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Scientific Reports volume 14, Article number: 21208 (2024 ) Cite this article motherboard heatsink
The variational autoencoder (VAE) architecture has significant advantages in predictive image generation. This study proposes a novel RFCNN-βVAE model, which combines residual-connected fully connected neural networks with VAE to handle multi-heat source arrangements. By integrating analytical solutions, polynomial fitting, and temperature field superposition, we accurately simulated the temperature rise distribution of a single heat source. We further explored the use of multiple equivalent heat sources to replace adiabatic boundary conditions. This enables the analytical method to effectively solve the two-dimensional conjugate convective heat transfer problem, providing a reliable alternative to traditional numerical solutions. Our results show that the model achieved high predictive accuracy. By adjusting the β parameter, we balanced reconstruction accuracy and latent space generalization. During the stable phase of the multi-heat source optimization iteration, 73.4% of the results outperformed the dense dataset benchmark, indicating that the model successfully optimized heat source coordinates and minimized peak temperature rise. This validates the feasibility and effectiveness of deep learning in thermal management system design. This research lays the groundwork for optimizing complex thermal environments and contributes valuable perspectives on the effective design of systems with multiple thermal sources or for applications like multi-beam lighting equalization.
The demand for high-power density power electronic devices is continuously increasing, leading to tighter arrangements of devices. This results in hot spots at different locations needing to be considered uniformly, making thermal management more urgent1. We expect each device to be controlled within the expected temperature rise range in the limited cooling space, as reducing the temperature rise can extend the lifespan of electronic devices2,3. Additionally, a reasonable layout can also reduce mechanical deformation caused by heat4.
Therefore, there is increasing attention on the optimization of multi-heat source arrangements5. Stanescu et al.6 studied the optimal spacing of cylinders in forced convection environments to achieve maximum thermal conductivity. Hajmohammadi et al.7 researched the arrangement of discrete heating sections on pipe walls to reduce hotspot temperatures. Aslan et al.8 used convex optimization (CO) algorithms to rationally plan heat source layouts under various boundary conditions. Chen et al.9 used bionic optimization (BO) to optimize the arrangement of two-dimensional heat sources. Chen et al.10 utilized gradients to propose an adaptive multiresolution FEA method to solve heat source layout optimization problems. Ghioldi et al.11 combined heat transfer equations with sequential quadratic programming algorithms to achieve optimal device positioning. Su et al.12 used Bayesian optimization to achieve co-optimization of thermal conductivity distribution and heat source layout.
Currently, machine learning has rapidly developed in feature extraction13 and image reconstruction14 fields, and it has also seen significant advancements in heat transfer and mass transfer15,16, and fluid dynamics17,18, being widely applied to improve prediction efficiency. Machine learning can also serve as a surrogate model in the analysis of physical problems19,20. To better serve physical problems with deep learning, Physics-Informed Neural Networks (PINN) were developed21,22,23, combining conservation laws, dynamic equations, etc., to constrain the model's possible output space, making it more consistent with physical laws when generalizing unknown data compared to traditional models. Furthermore, there have been numerous studies on deep learning for thermal optimization. Chen et al.20 and Sun et al.24 used the Feature Pyramid Network (FPN) to learn the intrinsic patterns of the temperature field, serving as a surrogate model to optimize thermal performance under different input layouts, achieving results superior to traditional algorithms. Qian et al.25 combined genetic algorithms and Generative Adversarial Networks (GAN) to optimize multi-heat source layouts, showing advantages in both accuracy and efficiency. However, current research mainly focuses on developing surrogate models for temperature fields in heat source layouts to replace time-consuming simulation processes. The typical approach involves generating temperature field data from layout diagrams26,27,28. Most surrogate models place heat sources at discrete spatial locations rather than continuously, with some studies addressing layout optimization29,30,31,32. The best layout accuracy found in the study was generating temperature fields on a 200 × 200 layout domain33, but that article focused solely on temperature field prediction without actual layout optimization. Additionally, studies that generate temperature fields by directly inputting non-graphic parameters34,35 also focus on inverse reconstruction of temperature fields without directly addressing layout optimization. Moreover, almost all surrogate models study multiple discrete heat sources collectively, lacking flexibility in placement.
In selecting the deep learning model, we considered that the temperature rise distribution maps in our study require dimensionality reduction encoding. We hope the model can extract features, reconstruct them through the decoder, have predictive capability, and allow control over reconstruction accuracy. Autoencoders are widely used in high-dimensional data compression36,37 and pattern detection38. The derived variational autoencoder (VAE) has been proven effective in restoring high-dimensional dynamics in a low-dimensional latent space39. Additionally, to balance reconstruction accuracy, Burgess et al.40 introduced β-VAE, which enhances the interpretability of the latent space and the separation between latent variables. Therefore, the model we use in our research is a modified version based on β-VAE, primarily utilizing its probabilistic framework41. The learned latent space has good continuity and structure, enabling the generation of new data similar to the training data.
