a solution for future building retrofit analysis using building operation data, instead of establishing a complex physics-based BEM or collecting detailed building characteristic data, which is difficult and time-consuming in real-world applications.

A real-world case study

We also evaluated the performance of our model using a real-world dataset collected from a student office from April to August 2021. The office had an area of 6 m × 15 m and was occupied by up to 12 people. Detailed information on this dataset can be found in Supplemental experimental procedures 1 (Figure S4). The training process is detailed in Supplemental experimental procedures 3.4. First, we evaluate the performance of the indoor environmental dynamics by comparing our ModNN model with a traditional 3R2C model (Supplemental experimental procedures 7, Figure S17), as shown in Figure 5A. The prediction horizon for this case study is 24 h (96 time steps), and for each time step, we plot the indoor air temperatures (96 steps) predicted by ModNN and 3R2C in red and blue, respectively. The green line represents the measured indoor air temperature. The ModNN prediction results closely align with the measured temperatures, with an average MAE of 0.43°C and a mean absolute percentage error (MAPE) of 1.93%. It outperforms the 3R2C model with an average MAE of 0.94°C and a MAPE of 3.9%. It is worth noting that ModNN shows poor performance with MAE of 1.52°C and MAPE of 5.74% in the first 3 days owing to the limited training data at the beginning of the simulation. Another reason is that a disruptive event (HVAC system shut down caused by a fire alarm) occurred on July 29 (Figure S13A), which deactivated the HVAC system and caused the indoor air temperature to fluctuate unpredictably. The 3R2C model performed slightly better, with an MAE of 1.08°C and MAPE of 4.3%, in this period with same limited training data. Interestingly, during other disruptive events (fire alarm) on July 13 and August 12 (Figures S13B and S13C), we observe significant performance improvement in ModNN, with a decrease in the MAE to 1.56°C and a decrease of the MAPE of 6.5%. This enhancement is due to the exposure of the model to a more diverse dataset. However, the performance of the 3R2C model does not improve significantly, and the temperature remains unbounded after occurrence of the disruptive event. The third HVAC failure occurred on Aug 11, where ModNN exhibits an MAE of 2.8°C and MAPE of 10.7%. Nevertheless, there is no notable difference between the latter two HVAC failures. It seems that increasing the size of the training data do not significantly improve the performance of the ModNN model in handling disruptive events beyond a certain point. Compared with the 3R2C model, ModNN can predict the space temperature more accurately. Particularly after the HVAC failure event, the space temperature from the ModNN model quickly reaches the set-point, while the space temperature from the 3R2C model has a temperature violation of 5°C.Figure 5 Performance of ModNN on a real-world dataset

(A) The 46-day space air temperature predicted by ModNN compare with the RC model and measurement results.

(B) Model performance with different prediction horizons.

(C) Impact of the encoder length on model performance.

(D) Impact of the model training data size on model performance.

(E) Peak load shift energy optimization result.

(F) Impact of the balance parameter for multi-objective optimization.

(G) Summary of the energy optimization results, where FF is the flexibility factor (values close to one indicate optimal load-shifting performance), Temp indicates temperature violations, Energy is the percentage energy savings, Cost is the percentage cost savings, and Peak is the ratio of peak load reduction.

The red line indicates the results from ModNN, the green line represents the baseline, and the blue line shows the 3R2C results. The boxplots display the distribution of the compared results, including the interquartile range (from the first quartile to the third quartile) shown in the central box, with the median indicated by a line within the box, and the error bars denote the corresponding standard deviation.

In addition, we conducted a sensitivity analysis of the impact of the encoder length, decoder length, and training data size on the model performance, as shown in Figures 5B–5D, respectively. We observe a decline in the model performance as the prediction horizon increases, which might be caused by the cumulative error and the uncertainty introduced by disturbance variables. The incorporation of an encoder notably enhances the model performance. However, extending the encoder length does not yield further improvements in model performance, and the best encoder length for this case study is 4 h. This is likely owing to the small size of the space being modeled, for which a shorter encoder is more effective for stabilizing the initial conditions. Regarding the size of the training dataset, the performance of the model shows rapid convergence within less than 10 days of training, indicating that only a limited dataset is necessary for effective training. The MAE is higher during the first disruptive event but decreases in subsequent disruptive events, corroborating our findings.

