Introduction

In recent years, increasingly small-sized but high-power electronic components are piled up on a circuit board, and the problem of thermal management inside the overall system is becoming more and more serious [1, 2]. The objective of thermal management is to reduce the maximum temperature and improve the heat distribution of the circuit board so as to provide a good thermal environment and keep the electronic components running properly [3]. One practical approach to improve the thermal performance of the circuit board through passive cooling is to optimize the positions of the electronic components [4, 5]. It can be regarded as a thermal layout optimization (TLO) driven by the thermal performance of the system. Due to its effectiveness, TLO has been extensively applied to chip floorplanning [6, 7] and integrated circuit design [8].

In fact, TLO can be formulated as a combinatorial optimization problem, where each decision variable is to indicate whether an electronic component is placed or not. For example, suppose k components need to be placed in a two-dimensional circuit board of the size \(n \times n\), and there are \(C_{nn}^{k}\) possible layout schemes for the optimization problem (assuming that all components have the same size and power). Such a substantial and discrete decision space makes the TLO problem challenging to be solved, especially for high-dimensional layout datasets [9]. Moreover, in many datasets, the thermal performance of the layout needs to be evaluated using computationally intensive numerical simulations (e.g., finite-element analysis) [10]. It is usually time-consuming, especially when the number of electronic components is huge in the practical engineering design. Such TLO problems become expensive combinatorial optimization problems [11].

Meta-heuristic algorithms, such as genetic algorithms (GA) [12] and local search [13], have been widely used in engineering design optimization in the past decades due to the powerful global search ability [14, 15]. With “survival of the fitness” from natural evolution, meta-heuristic algorithms generate new individuals via corresponding evolutionary operators and select individuals with better fitness as a new population into the next generation [16]. However, most existing meta-heuristic algorithms need a large number of fitness evaluations to achieve satisfactory results [17]. For the expensive TLO problems, a single evaluation of layouts usually takes a relatively long time [18]. The high evaluation cost forces meta-heuristic algorithms difficult to find an approximated optimal layout within an acceptable computational budget.

One feasible way for the expensive optimization problems is to reduce the number of expensive evaluations, such as surrogate-assisted optimization algorithms [19, 20], where surrogate models with a low computation cost are built to approximate and replace the expensive evaluations. A variety of machine learning models, including the Kriging model [21], radial basis function networks (RBFNs) [22, 23], and random forest [24], are used in surrogate-assisted optimization algorithms [25]. Most existing surrogate-assisted optimization algorithms assume that a certain number of real fitness evaluations can be made available during the optimization process, known as online data-driven optimization [26]. Thus, the online surrogate-assisted optimization algorithms focus on properly updating the surrogate model by taking full advantage of the real fitness evaluations allowed, i.e., model management [11], which improve the accuracy of surrogate models to find a satisfactory solution.

The online surrogate management strategies can be roughly classified into generation-based and individual-based strategies. For generation-based model management strategies [27, 28], the whole population in several generations is resampled as the new data used to update the surrogate models. Compared to generation-based strategies, individual-based strategies are more flexible [29, 30]. Typically, two types of samples can effectively improve the evaluation accuracy of the model. One is promising samples located around the optimum of the surrogate model [31]. The other is uncertain samples [30] located in the search space where the surrogate model is likely to have a large approximation error and has not been fully explored.

Although surrogate-assisted optimization algorithms have made remarkable achievements, they still suffer challenges in solving the expensive TLO problems [32]. In this work, there are two ways to build a computationally cheap surrogate model to replace the time-consuming numerical simulation [33, 34]. One is to directly build an end-to-end regression model between the layout scheme and the maximum value of the temperature field (or maximum temperature). For these surrogate modeling methods, the quality of the layout schemes is distinguished by the maximum temperature predicted by the end-to-end regression model. However, the decision space of TLO is discrete whose sample similarity between layout schemes is hard to measure accurately. The optimization direction will be seriously misled due to the large evaluation errors of the end-to-end regression models. The other way is to build a surrogate model to construct the mapping between the layout and the resulting temperature field. The quality of the layout schemes can be accurately distinguished by analyzing the temperature field constructed from the mapping model. However, in a two-dimensional domain of the size \(n \times n\), \(n^{2}\) design variables are needed to determine the layout schemes, and \((20n)^{2}\) output variables are required to accurately describe the resulting temperature field [35]. It is intractable to apply the generic surrogate models, such as the Kriging and RBFN, to construct such a high-dimensional mapping model.

Over the past 2 decades, deep neural networks (DNNs) have successfully handled various machine learning tasks [36, 37]. The powerful characterization capabilities of DNNs can represent and generalize complex functions or data to reveal unknown relationships between a large number of variables [38, 39]. Therefore, a large number of DNN-based surrogate modeling works are emerging in a variety of fields, such as fluid simulation [40,41,42], molecular dynamics simulation [43, 44], and uncertainty quantification [45, 46]. In the field of computational fluid simulation, Tompson et al. [40] propose a data-driven approach where the solution of the sparse linear system in the standard fluid solvers is replaced by a well-tailored CNN. After that, a state-of-the-art adaptive fluid neural network is proposed in [41] with more flexibility and generalization. In terms of molecular dynamics, an integrable DNN [43] is established to represent the free energy function, which is trained using the derivative data from atomic-scale models. In addition to the surrogates for accelerating the computational simulation process, uncertainty quantification is another vital application of surrogate modeling. In [45, 46], the surrogate models of partial differential equation systems are proposed to achieve effective uncertainty propagation in DNNs. Moreover, the algorithm [47] uses a DNN to predict soft tissue mechanical properties.

From the literature above, DNN has demonstrated the powerful potential for surrogate modeling. However, there are still some problems with applying DNN to solve the expensive TLO problems [48]. First, the performance of DNN is closely related to the training dataset. Specifically, the DNN with superior generalization usually requires a large number of samples for training [49]. For the expensive TLO problems, it is difficult to provide such abundant samples for training the deep surrogate model. Second, there is a significant distribution difference between the initial training layouts and the optimized layout generated during optimization process. The accuracy of the deep surrogate models will significantly decrease, making it difficult for the algorithm to achieve satisfactory optimization results.