The training of deep learning networks requires a sufficient dataset. Currently, in solid heat transfer research, analytical methods are relatively rare in multi-heat source environments. Most studies use numerical methods such as the finite difference method42, finite volume method43, or finite element method44. However, the accuracy of these methods is related to the density of the mesh, and obtaining accurate results requires denser meshes and more simulation time, making large-scale optimization iterations difficult. Some studies use polynomial surrogate methods45, Kriging models46, and support vector regression (SVR)47 as surrogate model methods. However, these methods can face the curse of dimensionality when dealing with high-dimensional nonlinear problems48, significantly reducing modeling accuracy.
In this paper, we start from the common symmetry in physical research49 and use symmetry to simplify the calculation difficulty in solid heat transfer. Combining the related research on mirrored heat sources50, we derive the method of equivalent heat sources after optimization and improvement. By integrating the temperature field superposition theory51, this method can be used to replace adiabatic boundaries, greatly simplifying the coupling problem between multiple heat sources and making it possible to solve multi-heat source conjugate heat transfer problems using analytical methods. Moreover, it ensures high accuracy of the training set and accelerates the generation of the training set.
In conclusion, the key innovations of this paper are as follows:
Replacing numerical simulations, we developed a new dataset generation method by applying analytical solutions and polynomial fitting, saving computational time while obtaining high-precision temperature rise data.
By establishing a new surrogate model, we enabled the optimization of heat source layouts by inputting only coordinate parameters, moving away from the image-to-image generation approach commonly used in previous research. This method builds a surrogate model for individual heat sources, differing from previous studies that analyzed all discrete heat sources together.
We adopted a new evaluation method, thoroughly and comprehensively studying how the surrogate model’s predictive capability varies with the heat source position and examining the impact of dataset sampling density on prediction performance.
Typically, the heat dissipation process of electronic components on a single substrate can be simplified to a two-dimensional plane. In normal applications, we dissipate heat from the entire substrate. To simplify the problem, the electronic components are simplified to heat sources, approximated as point heat sources. For a convenient illustration, we use circular heat sources as substitutes. The heat dissipation area where the heat sources are arranged is a square, as shown in Fig. 1. We do not consider heat transfer perpendicular to the plane and assume that the entire surface undergoes steady-state convective heat transfer with heat sources, excluding thermal radiation. The equation is as follows:
Schematic diagram of 2D discrete heat source layout.
The boundaries on all four sides are adiabatic:
where T represents the temperature within the region, \(k\) denotes the thermal conductivity of the plate, and \(\phi \left(x,y\right)\) represents the intensity of the internal heat source, which depends on the layout of the heat sources and can be expressed as:
where \({\phi }_{i}\) represents the heat generation intensity of the \(i\) th discrete heat source within the region \({\Gamma }_{i}\) it covers, the variation in the positions of the discrete heat sources will result in different intensity distribution functions, which in turn generate different temperature field distributions. We adopt the excess temperature field to predict the temperature rise during the steady-state operation of the heat sources, aiming to minimize the temperature rise at the highest temperature point as much as possible.
The mirror heat source leverages the property that independent temperature fields can be superimposed, allowing for the replacement of traditional boundary conditions and avoiding the complex analysis of the temperature field distribution around heat sources in an adiabatic boundary environment. To replace the adiabatic boundary, identical heat sources are placed on both sides of the adiabatic boundary, treating it as a symmetry line. Under an approximately infinitely long adiabatic boundary, the identical heat sources placed on either side can effectively form an equivalent adiabatic boundary in the middle.
For a square region with adiabatic boundaries, previous research mirrored a heat source along each edge50, as shown in Fig. 2a. In this setup, Q1, Q2, Q3, and Q4 are the mirrored heat sources of Q. Generally, to achieve better results, the number of mirrored heat sources is increased. However, some distant mirrored heat sources, such as Q4 in Fig. 2a, contribute significantly less to the temperature rise at the points shown in the figure. Therefore, reasonably reducing the number of mirrored heat sources can greatly decrease the computational load, which is sensible for specific requirements.
(a) Demonstration of the mirror heat source method; (b) demonstration of the equivalent heat source method.
However, our study requires high accuracy for the entire temperature field, necessitating a reasonable investigation of the arrangement of mirrored heat sources to determine the number of equivalent heat sources needed. Here, we only discuss the case where the boundary is adiabatic. Considering a common heat source arrangement model, the first subject of study is a single heat source within a two-dimensional square region. For convenience, the length units and coordinate positions are in meters by default. In Fig. 2a and b, the equivalent adiabatic boundary of the square has a side length of 1, with the square's center located at (0,0). The heat source is represented by a circular area with a radius of 0.1, centered at (0.3,0). We use COMSOL software for the simulation, setting the dissipation power of each heat source to 29.36 W. The entire plane's thickness is set to 0.001, and the material's thermal conductivity is set to 201 W/(m K). Both the upper and lower surfaces are subjected to a convective heat flux of 10 W/(m2 K). First, we added mirrored heat sources using the method shown in Fig. 2a. The specific arrangement and corresponding numerical simulation results are shown in Fig. 3a. To visually observe the differences from the actual square adiabatic boundary, we output the simulation result data and compared the temperature rise distribution using four mirrored heat sources with the standard adiabatic boundary temperature rise distribution.
(a) Simulation results of the mirror heat source method; (b) percentage results of mirror heat source method vs. normal adiabatic boundary simulation.