The physical consistency of the model was evaluated, as shown in Figures S11 and S12, where we systematically adjusted the control inputs and disturbances either upward or downward, starting from each time step. Figure S11A shows the indoor space temperature response to both the maximum and minimum cooling loads. When the HVAC system is shut off at various time steps, the indoor temperature begins to rise immediately, with a more rapid rate of increase during midday when the outdoor conditions are warmer. Conversely, when the HVAC system operates at full capacity, the space temperature begins to decrease. However, it never drops below 20°C, as each variable air volume (VAV) box is equipped with a reheat coil, and the minimum threshold is 20°C. This is an interesting finding because we did not give the ModNN model any information about the reheat coil, but the model could find this guarantee itself through the physical constraints. Figure S11B displays the sanity check results for solar radiation; for the maximal solar radiation input, the space temperature increases accordingly. However, for the minimal solar radiation input, the space temperature only decreases during the daytime because there is no solar radiation at night, which means that it can no longer decrease at night. These scenarios underscore the capability of the ModNN to grasp inherent physical principles. Figure S11C shows the sanity check results for the outdoor air temperature, highlighting a consistent correlation between the indoor space temperature and fluctuations in the outdoor air temperature.

Finally, we present the control performance evaluation results in Figures 5E and 5G, in which the objective was to minimize the energy cost according to a time of use price while maintaining thermal comfort. A detailed description of the control optimization setup is provided in Supplemental experimental procedures 5 (Figure S16). We designed another neural network similar to ModNN to address the energy optimization problem. Given that the proposed ModNN is highly nonlinear and challenging to formulate explicitly, we used a well-defined stochastic gradient descent solver from PyTorch to solve our optimization problem inexplicitly. This is also known as a differentiable predictive control strategy inspired by ref. 47 In summary, we developed another control law neural network and adjusted its loss function to convert the original problem into an unconstrained problem. The neural network was then trained to minimize the loss function, which is equivalent to solving the original optimization problem. Finally, the well-trained control law neural network could predict the optimal control policy based on given state variables and future disturbances. On average, an impressive energy reduction of 43.3% was achieved. A significant load shift can be observed during peak hours by pre-cooling during off-peak hours. Approximately 55.3%–95.1% of the peak load can be reduced, and energy costs can be reduced by 28.1%–79%. Additionally, the flexibility factor improved, from 0.56 to 0.69 on average, indicating more focused energy consumption during off-peak hours. However, a compromise in thermal comfort is noted, with an unmet degree hour of 2.41 and an average maximum temperature violation of 1.6°C. This compromise arises from the multi-objective function of the optimization, which balances thermal comfort against price signals. Further evaluation of the impact of multi-objective balance parameters revealed that lower parameters, prioritizing cost savings, resulted in a load reduction of more than 90%, but higher temperature violations. Conversely, parameters set above 104 led to temperature violations stabilizing around 2°C, with peak load reductions approximating 78%. This indicates an inherent trade-off between optimizing cost savings and maintaining thermal comfort. The detailed optimization methodology can be found in Supplemental experimental procedures 5.

Discussion

Comparative analysis of different BEMs

A modularized ModNN is developed for load prediction, building dynamic modeling, advanced building control, and retrofit design analysis, providing a scalable solution for data-driven large-scale BEM and optimization. As summarized in Table 1, we compared ModNN with three commonly used BEMs based on the following aspects.(1) Modeling effort. White box modeling often requires the greatest effort to develop because it requires a large amount of upfront information for model inputs. For example, a detailed building geometry model with a layer-by-layer building envelope is required to accurately predict the building temperature behaviors and subsystem operations. In contrast, gray box models use simplified physical models, such as RC models, with parameters identified by measured data, thereby reducing the modeling effort compared with white box models. Nonetheless, both models require expert knowledge and extensive trial-and-error (potentially weeks to months) for model tuning and calibration. However, black box models and ModNN are data-driven, which often allows direct model training based on measured data without the effort of modelers, and the model structure and data-driven knowledge can be transferred between buildings without necessitating model redevelopment.

(2) Data requirements. White box models require comprehensive building metadata such as the layout, orientation, and detailed envelope properties, which are often challenging to obtain in practical applications. Gray box models can identify the model parameters from the operation data; however, they still require metadata to determine the appropriate RC structure of the simplified physical model, e.g., the number of Rs (resistors) and Cs (capacitors). Black box models and ModNN require only building operation data, which significantly simplifies the metadata collection efforts.

(3) Computational efficiency. White box models, which rely on forward calculations, do not require model training but can have increased computational time requirements depending on the geometric complexity of the building and the simulation length. In real-world applications, the collection of detailed building data is extremely difficult. Consequently, sets of parameters of white box models also require calibration to align with operational data, significantly increasing the modeling effort. The computational costs of gray box models depend on their structure; for instance, the linear 3R2C model in this study required less than 1 s of CPU time for training and testing. However, tuning an RC model is much more time consuming because of its sensitivity to initial values and structure, which requires an expert background and repeated trial-and-error. ModNN, generally requires more training time than black box models, owing to its complex structure. Despite this, both the black box models and ModNN can reduce the computation time by more than 90% compared with white box models, demonstrating significant efficiency potential for large-scale applications.