To this end, we propose an online DNN-based surrogate-assisted optimization algorithm (NSU-OA) for solving the expensive TLO problems. In the NSU-OA, a lightweight DNN, U-Net [50], is used to construct an image-to-image mapping between the layout and the resulting temperature field. The maximum temperature can be accurately measured by analyzing the temperature field constructed from the mapping model. Moreover, an iterative local search and a model management strategy are designed to improve the efficiency and performance of NSU-OA. The main contributions of this work are as follows:

  • For the expensive TLO problems, we propose a DNN-based surrogate-assisted optimization algorithm (NSU-OA). To build an accurate surrogate model, a lightweight DNN is used to establish a mapping model from the layout scheme to the temperature field. The maximal operating temperature of a layout can be accurately determined via measuring the temperature field predicted by the mapping model.

  • An iterative local search (ILS) is developed to explore the discrete decision space of TLO. The randomly and the sequentially shaking are adopted as two search operators to efficiently find the layout with good thermal performance.

  • Considering the negative impact of approximation errors, a segmented fusion model management strategy (SFMMS) is designed to enhance the prediction accuracy of the DNN-based model. Specifically, the layout with the minimal predicted temperature is re-evaluated using the expensive simulation. Then, the layout with the maximal exact temperature is chosen to enter the dataset for fine-tuning the network parameters of the trained network models.

The rest of this paper is organized as follows: in “Problem description”, we describe the thermal layout optimization problem in detail. The proposed online deep surrogate-assisted optimization algorithm is introduced in “Proposed algorithm”. “Numerical experiments” provides sufficient comparative experiments and corresponding analysis. Finally, “Conclusion” concludes this paper and the future work is outlined.

Problem description

Generally, as each electronic component can be regarded as a heat source, the layout X formed by different component placement schemes will generate a corresponding temperature field Q as shown in Fig. 1. In this work, TLO is to design a layout scheme with a certain number of electronic components (termed k) for minimizing the maximum temperature \(T_{max}\) in Q. Therefore, assuming that the length and the width of the layout X are set to n, there are a total of \(n^{2}=D\) cells where components can be placed. The TLO problem of placing k electronic components can be defined as follows:

$$\begin{aligned} \begin{aligned}&\text {min:} \quad f(X),\\&\begin{array}{rl@{}ll} s.t. &{} h(X) = k, X \in V \\ \end{array} \end{aligned} \end{aligned}$$
(1)

where \(X= (x_{1},...,x_{D})(x_{i}\in \{0,1\}, 1\le i \le D)\) is a layout scheme with D decision variables, k is the number of electronic components needs to be placed, V is the set of all the possible layout schemes, and \(f(\cdot )\) is the objective function for performance, and \(h(\cdot )\) is used to count the number of placed electronic components in a layout scheme.

Fig. 1
figure 1

An example of the layout and the resulting temperature field. It is noticed that the size of the temperature field is \(200 \times 200\), and the maximum temperature of the layout is 339.46

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In this case, we mainly study how to optimize the component placement in the layout to minimize the maximum temperature of the temperature field. If k electronic components are placed on the layout X with D cells, there are \(C_{D}^{k}\) layout schemes X and their corresponding temperature fields Q. The maximum temperature \(T_{max}\) can be measured from Q. The quality of layouts is distinguished according to \(T_{max}\). Therefore, the objective function \(f(\cdot )\) is computed as follows:

First, given a layout scheme X with k electronic components, where \(x_{i}\) is to indicate whether an electronic component is placed or not. To clarify the location information of the components, the layout scheme X can be represented by a matrix, where the value of the element is 1 if an electronic component occupies it and 0 for no electronic component. The layout scheme can be expressed as:

$$\begin{aligned} X = \begin{pmatrix} 1 &{}\quad 0 &{}\quad \cdots &{}\quad 1\\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 1&{}\quad \cdots &{} \quad 0\\ \end{pmatrix}_{n \times n}, \end{aligned}$$
(2)

where X denotes the \(n \times n\) layout matrix. The temperature field Q can be obtained by performing expensive numerical simulations on X.

$$\begin{aligned} Q= F_{s} (X), \end{aligned}$$
(3)

where Q is a \(20n \times 20n\) temperature matrix, and \(F_{s}(\cdot )\) represents the expensive simulation. The maximum temperature \(T_{max}\) is obtained by measuring the maximum value of the temperature matrix.

$$\begin{aligned} T_{max} = \max \left( Q\right) , \end{aligned}$$
(4)

where \(T_{max} \) denotes the maximum temperature value in Q. Figure 1a is an example of the layout scheme X. The layout is divided into \(10 \times 10\) cells, and each electronic component is regarded as a square heat source to place in the layout. Figure 1b is the temperature field Q, and the maximum temperature \(T_{max}\) of Q is 335.1714.

In this work, it is time-consuming to obtain the temperature field by performing the expensive simulation on the layout. A feasible method is to build a computationally cheap end-to-end surrogate model between the layout X and its corresponding maximum temperature \(T_{max}\).

$$\begin{aligned} \hat{T}_{max} = \hat{f}(X), \end{aligned}$$
(5)

where \(\hat{f}(\cdot )\) represents the end-to-end surrogate model. Another method is establishing a mapping that predicts Q of the layout. Then, \(T_{max}\) is obtained by measuring the maximum value of Q.

$$\begin{aligned} \hat{T}_{max} = max(\hat{F_{s}}(X)), \end{aligned}$$
(6)

where \(\hat{F_{s}}(\cdot )\) represents the mapping between X and its Q.

One challenge of surrogate modeling for the TLO problem is that the sample similarity between the layout schemes is difficult to measure [24]. The prediction accuracy is poor when directly building an end-to-end regression model between the layout scheme and its corresponding maximum temperature. On the other hand, a surrogate model that predicts the temperature field of the layout will introduce high-dimensional design variables, increasing the difficulty of modeling.

Proposed algorithm

In this work, NSU-OA is proposed for solving the expensive TLO problems. Similar to existing surrogate-assisted optimization algorithms, NSU-OA consists of three parts: (1) an optimizer, (2) DNN-based surrogate models; and (3) an online model management strategy. In NSU-OA, we employ local search as the optimizer to explore the discrete decision space of TLO. The schematic diagram of our algorithm is shown in Fig. 2. NSU-OA updates the DNN-based surrogate model in every ten generations, where the offspring has been generated using the local search operators and evaluated according to the prediction of the DNN-based surrogate models. \(\eta \) layouts from the evaluated offspring are chosen for the expensive simulations. Those re-evaluated layouts are used to fine-tune the parameters of DNN-based surrogate models. NSU-OA can be further divided into six main steps.

Fig. 2
figure 2

The framework of NSU-OA

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  1. 1.

    Initialization: An initial population P and a certain number of layouts are generated by random sampling in the decision space. These layouts are evaluated by the expensive simulations and added to the training dataset. Then, they are used as the training data to build a DNN-based mapping model between the layout and the temperature field for the expensive simulation.