Using the standard adiabatic boundary as a benchmark, the effectiveness of replacing the adiabatic boundary with mirrored heat sources is expressed as a percentage, as shown in Fig. 3b. It can be seen that the temperature rise in the areas near the four corners deviates significantly from the standard values, while the effect improves closer to the center. Over 85.3% of the area has a deviation of less than 3%, and over 64.5% of the area has a deviation of less than 1%. This indicates that, in cases where high precision is not required and specific points are chosen, the adiabatic boundary can be effectively replaced with four mirrored heat sources, thereby reducing computational load. However, Fig. 3b shows areas with significant errors, indicating energy leakage at the corners of the adiabatic boundary. This suggests that the mirrored heat source method cannot adequately replace the square adiabatic boundary. Therefore, to better replace the adiabatic boundary with mirrored heat sources, this study adds new equivalent heat sources to supplement the previous mirrored heat sources.
To explore suitable positions for adding equivalent heat sources to suppress energy leakage at the corners of the adiabatic boundary, we placed equivalent heat sources centered at the corners, on the opposite side of the line connecting the heat source, as shown in Fig. 2b. The arrangement and simulation results are shown in Fig. 4a. After simulation, the results were compared with the standard results of the adiabatic boundary, as shown in Fig. 4b. It can be seen that the overall deviation can be controlled within 1.24%. Considering that the grid density in numerical calculations affects the results, it is evident that adding 4 equivalent heat sources, along with mirrored heat sources, can effectively replace the adiabatic boundary. We refer to this method as the equivalent heat source method.
(a) Simulation results of equivalent heat source method; (b) percentage results of using equivalent heat source method compared to normal adiabatic boundary simulation.
Next, we will use the Equivalent Adiabatic Boundary Method to perform analytical analysis of conjugate heat transfer in two-dimensional planar objects. Compared to traditional numerical methods, this approach allows us to quickly obtain accurate results in the regions of interest without needing to consider the independence of the grid as in numerical calculations.
The object of this study's deep learning optimization is the previously mentioned circular heat source with a radius of 0.1, which dissipates heat within a square region with a side length of 1 and adiabatic boundaries. The square region experiences steady convective heat transfer, which can be represented by a constant average convective heat transfer coefficient. Utilizing the mirror heat source and the principle of temperature field superposition, we first analyze a single circular heat source placed in an infinitely large planar space, categorizing it as non-uniform cross-sectional fin heat transfer. The specific analysis is as follows:
First, when performing conjugate heat transfer analysis on a two-dimensional plane, thickness must be considered. This can be achieved using the extended surface energy equation, which is generally formulated as follows:
Assuming the thickness is \(t\) , and \(t\) does not change with \(r\) , the cross-sectional area \({A}_{c}=2\pi rt\) will vary with \(r\) .
Assuming the surface area involved in convective heat transfer is represented as \({A}_{s}=2\pi \left({r}^{2}-{r}_{1}^{2}\right)\) , replacing \(x\) with \(r\) in Eq. (4), the general form of the fin equation becomes:
Let \({m}^{2}\equiv \frac{2h}{kt}\) and define the excess temperature as \(\theta \equiv T-{T}_{\infty }\) , then we get:
It can be seen that the above equation conforms to the modified zeroth-order Bessel equation, and its general solution is:
In the equation, \({I}_{0}\) and \({K}_{0}\) are the zeroth-order Bessel functions of the first and second kinds, respectively. If the temperature at the edge of the heat source, i.e., at the base of the fin, is given as \(\theta ({r}_{1})={\theta }_{\text{b}}\) , and the fin tip is assumed to be adiabatic, i.e., \(\text{d}\theta /\text{d}r{\mid }_{{r}_{2}}=0\) then \({C}_{1}\) and \({C}_{2}\) can be determined, leading to the temperature distribution equation.
In the equation, \({I}_{1}(mr)=\text{d}[{I}_{0}(mr)]/\text{d}(mr)\) and \({K}_{1}\left(mr\right)=-\text{d}\left[{K}_{0}\left(mr\right)\right]/\text{d}\left(mr\right)\) are the modified first-order Bessel functions of the first and second kinds, respectively.
The heat transfer rate of the fin is \({q}_{\text{f}}=-k{A}_{\text{c},\text{b}}{\text{d}T/\text{d}r|}_{r={r}_{1}}=-k\left(2\pi {r}_{1}t\right)\text{d}\theta /\text{d}r{|}_{r={r}_{1}}\) .
By using this equation to obtain \({\theta }_{\text{b}}\) , and then substituting it into the Eq. (8), we get the relationship between \(\theta\) and \(r\) :
We assume that the heat sources are independent of each other. Using the above formula, we can independently analyze the space outside the heat source region. By setting the value of \({r}_{2}\) , we can assume it to be in an infinitely large region. However, it is important to note that the formula is applicable when \(r{>r}_{1}\) , which does not include the region containing the heat source.