(4) Model complexity. The complexities of white box models and ModNN are greater than those of simplified RC models and purely data-driven models owing to the integration of physical principles. While these incorporated physical principles increase the computational costs, they also enhance the physical consistency. However, the trade-off between physical consistency and model complexity warrants further investigation. It is important to determine the appropriate situations for incorporating physics into data-driven models, the level of physics to be integrated, and various methods to integrate physics more efficiently.

(5) Physical consistency. White box and gray box models can maintain physical consistency as they are developed based on heat transfer equations. Similarly, ModNN ensures physical consistency for specific inputs, such as HVAC power. Traditional black box models, however, do not guarantee physical consistency, and good prediction does not mean a good response. For example, as shown in Figures 4A–4C, ModNN accurately captures the differences in energy consumption due to retrofitting, whereas traditional black box models do not. This observation is corroborated by the data in Figures 6A and 6B, where both ModNN and LSTM accurately predict the space temperature over 30 days of training. However, the LSTM model does not react correctly when the HVAC power is turned off at 8:00 a.m., as illustrated in Figure 6B. Despite the HVAC power being off, the temperature, indicated by the red line, incorrectly continues to drop instead of rising. This error shows the model’s lack of physical consistency, supporting findings from previous research.40,42,43 Nonetheless, with 60 days of training data, LSTM can respond appropriately to HVAC power, as shown in Figure 6D, suggesting that the physical consistency of a traditional black box model could largely depend on the training data quality, which ModNN can always satisfy without these data sensitivities. This remains a limitation of the current study and is a potential area for future research.Figure 6 Sanity checks of ModNN and LSTM

(A) ModNN with 30 days of training data.

(B) LSTM with 30 days of training data.

(C) ModNN with 60 days of training data.

(D) LSTM with 60 days of training data.

(6) Applications. White box models benefit from their physically structured framework, enabling the direct calculation of building loads and assessment of retrofit performance. However, EnergyPlus, a typical white box model, assumes that the temperatures can be perfectly maintained by a sufficient HVAC capacity. This assumption could lead to a mismatch with real-world temperature distributions. Moreover, its complex model structure cannot be directly used for control optimization; instead, it requires non-gradient solvers such as reinforcement learning or heuristic algorithms. Gray box models are control-oriented and can predict space temperatures based on control inputs and environmental disturbances. Their simplified structure makes them compatible with various optimization solvers. In contrast, black box models are well suited for load prediction and dynamic modeling. However, they cannot be used to predict the post-retrofit energy consumption from operational data because of the lack of physical consistency guarantees. ModNN stands out by leveraging its physical consistency, making it suitable for load prediction and dynamic modeling. It is also capable of handling basic retrofitting designs.

(7) Scalability. ModNN demonstrates superior scalability potential by leveraging its modularized structure, which can be adopted for various applications using different plug-and-play modules. This modular approach provides a future collaborative framework for various stakeholders. For instance, as discussed above in the “Predicting the building HVAC load and energy consumption” section, the HVAC load module can be integrated with a COP module to compute the HVAC energy consumption, effectively bridging zone-level models with equipment-level analysis. Similarly, individual zone models can be connected using a graph neural network, simplifying urban-scale BEM. Furthermore, the modular typology provides the potential to connect not only building models, but also distributed energy models, such as distributed energy resources and generation systems, in future work, which will significantly enhance the scalability to buildings-to-grid communities.

Table 1 Comparison of different BEM methods for a single-zone building

Modela	Modeling effort	Data requirement	Computational efficiencyb	Model complexity	Physical consistency	Applicationsc	
Training	Testing	L	T	R	C	
White box	high	metadatad	\	15.2 s
100.1 s	high	consistent	⁝	:	⁝	.	
Gray box	medium	metadata	0.08 s	0.02 s
0.24 s	low	consistent	\	⁝	\	⁝	
Black box	low	operation data	5 s	0.21 s
2.3 s	mid	no guarantee	⁝	⁝	\	\	
ModNN	low	operation data	223 s	0.72 s
6.6 s	high	consistent	⁝	⁝	:	:	
a We used the EnergyPlus, RC, and LSTM models to represent white box, gray box, and black box models, respectively.

b The Department of Energy single-family house dataset was used to evaluate the computational efficiency; the gray box model is trained on 3 days of data, the black box and ModNN models are trained on 1-month of data, and the white box model does not need training data; the models are tested on 1 month of data (top) and 1 year of data (bottom).

c L refers to load calculation, T refers to temperature calculation, R refers to retrofitting, and C refers to control optimization. ⁝indicates the difficulty, where ⁝means easy and . means hard; / means not available.

d Metadata include both operational and static data (such as building geometric data and envelope information).