  2. 2.

    Candidate layout generation: Four neighborhood search operators are employed to generate candidate layout schemes. These layouts are evaluated using the DNN-based mapping model. In particular, the algorithm obtains the temperature field \(\hat{Q}\) of the layout through the DNN-based mapping model and then measures the maximum temperature \(T_{max}\) from the temperature field \(\hat{Q}\) to evaluate the quality of the layout.

  3. 3.

    Selection: All layout schemes in P are ranked according to their \(T_{max}\). Based on this, NSU-OA selects \(\eta \) new layouts with the lowest temperature value for the expensive simulations.

  4. 4.

    Online sampling: After re-evaluation, \(\eta \) new layouts X and their real temperature fields Q are stored in an external set U. Then, the network parameters are updated by U to improve the prediction accuracy.

  5. 5.

    Model management: DNN-based mapping model is fine-tuned using U. Specifically, the parameters of DNN are iteratively updated through a large number of epochs. In order to improve the efficiency of NSU-OA, two DNN models (Net\(_{1} \) and Net\(_{2}\)) are constructed based on the same training data. The algorithm switches to Net\(_{2}\) for evaluation when Net\(_{1}\) is fine-tuned. Similarly, when Net\(_{2}\) needs to be updated, the algorithm switches the updated Net\(_{1}\) for evaluation.

  6. 6.

    Repeat steps 2)–5) until the computational budget runs out. NSU-OA outputs the layout scheme \(X_\mathrm{{op}}\) with minimum \(T_{max}\) as the optimal layout.

In the following sections, we will present the details of the local search, DNN surrogate modeling and the model management strategy.

Iterative local search

For a layout scheme with a fixed number of components, TLO aims to find an optimal layout scheme that minimizes the maximum temperature of the resulting temperature field. In order to solve TLO problems efficiently, the iterative local search (ILS) is developed as the optimizer to search for the optimal layout scheme. Algorithm 1 illustrates the proposed ILS, which mainly consists of four local search operators.

The ILS simultaneously explores different neighborhoods through four local search operators, and the algorithm always selects the best neighbor layout to update the current layout in the population. The details can be found in Algorithm 1. First, an initial population P is generated by randomly sampling in the decision space. Then, the new layouts are generated by performing local search operators on the initial layout in P. These neighbor layouts are evaluated by the mapping model constructed via the DNN. If the new layout is better than the current solution X, the new layout will replace the current layout in P. Finally, the algorithm continues to iterate until the computation budget runs out. NSI-OA outputs a layout with the lowest \(T_{max}\) as the best found layout \(X_\mathrm{{op}}\). In this work, four local search operators can be expressed as follows:

  • In Fig. 3, the randomly shaking operator is to randomly remove a certain number of electronic components from the current layout. In this case, ILS randomly selects m (\(1\le m \le D\)) decision variables with value 1 (\(x_{i} = 1\)) in the current solution X, and then changes their values to 0. After repeated iterations t times, the algorithm obtains a set of randomly shaking layouts \(\{X_{rd}^{1}, X_{rd}^{2},..., X_{rd}^{t}\}\).

  • In Fig. 4, the random repairing operator is to randomly insert the same number of electronic components into the randomly shaking layout to obtain the randomly repairing layout. Given a randomly shaking layout \(X_{rd}^{i}\), ILS randomly selects m decision variables with value 0 (\(x_{i} = 0\)), and then change their values to 1. After repeated iterations t times, NSU-OA obtains a set of randomly repairing layouts \(\{X_{rr}^{1}, X_{rr}^{2},..., X_{rr}^{t}\} (h(X_{rr}^{i})=k)\). Then, these randomly repairing layouts are evaluated using the DNN-based surrogate model, and the repairing layout with lowest \(\hat{T}_{max}\) is selected as the best neighbor layout \(X_{\alpha }\).

  • In Fig. 5, the sequentially shaking operator is to remove each electronic component separately from the current layout to generate a shaking layout. For the current layout X with k components, ILS sequentially selects one decision variable with value 1 (\(x_{i} = 1\)), and then changes the value to 0. After repeated iterations k times, the algorithm obtains a set of sequentially shaking layouts \(\{X_{sd}^{1}, X_{sd}^{2},..., X_{sd}^{k}\}\). These shaking layouts are evaluated using the DNN-based surrogate model. The shaking layout with lowest \(\hat{T}_{max}\) is selected as the best shaking layout \(X_{sd}\) entering the sequentially repairing operator.

  • In Fig. 6, the sequentially repairing operator is to insert a component separately at the remaining positions of the shaking layout to obtain a repairing layout. Given a best shaking layout \(X_{sd}\) with \(k-1\) components, ILS sequentially selects one decision variable with value 0 (\(x_{i} = 0\)), and then change the value to 1. After repeated iterations \(D-k+1\) times, ILS obtains a set of sequentially repairing layouts \(\{X_{br}^{1}, X_{br}^{2},..., X_{br}^{D-k+1}\}\). These repairing layouts are evaluated using the DNN-based surrogate model, and the repairing layout with lowest \(\hat{T}_{max}\) is selected as the best neighbor layout \(X_{\beta }\).

In each iteration of ILS, NUS-OU will obtain two best neighbor layouts \(X_{\alpha }\) and \(X_{\beta }\) according to the current layout \(X_{i}\) in P. In this work, the algorithm compares the current layout with two best neighbor layouts, and selects the layout with the lowest \(\hat{T}_{max}\) as the best layout \(X_{ne}\) to update the current layout \(X^{i} = X_{ne}\). NSU-OA outputs the layout scheme with minimum temperature in P as the optimal layout \(X_{op}\) until the computational budget runs out.

Fig. 3
figure 3

The randomly shaking operators

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Fig. 4
figure 4

The repairing operators

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Fig. 5
figure 5

The sequentially shaking operator

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Fig. 6
figure 6

The sequentially repairing operator

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Algorithm 1
figure a

The ILS algorithm

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DNN-based surrogate-assisted modeling

As mentioned in “Proposed algorithm”, due to the discrete decision space, the prediction accuracy is poor when directly building an end-to-end surrogate model between the layout scheme and its corresponding maximum temperature. Likewise, a mapping model that predicts the temperature field of the layout will introduce high-dimensional design variables, increasing the difficulty of modeling. In this work, we consider DNNs as a class of potential surrogate models that can construct the mapping between the layouts and the resulting temperature fields.