For the temperature rise inside the heat source, polynomial fitting can be used to handle it. We only need the temperature rise of a single heat source in an infinite space. It is important to note that in our research, we aim for the heat dissipated on the plate outside the heat source area to equal 20W. Since the heat source area also participates in heat dissipation, we need to compensate for the heat source's power. First, we establish a single heat source simulation model, as shown in the schematic diagram in Fig. 5a. Additionally, Fig. 5b shows the expected temperature rise distribution for the single heat source.
(a) Single heat source simulation schematic and main parameters; (b) temperature rise and fitted temperature rise curve for a single heat source in infinite space.
Specifically, in COMSOL software, we select a two-dimensional simulation and place a circular heat source with a radius of 0.1 at the center position (0,0). First, we set the heat dissipation power of the heat source to 20 W, and the thickness of the entire plane to 0.001, with the material's thermal conductivity set to 201 W/(m K). A convective heat flux of 10 W/(m2 K) is applied to both the upper and lower surfaces. The mesh division is controlled by the physical field, and extremely fine element sizes are selected to ensure mesh independence.
Since we cannot directly set an infinitely large space, and considering the rapid temperature rise drop during heat transfer, a finite space can also reflect the temperature field distribution in an infinite space. Therefore, we gradually increase the radius of the plane to observe the temperature rise at the highest point. Ultimately, we choose a heat dissipation plane space with a radius of 30, and further increasing the radius of the dissipation space shows that the impact on the maximum temperature rise becomes negligible.
Next, we extracted the data from the single heat source temperature rise simulation. Since the heat source's power needs to be compensated, we first extracted the temperature rise at the heat source boundary, which was 7.558 K. Then, by setting \({q}_{\text{f}}=20\text{W}\) and using the analytical method, we calculated the temperature rise at the heat source boundary to be 11.096 K. According to the principle of temperature field superposition, the compensated heat dissipation power of the heat source is 29.36 W.
After compensation, the simulation was recalculated, and data along the x-axis passing through the center of the heat source were selected, and sampled at intervals of 0.002 within the range of −0.2 to 0.2. These data points were then imported into MATLAB's curve fitting tool, where a polynomial was used for fitting. The results, shown in Eq. (11), had a sum of squared errors (SSE) of 5.997e−06, both R-squared and adjusted R-squared values of 1, and a root mean square error (RMSE) of 2.526e−04, indicating a perfect fit of the polynomial to the data. The fitting effect is shown by the black line in Fig. 5b.
According to the obtained fitted polynomial, the value of \(x\) corresponds to the distance \(r\) from the point to the center of the heat source.
The data used for training and testing consist of temperature rise distribution matrices within a square adiabatic region for heat sources at different positions. Each temperature rise distribution matrix is uniformly sampled, sized at 128 × 128, ensuring good data accuracy while controlling the size of the training set. Each file name is labeled with the corresponding coordinate positions. Since the coordinates of the heat source parameters do not directly correspond to the sampling coordinates of the temperature rise matrix, the maximum temperature rise of each heat source cannot be directly obtained. This affects the training process and further tests the predictive capability of the trained model.
Next, we will provide a detailed explanation of the dataset generation, written in Python. First, we will write the code for calculating the temperature rise of a single heat source, setting the parameters for the region outside the heat source based on the analytical method. The parameter \({r}_{2}\) in Eq. (10) is set to 30 because if \({r}_{2}\) is infinite, the calculation will produce erroneous values. Based on previous analysis, this can be considered equivalent to an infinite space, with a negligible impact on temperature rise calculation accuracy. Other parameters are set as follows: h = 10 W/(m2 K), r1 = 0.1, k = 201 W/(m K), t = 0.001. For the interior of the heat source, where r ≤ 0.1, the fitted polynomial is used to calculate the temperature rise. This allows for the calculation of the temperature rise at a given point by inputting the distance r.
The following are the calculation steps to generate a set of single heat source temperature rise data:
Determine the coordinates of the heat source within the square adiabatic boundary \((xi,yi)\) . Using the equivalent heat source replacement method, draw 8 equivalent heat sources based on boundary symmetry or boundary corner center symmetry.
Generate a uniform 128 × 128 grid of matrix points within a square with a side length of 1 to store the temperature rise data for each point.
Perform the following operations for each grid point:
Calculate the distance r from each grid point to the actual heat source center and the centers of the 8 equivalent heat sources;
Substitute the r values corresponding to the 9 heat sources into the analytical solution or the fitted polynomial to calculate the excess temperature \(\theta\) produced by each heat source at that point;
Next, sum the results of the 9 excess temperatures to obtain the temperature rise data for that grid point.
Calculate all grid points to generate the temperature rise distribution data for the heat source at coordinates \((xi,yi)\) .
According to the task requirements, when the center coordinates of the square region are (0,0), the range of \(xi\) and \(yi\) is controlled within (−0.4, 0.4), and n points are generated in both the \(x\) and \(y\) directions. The size of the dataset is \(n\times n\) . The dataset generation speed is fast; using an Intel i9-13900KF, 32 × 32 sets of data can be output in 40 min.
Considering that the β-VAE model has good generalization capabilities, we need to test the impact of data density on the model's generalization ability and select an appropriate amount of data. Therefore, we generated training datasets with sizes of 1024 (32 × 32), 4096 (64 × 64), and 16,384 (128 × 128). Additionally, we can generate the test dataset by directly inputting coordinates or use training datasets with different data densities for cross-validation to evaluate the model's generalization capability.
