Predicting Unsteady Indoor Temperature Distributions by POD-DNN

. In this study, to predict unsteady temperature distributions, POD-DNN was utilized, where DNN was trained to predicted coefficients of POMs. Two strategies, flatten POD-DNN and nested POD-DNN were compared. The flatten POD-DNN provided high accuracy if training data is sufficient, but otherwise very inaccurate. The nested POD-DNN roughly predicted the development of temperature fields even training data was small. The results showed their different sensitivities to the training data size.


Introduction
For efficient energy consumption of buildings, predicting the detailed indoor thermal environment, such as temperature and velocity distributions, is essential. Computational fluid dynamics (CFD) has been widely utilized to simulate the detailed building environment. But the time cost of CFD is expensive.
Some improved methods have been developed to reduce the computing time of CFD, such as FFD [1] and rCFD [2]. And recently, machine learning (ML) is introduced to speed up the prediction for its ability to build nonlinear relationships between boundary conditions and the indoor thermal environment rapidly [3]. The basic practice is training a deep neural network (DNN) by the results of CFD, and the well-trained DNN is hoped to predict temperature and velocity distributions of cases with other boundary conditions. But, as a data-driven method, DNN only fits data without understanding the flow structure of airflow development. Ottino et al. [4] introduced an ML-based reduced-order model (ROM) approach for indoor environment prediction. In that paper, proper orthogonal decomposition (POD) was used to extract an optimal set of orthonormal bases (modes) and calculate their combination coefficients from CFD results. A DNN was trained to predict combination coefficients of modes. Because these modes produced useful information of flow structures [5], by combining modes according to predicted coefficients, the building environment can be predicted under the guidance of some flow characteristics. The approach did well in steady-state indoor flow prediction. But the unsteady prediction application is remained.
In this study, the POD-DNN approach is utilized to predict unsteady state indoor temperature distributions with alternative boundary parameters. POD was used to extract characteristic modes. DNN is trained to output * Corresponding author: c-wei@iis.u-tokyo.ac.jp combining coefficients of these modes to predict temperature distributions. For an unsteady problem with alternative parameters, by taking time and parameters into consideration, different strategies used to extract modes influenced the accuracy of predictions. In this study, two kinds of POD strategies, flatten POD and nested POD, were researched to investigate their influence on the prediction accuracy.
where ߮ is eigenvectors and ܽ is coefficients. By the POD analysis, a reduced set of data [ () , … , () ], namely POD modes (POMs), can be extracted from an original temperature field. These POMs provide useful information on temperature field structures. In practice, the coefficient relates to time and boundary condition parameters. If coefficients are precisely predicted from time and boundary parameters, complex temperature fields can be reconstructed quickly by combining them with only a few basic modes.

POD-DNN
DNN is good at fitting non-linear relationships between input values and output values. In this study, DNN is introduced to predict coefficients from inputs of time or https://doi.org/10.1051/e3sconf/202235604028 E3S Web of Conferences 356, 04028 (2022) ROOMVENT 2022 boundary parameters. Unsteady temperature fields are predicted by combining POMs and coefficients predicted by DNN. The POD-DNN prediction process is mainly divided into four parts: (1) conducting unsteady CFD simulation of different boundary parameters to build a database, and dividing them into a training set, validation set, and testing set, (2) extracting POMs and coefficients from the training set, (3) training DNN by the coefficients of the training set and predicting coefficients of testing set, (4) combining POMs according to the predicted coefficients to reconstruct testing set.
For an unsteady problem with alternative boundary parameters, taking time and parameters into consideration makes it a multidimensional problem. it is important to find a proper way to extract POMs. In this study, two POD strategies combined with DNN are compared: flatten POD-DNN and nested POD-DNN.

Flatten POD-DNN
The process of flatten POD-DNN is shown in Fig. 1. Considering a CFD model with grid numbers of ݉ , temperature field T (pn, ts) is simulated, where pn and ts mean boundary parameter vectors and time, respectively.