Data preparation

In U-Net modeling, the input is the layout scheme, and the output is the corresponding temperature field. As described in “Problem description”, the layout scheme is presented by a 2-dimensional matrix X of the size \(n\times n\). Then, the corresponding temperature field is numerically simulated using the finite-element analysis. When performing the finite-element analysis solver, the layout is meshed as a \(20n\times 20n\) grid system, thus presenting a \(20n\times 20n\) numerical matrix as the output describing the temperature field. This matrix is defined as the temperature matrix and denoted by Q.

For a easy sampling, the layout is divided into \(n\times n\) cells, and one electronic component can just fit the size of one smallest cell. In this work, the sampling plan is to randomly generate various layout schemes X and then use the finite-element analysis to obtain their temperature matrix Q, which is provided as the set of general training examples. Limited by the time-consuming finite-element analysis, we only generate 1000 sets of training examples for network training in the initial stage. It is worth noting that the initial population P with 50 layouts is randomly selected from the 1000 training examples.

As stated above, the layout can be exactly described using a 2-dimensional matrix X of the size \(n\times n\) while the temperature field of the layout is represented by a \(20n \times 20n\) temperature matrix Q. To formulate a proper mapping model, the layout also needs to be represented by constructing a \(20n \times 20n\) matrix. Based on this, we further mesh each cell of the layout into smaller sub-girds with the size of \(20\times 20\) in order to match the scale of Q. The values of these sub-grids are consistent with their parent cell, which means that if there is one electronic component placed in one cell of \(10\times 10\) layout, all \(20 \times 20\) sub-grids maintain the value of 1. Therefore, we can obtain a fine-grained layout matrix C with the size of \(20n \times 20n\), which formulates the \(20n \times 20n\) grid image containing the electronic component layout information. This layout image C will be directly used as the input of DNN-based surrogate models.

Surrogate modeling based on U-Net

For TLO problems, the DNN-based surrogate model aims to build an image-to-image mapping model between the layout image C and its temperature field Q. The mapping model aims to predict the temperature field of the layout, and then the algorithm measures an accurate maximum temperature \(T_{max}\) for the predicted temperature field. With regard to this aspect, the U-Net [50], as a lightweight network model, is initially presented for image segmentation tasks. Its has demonstrated high performance in various image tasks, particularly in the medical imaging field. Due to the symmetric structure and the use of skip connections, U-Net allows information transfer between the encoder and decoder and improves the accuracy of the model [51]. In U-Net, each decoding layer is connected to its corresponding encoding layer through skip connections. These skip connections facilitate the fusion of features from lower and higher layers, thereby enhancing the network’s ability to capture both details and contextual information. Moreover, U-Net is highly scalable, as it can be extended or compressed by adding or removing layers to adapt to different tasks and datasets [52].

In this work, the layout scheme can be regarded as a two-dimensional layout matrix, and the temperature field is also a two-dimensional temperature matrix. The layout matrix and the temperature field can be viewed as a pair of images with the same size through the data pre-processing. U-Net is good at the image-to-image task. The lightweight structure allows the model to achieve excellent prediction results even with a small number of samples. Under the U-Net framework, the layout is regarded as a \(20n\times 20n\) semantics-variable image captured by U-Net at multiscale hierarchies, where semantics represents how the temperature field varies as the layout scheme changes. Based on the abovementioned considerations, it is decided to try U-Net to implement the image-image mapping model.

Figure 7 shows the overall architecture of U-Net. There are four max-pooling layers to reduce feature map size, and skip connections are conducted before each pooling layer to concatenate feature maps in the channel dimension. The U-Net supplements a usual contracting network by successive layers, where pooling operations are replaced by upsampling operators to increase the resolution of the output. The successive convolutional layer can then learn to assemble a precise dense prediction based on this information. To fit the TLO problem, the \(3\times 3\) convolutions with one padding are adopted in the U-Net. In addition, normalization layers are performed after the \(3\times 3\) convolution layer. The detailed size changes of all feature maps are shown in Fig. 7.

Training method

Given an arbitrary layout scheme X, we can get the output, which is the predicted temperature field \(\hat{Q}\) of the size \(20n\times 20n\), from the U-Net model. The training objective of U-Net is to minimize the margin between the predicted temperature field \(\hat{Q}\) and the ground-truth simulated one Q so that the U-Net model can fit the provided data. Once the training of the U-Net model finishes, the U-Net can play as a surrogate model for inference, which means predicting the corresponding temperature field with respect to arbitrary layout schemes.

The absolute error (AE) between the predicted value \(\hat{q}\) of \(\hat{Q}\) and the ground-truth value q of Q is defined as:

$$\begin{aligned} \text {AE}(\hat{q}, q)= |\hat{q}-q |. \end{aligned}$$
(7)

Then, the mean absolute error (MAE) between the predicted temperature field \(\hat{Q}\) and the ground-truth temperature field Q is defined as:

$$\begin{aligned} \text {MAE}(\hat{Q}, Q)= \frac{1}{20n\times 20n} \sum ^{20n}_{i=1}\sum ^{20n}_{j=1}\mathrm{{AE}}(\hat{Q}_{ij}-Q_{ij}). \end{aligned}$$
(8)

In this image-to-image mapping model, the MAE defined above is used as the loss function to train the U-Net model. For inference, we simply use the resulting \(\hat{Q}\) from U-Net as the predicted temperature field.

Fig. 7
figure 7

The flowchart of NSU-OA

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Segmented fusion model management strategy

For the TLO problem, we employ ILS to explore the decision space of the layout to generate the neighborhood solutions, and evaluate the quality of solutions via a DNN-based surrogate model. Due to the expensive simulations, NSU-OA only uses 1000 sets of layout samples for model training. The DNN-based surrogate model can achieve a good prediction performance when test layouts and training layouts have similar component distributions. However, as the optimization process proceeds, there is a large difference between the initial random layouts and the optimized layouts generated by ILS. The distribution differences between the optimized layouts and initial layouts lead to the significant increased prediction errors of the deep surrogate models. Furthermore, the training process of the network is more time-consuming than generic surrogate models. It is inadvisable to re-train the network model during the optimization process. Therefore, we propose an segmented fusion model management strategy (SFMMS) to fine-tune the trained network parameters, enabling the model to be incrementally updated according to the execution results of tasks.