Figure 6 illustrates the generation process of the surrogate model in this study. Directly using the final RFCNN-βVAE model often leads to training convergence difficulties. To accelerate convergence, we first preprocess the decoder part. The main difference between the autoencoder (AE) and the variational autoencoder (VAE) lies in the handling of the latent space, and the decoder part of the AE model corresponds exactly to that of the β-VAE. Therefore, we first use the AE to reconstruct the temperature rise matrix, then input the pre-trained AE model’s decoder into the subsequent RFCNN-βVAE model. Specifically, we first select a basic AE architecture and adjust the latent space dimension. We choose one of the three training sets, 32 × 32, 64 × 64, or 128 × 128, for training and adjust the latent space dimension based on reconstruction performance. The decoder of the best AE model is extracted and used as the pre-trained part in the subsequent RFCNN-βVAE model. After establishing the RFCNN-βVAE model, we mainly adjust the β parameter. The dataset used is the same as that for training the AE, but the input parameter becomes the heat source location parameter instead of the temperature rise map. After training, the global maximum temperature rise point prediction performance and new data are used for testing. The best results are then used as the surrogate model for single heat source temperature rise, for subsequent multi-heat source optimization layout.
Workflow of layout optimization through surrogate model construction.
The input to the model is 128 × 128 temperature rise data. The training process of the AE can be represented by the following Eq. (12). It is a mapping fitting process, divided into the encoding function \({f}_{EC}\) and the decoding function \({f}_{DC}\) , aimed at reducing the difference between the input \(x\) and the output \(\widetilde{x}\) .
The structure of the AE model is shown in Fig. 7, consisting of an encoder and a decoder. The encoder compresses the features of the input temperature rise matrix data and inputs them into the latent vector space. By adjusting the dimensionality of the latent vector space, we can determine the appropriate dimension for accurately representing heat transfer physical problems. In the decoder part, the latent vector is decoded back into the input temperature rise matrix data.
Schematic of the AE model Schematic.
The encoder of the AE model is based on ResNet, designed to extract features from the input data using a deeper network and achieve better convergence during training. The input to the encoder is a 128 × 128 × 1 temperature rise matrix. The input data is processed through 5 ResNet encoding blocks, each containing 3 convolutional layers. As shown in Fig. 7, the convolution strides of Conv1 in the encoder and Conv3 in the skip connection are set to 2, acting as compression units, reducing the feature map size to a quarter of the input. The number of filters in each layer gradually increases. The kernel sizes of Conv1 and Conv2 are 3 × 3, while Conv3 has a kernel size of 1 × 1. The input data is ultimately compressed to a size of 4 × 4 × 512 by the encoder.
The compressed data from the encoder needs to pass through a fully connected layer to enter the latent space. The dimensionality of the latent space is a key parameter that determines the structure of the AE and VAE. We set the dimensionality of the latent space to \(z\) , with values of 4, 8, 16, 32, 64, 128, 256, 512, and 1024. We need to choose the appropriate dimensionality to ensure good reconstruction performance.
The main task of the decoder part is to restore the feature vectors compressed by the encoder into the latent space back to the original input shape. To balance the performance and size of the model, we chose a decoder structure consisting of one fully connected layer and five transpose convolutional layers, gradually enlarging the feature map size from 4 × 4 to 128 × 128, while reducing the number of channels from 512 to 1, and outputting the decoded temperature rise matrix.
In the training process of the AE model, we chose the Mean Squared Error (MSE) as the loss function to evaluate the difference between the reconstructed output temperature rise matrix and the original temperature rise matrix. MSE is a smooth quadratic function with good mathematical properties for most optimization problems. It is differentiable over its entire domain, which is crucial for optimization methods such as gradient descent. Additionally, MSE is more sensitive to large errors because the square of the error magnifies larger errors. This helps the model reduce large prediction errors more quickly during training. The calculation is as follows:
Traditional AEs do not impose constraints on the latent space, resulting in a simple latent space structure. This leads to insufficient generalization performance, heavy reliance on training data, and potential overfitting52. Therefore, the variational principle is introduced53, mapping the input data through the encoder to a normal distribution in the latent space, parameterized by mean and standard deviation. Using the reparameterization trick, latent variables are randomly sampled from this normal distribution, generating representations of the latent space.
To facilitate the temperature rise prediction of a single heat source using only coordinate data, we propose a composite model that outputs the corresponding temperature rise distribution map by inputting the heat source's coordinate parameters. The model architecture is shown in Fig. 8. The coordinate parameters p are input into a fully connected neural network with residual connections (RFCNN) to obtain the mean and standard deviation defining the latent variables. Then, a latent space vector \(z\) is sampled, which is passed through the decoder to predict the 128 × 128 temperature rise distribution map. The dimensionality of the latent vector needs to be determined through comparison.
Schematic of the RFCNN-βVAE Model.
At this stage, the model's encoder consists of a series of residual linear layers. Each residual linear layer includes a main linear layer and an optional skip connection, using ReLU as the activation function. This design draws on the concept of residual networks, effectively mitigating the vanishing gradient problem and enhancing the stability of model training.