Practical implications and potential applications

In this section, we summarize the potential applications of our model in real-world scenarios.(1) Building energy system control optimization. ModNN is a data-driven model characterized by its physical consistency. It can predict building dynamics (e.g., indoor temperature and humidity) over a period of time in the future using a given control input; thus, it can be used for model-based predictive control to automatically adjust control strategies to achieve indoor thermal comfort while reducing energy consumption.

(2) Buildings-to-grid integration. During the planning stage, the predicted building load from ModNN can be used to determine the appropriate size of distributed energy resources (DERs), such as local generation systems, energy storage systems, and HVAC systems. During the real-time operation stage, it enables real-time control of DERs and building systems to provide flexibility to the grid.

(3) Building retrofit design. Building retrofitting involves modifying existing buildings to improve energy efficiency or decrease energy demand. A building energy model is required to accurately predict building performance, even when dealing with unseen data. ModNN demonstrates robust generalization capabilities, such as predicting building performance under future weather conditions and various building retrofit measures, which provides a potential solution for large-scale building retrofit analyses. Additionally, ModNN enables the prediction of building performance from building operation data, which significantly reduces data collection costs.

(4) Digital twin and urban-scale BEM. Benefiting from the modularized model typology, the module connections of ModNN can easily be adjusted to adapt to the interests of different stakeholders, which can contribute to building digitalization and its application in energy monitoring and management. In addition, we demonstrate that a single model can be integrated using a graph neural network to construct a multi-zone model. This approach provides a potential solution for urban-scale BEM through a combination of ModNN models.

(5) Policy formation and energy code revisions. By estimating the energy consumption across different building types, control strategies, and weather conditions, ModNN enables policy makers to better understand building energy demands on a large scale. This supports urban building energy management and planning as well as the development of energy efficiency measures and energy codes.

Limitations and future studies

Finally, we summarize the limitations of this study and directions for future studies.(1) Additional model validation studies based on different datasets from various building types should be explored. These datasets should include different envelope properties, temperature set-points, and climate zones to demonstrate the robustness and generalizability of the model.

(2) Customized optimization methods for ModNN will be investigated in future studies. Unlike traditional data-driven models, ModNN has more complex structures, which makes solving optimization problems challenging. It is important to discover efficient ways to address energy optimization problems integrated with highly complex neural networks. Furthermore, the trade-off between model complexity and optimization efficiency remains an open question.

(3) Future studies should test more complex building retrofitting measures using the proposed model. This will require the development of various heat balance neural network modules and their integration with ModNN.

(4) A data-driven modular library should be developed and evaluated in future studies, incorporating multi-scale (equipment, system, building, community, and city) and multi-component (building, energy storage, renewable generation, electric vehicles, etc.) parameters to test the scalability of the proposed modularized model structure.

Conclusion

A modularized encoder–decoder neural network incorporating heat balance principles was successfully developed for BEM. By incorporating physical constraints, the proposed ModNN model demonstrated superior performance in terms of load prediction, dynamic building modeling, advanced building control, and retrofit design analysis. By leveraging the advantages of the data-driven approach, ModNN offers a scalable solution for future BEM without substantial modeling efforts and customized case-by-case calibrations. In addition, the incorporation of physical knowledge not only ensures correct model responses but also enhances the ability of the model to generalize across different user cases. Furthermore, the modularized hierarchical structure of the model introduces plug-and-play capabilities, paving the way for advanced multi-scale, multi-component, and multi-task BEM, marking a leap forward in the field.

Experimental procedures

Resource availability

Lead contact

Further information and requests for resources should be directed to and will be fulfilled by the Lead Contact, Bing Dong (bidong@syr.edu).

Materials availability

This study did not generate new unique reagents.

Data and code availability

Our source code is available at GitHub (https://github.com/Bugs-Owner/Modularized-Neural-Network-Incorporating-Physical-Priors-for-Future-Building-Energy-Modeling.git) and has been archived at Zenodo.68

Supplemental information

Document S1. Figures S1‒S7, Tables S1–S5, and Supplemental experimental procedures 1–7

Document S2. Article plus supplemental information

Acknowledgments

This work was supported by the U.S. National Science Foundation (Award No. 1949372 ).

Author contributions

Conceptualization: Z.J. and B.D.; methodology: Z.J. and B.D.; investigation: Z.J.; software: Z.J.; visualization: Z.J.; writing – original draft: Z.J. and B.D.; writing – review and editing: Z.J. and B.D.; funding acquisition: B.D.; project administration: B.D.; supervision: B.D..

Declaration of interests

The authors declare no competing interests.

Supplemental information can be found online at https://doi.org/10.1016/j.patter.2024.101029.
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