In SFMMS, two DNN-based surrogate models Net\(_{1}\) and Net\(_{2}\) are constructed based on the same initial training layouts. In each generation of NSU-OA, \(\eta \) new layouts with the lowest temperature value from the evaluated offspring are chosen for the expensive simulations. After re-evaluation, these new layouts X and their real temperature fields Q are stored in an external set U. NSU-OA updates the parameters of surrogate models via U in every ten generations. To improve the efficiency of NSU-OA, the two models alternate between the prediction and the update during the optimization process. One model is used for prediction, and the other model fine-tunes the network parameters every 10 generations until the computational budget runs out.

Taking a TLO problem with k components as an example, at the 0th generation, Net\(_{1}\) is used to predict the temperature field Q. At the 11th generation, an external set U contains \(10\eta \) real evaluation layout samples. Then, SFMMS starts to fine-tune the parameters of Net\(_{1}\) by U. Since Net\(_{1}\) is fine-tuned, the algorithm cannot continue to use Net\(_{1}\) for evaluation. Based on this, SFMMS switches Net\(_{2}\) to predict the temperature field of the optimized layout. At the 21th generation, \(10\eta \) new real layouts are also selected to U after the expensive simulations. Similarly, the set U is used to fine-tune the parameters of Net\(_{2}\), and the updated Net\(_{1}\) is used to evaluate offspring. Repeat the above steps until the computational budget runs out. The network can predict an accurate temperature field distribution of the optimized layouts by SFMMS, which will significantly improve the quality of the optimization results.

Numerical experiments

In this section, we first evaluate the performance of the DNN-based surrogate model on the prediction accuracy of the temperature field. Then, the layout optimization based on the DNN-based surrogate model is conducted on TLO problems in order to validate the feasibility and efficacy of NSU-OA.

Experiment settings

In this work, three TLO problems are used to verify the performance of the proposed algorithm. Specifically, the size of the circuit board is \(10\times 10\), and there are \(D=100\) cells that can be used to place electronic components. For the three TLO problems, the number of electronic components k is 20, 40, and 60, respectively. In NSU-OA, ILS is employed to explore the discrete layout space. The population size of NSU-OA is 50. In each iteration of ILS, the number t of random shaking layouts \(X_{rd}\) is set as 50, and the corresponding number of random repairing layouts \(X_{rr}\) is 50. The number of sequentially shaking layouts \(X_{sd}\) is 20, 40, and 60 corresponding to electronic components is \(k=20\), 40, and 60, respectively. At the same time, the number of sequentially repairing layouts \(X_{br}\) (\(D-k+1\)) is 81, 61, and 41 (Table 1).

Table 1 Values of experiment hyperparameters
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NSU-OA builds the U-Net model to predict the temperature field Q of the layout scheme X. Then, the maximum temperature \(T_{max}\) is obtained by measuring the maximum value of Q. Specifically, the U-Net model is implemented using Pytorch 1.6. Adam [53] is chosen as the optimizer method, and 1000 random layout samples are generated to train the parameters of the network. We set the training epoch to 80, the learning rate to \(10^{-3}\), and the batch size to 32 for each model. In order to verify the performance of U-Net, two physical reconstruction networks, FCN [54] and FPN [55] are developed for comparison. In addition, RBFN [56] is also selected to build an end-to-end regression model between the layout scheme X and the maximum temperature \(T_{max}\). To be fair, we also use the same 1000 training layouts for model training. Specifically, for RBFN, we use the layout X and its corresponding maximum temperature \(T_{max}\) of the temperature field for training. In RBFN, the Gaussian function is used as the kernel, and the Hamming distance is used to calculate the distance between layouts. The center points of RBFNs are first obtained by the k-means clustering algorithm and then presented in the binary representation based on the threshold of 0.5.

During the optimization process, a certain number of optimized layouts are sampled to update the model parameters. In each iteration of NSU-OA, \(\eta \) layout schemes are selected and perform expensive simulations to obtain their temperature fields. Due to the deep surrogate model requires a large number of layout samples for training, NSU-OA selects two optimized layouts with minimum \(T_{max}\) (\(\eta = 2\)) in each generation and adds them to the dataset U after expensive simulations. In particular, a total of 50, 70, and 100 new optimized layouts are selected corresponding to the TLO problems with \(k=20\), 40, and 60, respectively.

All the compared algorithms are performed over thirty independent runs in three TLO problems. The Wilcoxon rank-sum test is also executed to compare the results by NSU-OA and other algorithms at a significance level of 0.05. Symbols “+” and “−” indicate that the NSU-OA is significantly superior to and inferior to the compared algorithm. The symbol “\(\approx \)” denotes no statistically significant difference between NSU-OA and the compared algorithms.

Ablation experiments

In NSU-OA, there are three main components: the DNN-based surrogate model, iterative local search, and SFMMS. In this section, we conduct ablation experiments to illustrate the effectiveness of the three components. In particular, we prepare two types of layout datasets to visualize the predictive performance of deep surrogate models.

Performance of surrogate models

Table 2 The Prediction results of different surrogate models on two types of layout datasets
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In this work, two types of layout datasets are used to verify the prediction accuracy of the surrogate models. One is a random layout dataset, which consists of 110 random layouts randomly sampled in a discrete layout space. The other is an optimized layout dataset consisting of 110 optimized layout samples collected in an optimization loop using expensive simulations as the evaluation function. It is worth noting that we build two types of layout datasets for three TLO problems, respectively. Moreover, in order to further verify the effectiveness of SFMMS, two types of models are prepared for testing. One is an initial surrogate model trained on 1000 random layouts, and the other is an updated model fine-tuned by SFMMS (we only use the last updated model for testing).

The results are shown in Table 2, where the best results are highlighted in bold. The \(\varDelta T_{max}\) represents the absolute error of the maximum temperature \(T_{max}\) obtained by measuring the temperature field predicted by the deep surrogate models. MAE represents the mean square error between the predicted temperature field \(\hat{Q}\) and the ground-truth temperature field Q as described in “Training method”. Due to RBFN directly predicting a maximum temperature \(T_{max}\) of the layout, the MAE of the temperature field is denoted by \(``/''\) in Table 2.

In Table 2, the deep surrogate models achieve better experimental results than RBFN on all test datasets. The prediction accuracy of the deep surrogate models maintains at around 0.1 on the different random layout datasets. However, the prediction error \(\varDelta T_{max}\) of RBFN is as high as 0.82 on the 20-component random layout dataset. Instead, the prediction error of FCN is only 0.15, which is the worst among the deep surrogate models. The U-Net is 0.06 in the 20-component random layout dataset. These results demonstrate that the DNN-based surrogate model can effectively represent and generalize the unknown relation among massive variables between the layout matrix and the temperature field. By reconstructing the temperature field through the DNN-based surrogate model, the algorithm can accurately measure the maximum operating temperature of the layout.