First, using the previous AE training dataset, extract the 2D coordinate information of the heat source from each temperature rise matrix file as input. The 2D parameters are projected into a 2048-dimensional space after passing through a residual linear layer. Next, keeping the 2048 dimensions unchanged, the feature vector is processed through three subsequent residual linear layers. Then, it passes through two fully connected layers to generate the mean and log variance of the latent distribution, with the dimensionality determined by the latent space dimension.
Next, we implement the reparameterization trick of the variational autoencoder, which requires sampling the latent vector \(z\) from the latent distribution, where \(\mu\) and \(\sigma\) represent the mean and standard deviation, respectively. Additionally, \(\epsilon \sim \mathcal{N}(0,I)\) is the random noise sampled from the standard normal distribution.
The decoder part uses a pre-trained AE decoder to decode the reparameterized latent vector z into a reconstructed version of the original data and compares it with the original temperature rise matrix corresponding to the input coordinates. The training process of the VAE involves using a composite loss function that includes reconstruction loss and KL loss. The expression is as follows:
The first term of the loss function is the reconstruction error, which measures the difference between the generated data and the original data, using the Mean Squared Error (MSE). The second term is the KL divergence, which measures the difference between the distribution of the generated latent variables and the standard normal distribution, ensuring the generative capability and stability of the VAE model during training. Therefore, the overall training objective is to ensure accurate reconstruction and prediction while maintaining the latent space distribution close to a standard normal distribution.
Of course, the original VAE architecture does not sufficiently control the model's latent space. Higgins et al.54 introduced the scalar hyperparameter \(\beta\) in the VAE to balance reconstruction accuracy and latent space disentanglement. The β-VAE loss function is defined in Eq. (16):
Generally, a lower \(\beta\) value may result in higher reconstruction accuracy but lower disentanglement of the latent space, while a higher \(\beta\) value increases latent space disentanglement but may not improve reconstruction accuracy. In our study, we primarily aim for high reconstruction accuracy and smooth transitions in the predicted data, with low requirements for latent space disentanglement. Therefore, we need to adjust the \(\beta\) parameter based on the prediction performance.
Overall, by combining RFCNN with β-VAE, we ultimately achieved the ability to quickly predict the temperature rise distribution for a heat source at a given location by inputting just two coordinate parameters of the single heat source. This lays the foundation for using the model to optimize multi-heat source arrangements.
We used the Pytorch deep learning framework to implement the model construction and training process. The training and testing of the model were conducted in an NVIDIA GPU-accelerated environment to ensure efficient computational performance, using the NVIDIA GeForce RTX 4090 graphics card.
During the pre-training process, because the AE model often encounters convergence issues in actual training, we chose to monitor the loss function value during the AE model training phase. If the MSE does not decrease significantly, the training will restart. The optimization uses the Adam optimizer with an initial learning rate set to 1 × 10–4 and a learning rate scheduler (StepLR) to adjust the learning rate by a factor of 0.7 every 10 epochs. We compare the original temperature rise images with the reconstructed temperature rise images using the MSE loss function. To ensure the model fully trains on all data, we randomly shuffle the original data multiple times, forming a dataset of 16,384 samples for AE training. The training lasts for 51 epochs with a batch size of 32, allowing for rapid iteration, and the converged results are saved as the pre-trained model.
During the RFCNN-βVAE training process, we use a composite loss function similar to the β-VAE, as shown in Eq. (16). The decoder is preloaded with the previously trained AE model. By randomly shuffling the original data multiple times, we use a dataset of 212,992 samples to train the RFCNN-βVAE model. Each temperature rise distribution file contains temperature rise data and the corresponding heat source coordinates. Unlike AE model training, the input data for the RFCNN-βVAE model consists of 2 coordinate parameters, and \({\mathcal{L}}_{rec}\) in Eq. (16) uses MSE to compare the difference between the original temperature rise distribution matrix and the reconstructed prediction results at the corresponding coordinates. The model optimization uses the Adam optimizer with an initial learning rate set to 1 × 10–4 and a training period of 151 epochs. StepLR is used to decay the learning rate by a factor of 0.7 every 10 epochs. Different latent space vector dimensions and \(\beta\) parameters are adjusted to obtain various training models, and the prediction performance differences under different data sampling densities are compared.
Since we used analytical methods, polynomial fitting, and the superposition of temperature fields to solve the problem, we need to assess the accuracy of the results. As shown in Fig. 9a, the position of a single heat source on a two-dimensional plane is illustrated. Figure 9b shows the simulation results calculated by COMSOL, Fig. 9c presents the results obtained by the analytical formula, and Fig. 9d shows the percentage comparison of the two. It can be seen that the two results maintain a high degree of consistency, with an error of less than 1.21%, meeting the conditions for being used as a training set.
(a) Schematic of the position of a single heat source on a two-dimensional plane; (b) simulation results calculated by COMSOL; (c) results obtained by formula calculation; (d) percentage results of the comparison between the two.