Nested POD-DNN
This method is introduced by Wang et al. [6]. Fig. 2 shows the process. In this method, POMs are extracted by applying POD two times. The first time, POMs (pn) are extracted from ॻ( ) for each boundary parameter pn. The second time, final POMs [ ᇱ , … , ᇱ ] are extracted from [( )| … |( )]. Then, coefficients relating to time and parameters are calculated from T (pn, ts) and the POMs. The set of coefficients are used to train a DNN. The trained DNN is applied to predicted coefficients of testing cases by inputting parameters and time. Finally, the POMs are combined according to predicted coefficients to predict temperature fields of testing cases.

A two-dimensional indoor problem
Temperature distributions of a two-dimensional indoor space, shaped as 6.0 m (x) × 3.0 m (y), were investigated. Fig. 3 shows the outline of the target space. One inlet boundary and one outlet boundary were set near the floor and at the centre of the ceiling, respectively. And one window was across from the inlet boundary. The lengths of the inlet boundary, outlet boundary, and window were 0.2, 0.2, and 2 m, respectively. The window was set to a fixed temperature to exchange heat by convection. Other envelopes were assumed to be adiabatic.
The database was built by CFD simulation. Settings of the CFD simulation are shown in Table 1. STAR CCM+ v14.06.013 was employed for the CFD simulation in this study. The initial values of the temperature and velocity fields were set to 27 °C and 0 m/s, respectively. The alternative boundary parameters space consists of three kinds of boundary conditions: inlet air velocity Uin, inlet air temperature Tin, window surface temperature Tw. Values of them are provided in Table 1. Cases are named by parameter set: (Uin, Tin, Tw), for example, (1.0, 18, 35) means the case whose Uin, Tin, Tw is 1.0 m/s, 18 ˚C, and 35 ˚C. The time of each simulation was in a range of 0 to 300 s, where the time step was set to 0.1 s. But only results of selected time points were used to obtain the dataset (the first second, every 5 seconds from 5 to 100 s, every 10 seconds from 110 to 200 s, and every 20 seconds from 220 to 300 s).
The CFD database was separated to training set, validation set, and testing set by boundary parameters. Regarding DNN prediction, the process is mentioned in Section 2.2. Parameters Uin, Tin, Tw were input to the flatten POD-DNN, on the other side, time ts and parameters Uin, Tin, Tw were input. Coefficients of POMs were output. Both input and out data were normalized by transferring them to a range of 0 to 1. A dense DNN with 34 hidden layers was used, where there were 64 nodes in each hidden layer. Activation function, optimization algorithm, and loss function were set to ELU, Adam, and mean squared error, respectively.

Results
Contour figures of a testing case (0.5, 16, 40) were shown in Fig.4 to check prediction accuracy. In the results of CFD, a vortex in low temperature gradually became larger and moved away from the inlet until it filled the entire space. Regarding results of the flatten POD-DNN, when the training set was large, the prediction results were almost the same as the CFD results. However, when the training set was small, there were obvious multiple vortexes, different from the results of CFD, showing low accuracy. For the nested POD-DNN, when the training set was large, compared with the flatten POD-DNN, there was more noise, and small vortexes near the ground were not reproduced clearly. But generally, the development of the dominant vortex was well reproduced. When the training set was small, not like the results of the flatten POD-DNN, although accuracy dropped somehow, the development of the dominant vortex was reproduced. As a summary, if training data was sufficient, the flatten POD-DNN predicted accurately. But this method is sensitive to the training data size, less training data leading to obvious larger errors. Although there are some noises and errors, the nested POD-DNN can roughly predict the development of airflow regardless of the training data size, showing its less sensitivity to the training data size.

Conclusion
In this study, to predict unsteady temperature distributions, POD-DNN was utilized, where DNN was trained to predicted coefficients of POMs. Two kinds of POD strategies were compared. The results showed different their sensitivities to the training data size. The flatten POD-DNN provided high accuracy if training data is sufficient, but otherwise very inaccurate. The nested POD-DNN roughly predicted the development of temperature fields even training data was small. Reasons for the different sensitivities should be investigated in future works.