Although the deep surrogate models have a good prediction accuracy on random layout datasets, it can be observed that the prediction error of the surrogate models is relatively high on the optimized layout dataset. As the number of components increases, the MAE of the surrogate model further increases. The MAE of FPN is 0.41 and the \(\varDelta T_{max}\) is 0.96 on the 20-component optimized layout dataset. Furthermore, the prediction error \(\varDelta T_{max}\) of FPN is as high as 2.12 on the 60-component optimized layout dataset. Even the prediction error \(\varDelta T_{max}\) of U-Net is 1.02. The prediction error of the deep surrogate models will seriously affect the search direction of the algorithm and obtain a layout with a poor thermal performance.

To maintain a high prediction accuracy of surrogate models during the optimization process, SFMMS fine-tunes the parameters of surrogate models via a certain number of online optimized layout samples. The updated FCN achieves a 0.25 prediction error on the 40-component optimized layout dataset. Even the prediction error \(\varDelta T_{max}\) of RBFN decreases from 6.41 to 4.88. With the SFMMS, the performance of U-Net is significantly better than other surrogate models in all optimized layout datasets. The results indicate that SFMMS is beneficial for the surrogate models to obtain accurate prediction results on optimized layouts. However, the updated surrogate models fail to achieve good results in the random layouts. MAE and \(\varDelta T_{max}\) of U-Net are better than the updated U-Net in all random layout datasets. This result indicates that the prediction performance of the deep surrogate model on random layout samples degrades after the parameter fine-tuning. This is because DNN will happen task forgetting when performing incremental learning, i.e., it will forget part of the previously learned knowledge after learning some new knowledge.

Fig. 8
figure 8

The prediction results of surrogate models in the 20-component optimized layout dataset, which aims to show the changes of the prediction accuracy for different surrogate models. It is worth that the optimized layout dataset was collected from 110 historical optimal solutions generated by a real simulation-based optimization process

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In order to visually verify the prediction performance of the surrogate models, we deliberately prepare a set of optimized layout datasets to demonstrate the prediction performance of the surrogate models. The optimized layout dataset was collected from the historical optimal solutions generated by an expensive simulation-based optimization process. In Fig. 8, we show the prediction curves of the initial and updated surrogate models (including RBFN, FCN, FPN, and U-Net ) in the 20-component optimized layouts, respectively. In Fig. 8a, there is a significant gap between the prediction curves of RBFN and the curve of expensive simulations. These results indicate that RBFN fails to learn the task features from the training dataset. The poor performance of RBFN can seriously mislead the search direction of the optimization algorithm. In terms of RBFN and updated RBFN, the performance of RBFN was not improved by adding the optimized layout samples. It further shows that the generic surrogate model is difficult to solve the TLO problems.

In Fig. 8b and c, FCN and FPN have similar evaluation performance on some optimized samples (e.g., the 60th and 100th layout sample). The deep surrogate models have excellent prediction accuracy on the first 20 layout samples. It can be seen that the prediction result curves of FCN and FPN highly coincide with the real simulation curve. The results show that the deep surrogate model can learn the characteristics of the training samples and reconstruct the corresponding temperature field. However, the prediction performance of FCN degrades significantly on the high-quality layout samples. During the training stage of the network, the initial training dataset is constructed by random layout samples. As the optimization proceeds, there is a large difference between the component distribution of the optimized layout and the random layout. It is difficult for the deep surrogate models to obtain an accurate temperature field when faced with such layout samples. The prediction curves of the updated FCN and FPN have been significantly improved by observing Fig. 8b and c. Especially for the updated FCN, the curve is approximately close to the simulation curve. The prediction results provided by the deep surrogate model after the update significantly improve the quality of the optimal solution obtained from the optimization algorithm. However, the updated FCN and FPN also have shortcomings. In Fig. 8b, the updated FCN has an opposite trend to the simulation curve after the 100th generation. In Fig. 8c, the curve of updated FPN has a large gap with the simulation curve at the 50th generation. After learning newly optimized layout features, the updated FCN and FPN forget the knowledge learned by offline layout samples, which restricts the optimization algorithm from obtaining high-quality optimal solutions. Finally, in Fig. 8d, the prediction curve of U-Net is a high-quality prediction curve without model management. After the 40th generation, the prediction performance of U-Net is significantly higher than other surrogate models. The good generalization of U-Net indicates that the unique U-shaped structure of the network preserves the shallow low-level features. The four cascaded branches of U-Net enormously extend the width of the network and achieve a good balance between fit and generalization. However, the lack of optimized layout samples also makes the prediction results of U-Net inaccurate in the 70th generation.

Performance of SFMMS

To verify the performance of SFMMS, three different surrogate models, including RBFN, FCN, FPN, and U-Net are tested on the TLO problems with the same parameters. The optimization results obtained with ILS over twenty-five independent runs are collected in Table 3, where the best results are highlighted in bold. The \(T_{max}\) represents the maximum temperature, which can be measured by the temperature field obtained by expensive simulations.

  1. 1.

    RBFN-ILS includes RBFN, ILS and SFMMS. In RBFN-ILS, the algorithm uses ILS as an optimizer to generate candidate layouts. Then, the maximum temperatures \(\hat{T}_{max}\) of these candidate layouts are directly predicted by RBFN. In each generation of RBFN-ILS, two layouts with the minimum \(\hat{T}_{max}\) are selected to perform expensive simulations to obtain the real maximum temperature \(T_{max}\) and add them to the dataset U. Every ten generations, the obtained dataset U is combined with the initial training dataset to rebuild RBFN for evaluation.

  2. 2.

    RBFN-ILS-u includes RBFN and ILS without SFMMS. In RBFN-ILS-u, the algorithm also uses ILS as an optimizer to explore the discrete layout space. Then, RBFN is used to directly predict \(\hat{T}_{max}\) of the layouts. Although the best two samples in each generation are selected to perform expensive simulations, these online optimized layouts are not used to update RBFN.

  3. 3.

    FCN-ILS includes FCN, ILS and SFMMS. In FCN-ILS, the algorithm predicts the temperature field of the layout through FCN, and then the maximum temperature \(\hat{T}_{max}\) is obtained by measuring the predicted temperature field. Similar to RBFN-ILS, two layouts with the minimum \(\hat{T}_{max}\) are selected to perform expensive simulations to obtain the real \(T_{max}\) and add them to the dataset U. However, FCN-ILS only uses U to fine-tune the parameters of the network.