Secondly, to verify the accuracy of the temperature rise distribution when using the analytical formula combined with the temperature field superposition principle for multiple heat sources, we selected a scenario with three heat sources for validation. Figure 10a shows the results calculated by the formula, and Fig. 10b shows the percentage comparison with numerical calculation methods. The overall deviation is less than 0.5‰, indicating highly accurate simulation and proving the applicability of the above method in multi-heat source scenarios.
(a) Temperature rise distribution of three heat sources obtained by formula calculation; (b) percentage comparison between formula calculation and numerical calculation results in a multi-heat source scenario.
When creating the dataset, we used different point intervals to establish training datasets with sizes of 32 × 32, 64 × 64, and 128 × 128. We selected an RFCNN-βVAE model structure with a latent vector dimension of 512 and \(\beta\) =0.001 to compare training results across different datasets.
Ratio map of predicted effects to calculated maximum values within 0.8 × 0.8 and 0.6 × 0.6 areas under different datasets.
Changes in MSE during the AE model training iteration process under different latent space dimensions.
Overall, when the latent space dimensions are larger, specifically 512 and 1024, the decline is the fastest, and the AE model converges better, allowing for a more accurate simulation of the actual physical temperature rise distribution. After comprehensive comparison, we chose a vector dimension of 512 for the subsequent RFCNN-βVAE training process, which ensures the model's prediction accuracy while also controlling the model size.
(a) Loss evolution during the training of the RFCNN-βVAE model with a latent dimension of 512 and β = 0.001. (b) frequency plot of the ratio between the predicted maximum value and the calculated maximum value at different β values (c) correlation matrix of latent variables for a latent space dimension of 512 at different β values.
We also evaluated the correlation of latent space vectors41, with the correlation matrix \(R={(R)}_{z\times z}\) defined as follows:
where \({C}_{ij}\) represents the covariance matrix between latent space dimensions \(i\) and \(j\) . For all \(1\le i\ne j\le z\) , when \({R}_{ij}=1\) , it indicates the complete correlation between variables, and when \({R}_{ij}=0\) , it indicates no correlation between variables. Figure 13c shows the disentanglement degree of the latent space, where larger \(\beta\) values lead to greater disentanglement between latent space variables. However, this results in a decline in generalization accuracy. For higher prediction accuracy, we chose a smaller β value.
Our assessment of reconstruction performance primarily focuses on the differences between the temperature rise distribution across the entire region and the actual values. Typically, the extremum of physical quantities is our main concern. However, considering the temperature distribution characteristics of multi-heat source transfer, which includes both steep gradient drops and smooth transitions, we require not only the accuracy of the extrema but also the accuracy of the temperature distribution outside a single heat source. We have selected two evaluation methods: absolute error distribution and percentage error distribution.
Since it is difficult to use data from the training set when applying the model in practice, we use the test dataset to evaluate the model's predictive capability. Some typical prediction results are shown in Fig. 14. It can be seen that the areas with larger absolute errors are located at the edges of the circular heat sources, where the heat sources meet the cooling space, which is expected to produce significant changes. The temperature rise prediction at the center of the heat sources remains quite accurate, with deviations less than 0.01 °C. The percentage error shows larger deviations at locations far from the heat source, but considering that the temperature rise at these distant locations is very small, typically less than 1/50 of the central temperature rise beyond four heat source radii, the influence between heat sources becomes minimal at such distances. Consequently, the contribution to the highest temperature rise point in the system is very low, having minimal impact on the optimization results of multi-heat source arrangements.
Typical prediction effect comparison chart.
Our main objective is to determine the feasibility of using the trained RFCNN-βVAE model to optimize multi-heat source arrangements, focusing on the stability of the model's predictions at different spatial positions. First, we use the trained network to predict the temperature rise of a single heat source, then directly superimpose multiple prediction results according to the principle of temperature field superposition to simulate the temperature rise distribution under multiple heat sources. Next, we use the Adam optimizer to minimize the maximum temperature rise across the entire temperature field as the training objective, iteratively optimizing the coordinate positions.
We examine the optimal arrangement of three heat sources with equal heat generation, where their optimal positions are difficult to calculate directly. In this case, the trained RFCNN-βVAE model can be used for predictive optimization. We optimize the input parameters of three variational autoencoders, which involves individually generating three temperature rise matrices. Our goal is to minimize the maximum value in the temperature rise distribution map synthesized from these three matrices. To ensure smooth iteration, we restrict the range of x and y values during the iteration process and set an initial learning rate of 0.25, along with a learning rate scheduler that multiplies the learning rate by 0.6 every 100 steps throughout a total of 1000 iterations, with a minimum learning rate set at 4 × 10–3. The starting points for all three heat sources are placed at the center of a square region.
Due to the asymmetrical arrangement of three heat sources in a square space, multiple advantageous solutions emerge during the iteration process, but performing a full permutation on the 128 × 128 dataset takes too long. Therefore, we initially chose a 32 × 32 dataset to perform a full permutation to obtain preliminary advantageous solutions. Then, based on these advantageous solutions from the 32 × 32 dataset, we extract data from the corresponding area in the 128 × 128 dataset. The selection range is a rectangular area with a side length of 0.2 centered on each heat source coordinate. Next, we perform a full permutation using the data within these areas to establish a standard for evaluating the model's predictive performance, with the maximum temperature rise at this point being 18.86737 K. If we can achieve better results through iteration, it will demonstrate that deep learning can predict better outcomes than the densely arranged dataset.
We iterate with the objective of minimizing the maximum temperature rise point, inputting the predicted coordinates from the model during the iteration into the formula to calculate the corresponding temperature rise results for the three heat source coordinates, and extracting the maximum value among them. The main focus is on observing changes in the optimization effect during the iteration process. The typical prediction results obtained are shown in Fig. 15, which illustrates the entire iterative process starting from the origin and highlights the changes in predicted values during the stabilization phase after more than 500 iterations. It can be seen that in the stabilization phase after more than 500 iterations, over 73.4% of the data outperforms the results obtained from the dataset, and in the stable state, the worst result is less than 0.2% higher than the best temperature rise data.
(a) Iteration process of the highest temperature rise in multi-heat sources; (b) display of advantageous results during the iteration process.
To further assess the prediction performance of the surrogate model, we selected four identical discrete heat sources to test the model's optimization performance. According to our research objective, the optimal arrangement is at the positions shown in Fig. 16a. Since the optimal result is known, we can evaluate the effectiveness of the surrogate model in optimizing the layout. The obtained results are shown in Fig. 16b. We performed a total of 2000 position optimization iterations, reducing the learning rate by a factor of 0.6 every 100 steps, with the minimum learning rate set to 8 × 10−3. The starting points of the four heat sources were placed at the center of a square region. The last 1000 iterations were selected for evaluation, and the evaluation results are shown in Table 2.
(a) Schematic diagram of the optimal positions for four discrete heat source layouts, (b) scatter plot of the predicted positions after the last 1000 iterations of layout optimization using the surrogate model.
The deviation can be kept within 2.6%, demonstrating good prediction accuracy. However, it can also be observed from the figure that the trained model tends to get stuck in a local optimum during the prediction process, making it difficult to escape. This reflects a limitation of the model constructed in this study. Despite this, the hotspot temperature rise obtained using the average coordinates is only 0.021%, or 0.004 K, higher than the standard value.
This study developed and validated the application of the RFCNN-βVAE model based on Variational Autoencoders for multi-heat source layout optimization. Through model training, we validated the model’s predictive stability and accuracy across different spatial positions. By integrating analytical solutions, data fitting, and temperature field superposition methods, we accurately simulated the temperature rise distribution of a single heat source and demonstrated the potential of the surrogate model for multi-heat source optimization. The work is summarized as follows:
To quickly and accurately obtain training datasets, we innovatively proposed the use of equivalent heat sources to replace adiabatic boundaries, enabling the generation of temperature rise data using analytical methods combined with polynomial fitting. This approach ensures precision while avoiding the time-consuming nature of numerical simulations.
We proposed a novel surrogate model, RFCNN-βVAE, and innovatively applied it to the process of optimizing heat source layouts. This is the first surrogate model capable of generating temperature rise distribution maps by directly inputting the position parameters of individual heat sources. It is easy to operate for heat source layout optimization and offers great flexibility.
Based on task requirements, we established a new method for evaluating model performance by assessing the surrogate model’s prediction of hotspot temperature rise, thereby demonstrating the model’s predictive performance across the entire space. We found that with a latent space dimension of 512 and β set to 0.001, the model achieved a good balance between prediction accuracy and generalization capability.
Based on the constructed surrogate model, we tested its ability to optimize multi-heat source systems and validated the effectiveness of layouts for 3 and 4 heat sources. During the optimization of 3 discrete heat sources, over 73.4% of the data results in the stable phase of late iterations outperformed those from the densely arranged datasets. In the optimization of 4 discrete heat sources, we verified the difference between the model’s predictions and the optimal values, finding that the hotspot temperature rise was 0.021% or 0.004 K higher than the standard value, further demonstrating the model’s reliability.
Overall, this paper demonstrates the tremendous potential of deep learning in optimizing layouts within the field of heat and mass transfer. The simplicity of parameterization allows for easy control of the number of heat sources and further optimization for cases involving unequal heat generation. The model can also be used for real-time inversion and monitoring of temperature fields. The research methodology in this paper can also be extended to other fields. For example, by replacing the temperature field generation in the surrogate model with light intensity distribution generation, it could optimize the uniformity of multi-light source backlighting in displays. Similarly, replacing it with stress distribution generation could optimize the layout of critical structural components in buildings or assemblies.
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Fully connected neural network with residual connections
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This work was supported by the National Key Research and Development Program of China (2022YFB2804302).
Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen, 518055, China
Yide Yang & Jianshe Ma
Department of Precision Instruments, Tsinghua University, Beijing, 100084, China
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Y.Y. designed the experiments, performed the experiments, and analysed the results. Y.Y. wrote the manuscript. M.G. and J.M. revised the manuscript. All authors reviewed the manuscript.
The authors declare no competing interests.
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Yang, Y., Gong, M. & Ma, J. Research on multi-heat source arrangement optimization based on equivalent heat source method and reconstructed variational autoencoder. Sci Rep 14, 21208 (2024). https://doi.org/10.1038/s41598-024-71284-8
DOI: https://doi.org/10.1038/s41598-024-71284-8
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