  4. 4.

    FCN-ILS-u includes FCN and ILS without SFMMS. The basic settings of FCN-ILS-u are the same as FCN-ILS. The only difference is that the network parameters are not updated via U in FCN-ILS-u.

  5. 5.

    FPN-ILS includes FPN, ILS and SFMMS. The basic settings of FPN-ILS are the same as FCN-ILS. The only difference is that the mapping model used is FPN in FPN-ILS.

  6. 6.

    FPN-ILS-u includes FPN and ILS without SFMMS. The basic settings of FPN-ILS-u are the same as FPN-ILS. The only difference is that the network parameters are not updated via U in FPN-ILS-u.

  7. 7.

    NSU-OA-u includes U-Net and ILS without SFMMS. In NSU-OA-u, the algorithm only uses the U-Net to predict the temperature field of the layout during the optimization process. In each generation of NSU-OA-u, two optimized layouts with the lowest \(\hat{T}_{max}\) are selected to perform the expensive simulation to obtain the real maximum temperature \(T_{max}\). Finally, the layout with the lowest \(T_{max}\) is output as the optimal layout.

  8. 8.

    NSU-OA includes three components, U-Net, ILS, and SFMMS.

Table 3 Mean (standard deviation) maximum temperature of NSU-OA and other seven comparison algorithms
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Table 3 shows the mean and the standard deviation of the maximum temperature obtained by the proposed algorithm NSU-OA variants with different components. We chose three TLO problems: 20-component, 40-component, and 60-component problem to verify the effectiveness of our proposed components. As can be seen from Table 3, NSU-OA with the three components achieves the best performance on all test problems, followed by FCN-ILS, FPN-ILS, and RBFN-ILS performs the worst.

In terms of NSU-OA and NSU-OA-u, the algorithms use the same optimizer to search for optimal layout schemes. Table 3 shows that NSU-OA-u without SFMMS is significantly worse than NSU-OA in all tasks. It shows that SFMMS can effectively improve the optimization results of surrogate-assisted algorithms except for RBFN. In terms of RBFN-ILS and RBFN-ILS-u, both algorithms get poor optimization results on the different thermal layout problems. It is mainly caused by two reasons. First, the TLO problem is a COP. RBFN is hard to measure the mapping relationship between the layout scheme and its maximum temperature. Second, the input variable of RBFN is a \(100\times 1\) dimensional vector. The high-dimensional decision space makes the training process of surrogate models more difficult. Finally, the poor prediction accuracy of RBFN seriously misleads the search direction of RBFN-ILS. It also explains why the optimization results of RBFN-ILS are slightly improved compared to RBFN-ILS-u.

In terms of FCN-ILS and FPN-ILS, both algorithms achieve better optimization results than RBFN-ILS in all test problems. The deep surrogate models can approximately reconstruct the temperature field of the layout based on the powerful function fitting capabilities. Then, the accurate maximum temperature is obtained by measuring the predicted temperature field of the layout. FCN-ILS and FPN-ILS have an accurate search direction that benefits from the accurate prediction performance of the deep surrogate models. In terms of FCN-ILS and FCN-ILS-u, FCN-ILS achieves better optimization results than FCN-ILS-u with a small number of online optimized samples. This indicates that the proposed SFMMS improves the prediction accuracy of the deep surrogate models by fine-tuning the parameters of the DNN model. Especially for the 60-component problem, the maximum temperature of FCN-ILS reduces from 399.23 to 397.30, and the optimization result of FPN-ILS is as low as 397.53. The performance of SFMMS is attributed to two aspects. On the one hand, the small amount of online data improves the generalization ability of the network on optimized layout samples. On the other hand, the segmented update of the surrogate model avoids the task forgetting [57] caused by the incremental learning of a single model. Since NSU-OA contains all three components, it achieves the best task performance on all test TLO problems. The results show that all three components included in NSU-OA are indispensable.

Thermal layout optimization results

In this study, we demonstrate the optimization results of different algorithms for the TLO problems. To compare the performance of our proposed algorithm, we provide an approximately optimum layout obtained directly using the expensive simulation as the evaluation function for the optimization algorithm. Furthermore, we also compare the performance of ILS and GA on three different surrogate models, including RBFN, FCN, and FPN. The Wilcoxon rank-sum test (0.05 significant level) also calculates between these algorithms symbolized by \(+\), − and \(\approx \).

  1. 1.

    GA is widely used in optimizing black-box functions by relying on biologically inspired operators, such as mutation, crossover, and selection. Therefore, we develop a GA-based layout optimization algorithm GA-limited as a comparison algorithm. In GA-limited, we implement a GA using elitism selection, a mutation and a crossover operator to optimize the population of layouts. Specifically, the population size of GA-limited is 50, the crossover probability is 0.8 and the mutation probability is 0.1. The elitism selection works by keeping the top \(50\%\) of the layouts from the current population as survivors and discarding the rest. In each generation of GA, the expensive simulation is directly integrated into the optimization loop as a fitness function to evaluate the quality of the layout. In particular, the expensive simulations are used as the fitness function of GA-limited to evaluate the quality of candidate layouts. To be fair, the number of expensive evaluations is set to 1050, 1070, and 1080 for TLO problems with 20 components, 40 components, and 60 components, respectively (the number of expensive evaluations is equal to the number of training layouts and the number of online optimized layouts for model management).

  2. 2.

    As described in “Iterative local search”, ILS explores the discrete layout space of TLO problems by developing four local search operators. Similarly, we develop an ILS-based layout optimization algorithm ILS-limited as a comparison algorithm. Specifically, the population size of ILS-limited is 50. The expensive simulations are also employed to evaluate the quality of the candidate layouts. The number of expensive evaluations is also set to 1050, 1070, and 1080 for TLO problems with 20 components, 40 components, and 60 components, respectively.

  3. 3.

    RBFN-GA, FCN-GA, and FPN-GA are three surrogate-assisted comparison algorithms where GA is employed to generate candidate layouts. The setting of there three algorithms are the same as GA-limited (including population size, crossover probability, mutation probability, and elitism selection). Specifically, the number of expensive evaluations is also set to 1050, 1070, and 1080 for TLO problems with 20 components, 40 components, and 60 components, respectively. 1000 layout schemes are used to train the surrogate model, and 50 (or 70 and 80) expensive simulations are used to evaluate the online optimized samples.

  4. 4.

    RBFN-ILS, FCN-ILS, and FPN-ILS are also three surrogate-assisted optimization algorithms, where ILS is employed to generate candidate layouts. The setting of these three algorithms is the same as ILS-limited. Specifically, the number of expensive evaluations is also set to 1050, 1070, and 1080 for TLO problems with 20-component, 40-component, and 60-component, respectively. One thousand layout schemes are used to train the surrogate model, and 50 (or 70 and 80) expensive simulations are used to evaluate the online optimized samples.

  5. 5.

    In order to further verify the performance of the proposed algorithm, we designed a set of optimization results that do not limit the number of real simulations, named Optimum. In Optimum, ILS is used as an optimizer to explore a discrete design space, and the expensive simulations are directly integrated into the optimization loop to evaluate the quality of the candidate layouts. More importantly, we do not limit the number of expensive simulations until the algorithm converges. Therefore, the optimization results of Optimum can be regarded as approximate optimal solutions.

Table 4 Mean (standard deviation) maximum temperature of NSU-OA and other seven comparison algorithms
Full size table

In Table 4, it can be observed that Optimum achieves a good optimization result on all TLO problems. On the 20-component problem, the maximum temperature of Optimum is as low as 326.67. However, it is very time-consuming to directly use the expensive simulation to evaluate the layout schemes generated in the optimization loop. Our proposed NSU-OA uses the U-Net to replace the expensive simulations to reconstruct the temperature field of the layout. On the 20-component problem, the maximum temperature of NSU-OA is 327.08, which is very close to the optimization result of Optimum. Especially for the 60-component problem, the maximum temperature of Optimum is 396.28, and the maximum temperature of NSU-OA is 396.84. However, in terms of NSU-OA and ILS, the maximum temperature of ILS (the same number of expensive simulations as NSU-OA) is 328.61 on the 20-component problem, and the maximum temperature of ILS is as high as 361.73 on the 40-component problem by observing Table 4. These results can be concluded that NSU-OA can thoroughly search the decision space by building a surrogate model with a low computation cost to obtain an excellent approximate optimal layout.

In terms of FPN-ILS and FPN-GA, it can be observed that FPN-ILS performs better than FPN-GA on all test TLO problems. In addition, for the 40-component problem, the maximum temperature of FCN-GA is as high as 362.32, and the maximum temperature of FCN-ILS is 360.70 by observing Table 4. The maximum temperature of FPN-ILS is as low as 397.53, and FPN-GA is 399.36 on the 60-component problem. ILS achieves better optimization results than GA under all deep surrogate models. These results indicate that ILS can effectively explore the decision space of the layout via shaking and repairing operators. However, RBFN-ILS has poor optimization performance in all TLO problems. The reason is that RBFN is difficult to predict the accurate maximum temperature for layout schemes. The low prediction accuracy makes it impossible for the algorithm to maintain the correct search direction. It can be observed that all deep surrogate models have achieved better optimization results than RBFN. Compared with RBFN, the deep surrogate models predict the temperature field to measure the maximum temperature. This allows the algorithm to predict the maximum temperature more accurately. Moreover, NSU-OA performs better than the other deep surrogate models. The deep surrogate models predict the temperature field to measure the maximum temperature. For U-Net, the lightweight network can better summarize the internal relationship of the samples in the case of a small amount of data to learn robust network parameters. In addition, SFMMS can fine-tune the network parameters during optimization loops to ensure that the surrogate model accurately predicts the temperature field of the optimized layout. This may also be the reason why NSU-OA can achieve the best performance on all test TLO problems.

In Fig. 9, four images show the optimization results of the comparison algorithms on the 20-component problem. The left side of the image is the layout scheme, and the right side is its corresponding temperature field. It can be observed that the NSU-OA only optimizes the placement of components in the layout and its operating temperature field is significantly improved. However, in Fig. 9c, the temperature field of the ILS has two hot spots. This shows that under the same number of expensive simulations, NSU-OA achieves better optimization results. Finally, Optimum exhibits a smooth temperature field with a relatively low maximum temperature by observing Fig. 9d. The temperature field of the NSU-OA in Fig. 9b is very close to the approximate optimum in Fig. 9d. The results show that our proposed NSU-OA can achieve well-optimized results while reducing a lot of computation.

Fig. 9
figure 9

Thermal layout images on 20-component TLO problem. The left side of the image is the layout scheme, and the right side is the corresponding temperature field. Specifically, a is a random layout sample. b–d are optimized layouts obtained by NSU-OA, ILS-limited and Optimum, respectively. The maximum operator temperature for these layouts are 337.05, 327.08, 328.61, and 326.67

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Overall, the performance of NSU-OA can be attributed to the following three aspects:

  1. 1.

    The DNN-based surrogate model can accurately predict the temperature field of the layout, which can well replace the expensive simulation process in the TLO problems.

  2. 2.

    ILS can efficiently explore the decision space of the layout to obtain a high-quality optimal solution.

  3. 3.

    SFMMS segments fine-tune the parameters of the deep surrogate models to accurately predict the temperature field of the optimized solution and avoid the problem of historical forgetting caused by the continuous learning of the model.

There are also some limitations of NSU-OA. Since this work only selects a deep surrogate model to reconstruct the temperature field distribution of the layout. The performance of the NSU-OA is heavily dependent on the prediction accuracy of the DNN. Furthermore, the deep surrogate models require fine-tuning network parameters to ensure prediction accuracy.

Conclusion

We propose an online deep surrogate-assisted optimization algorithm using DNN as a surrogate model to replace expensive simulations for solving the expensive TLO problems. NSU-OA builds an image-image mapping model based on offline data and introduces SFMMS to online update the network parameters during the optimization process. Moreover, NSU-OA compares with FCN, FPN, and RBFN on three layout optimization problems and two kinds of layout sample datasets. The results indicate that NSU-OA has the best overall performance on most problems.

This work demonstrates the promising results of a deep surrogate-assisted optimization algorithm in handling the TLO problems. Given the powerful nonlinear approximation capability of DNNs, we believe that the proposed NSU-OA can effectively learn the mapping relationship for various optimization tasks (such as image-to-image, image-to-vector, and image-to-value). However, our algorithm still has much room for improvement. First, the algorithm is highly dependent on the prediction performance of the network. Likely, the network cannot guide the current solution toward the optimal solution. Ensemble learning may be a better solution. Second, NSU-OA needs to train corresponding deep surrogate models for different thermal layout problems, which is inconvenient for transferring new tasks. In future work, we aim to build a task-independent pre-trained model from large-scale data via self-supervised learning and then fine-tune the network for the specific task to achieve excellent performance.