Problem formulation

The operating-parameter data $X=\left\{x_1\mathrm{,}{x}_2\mathrm{,}\cdots ,x_T\right\}\in {ℝ}^T$ are given hypothetically, where T denotes the length of the time series. In the prediction task in nuclear reactors, a sliding window is applied to construct the dataset $\mathcal{D}=\left\{\left(X_t\mathrm{,}{Y}_t\right)\right\}\in {ℝ}^{t\times n}$, where t and n represent the t-th temporal window and total number of windows, respectively. More specifically, $X_t=\left\{x_{t-\omega -1}\mathrm{,}\cdots \mathrm{,}{x}_{t-1}\mathrm{,}{x}_t\right\}$ represents the input window of length ω and $Y_t=\left\{x_{t+1}\mathrm{,}{x}_{t+2}\mathrm{,}\cdots ,x_{t+\tau }\right\}$ represents the forecasting window in the future τ steps (τ≥1). The process can be viewed as a function F_P as follows: $\left\{x_{t+1}\mathrm{,}{x}_{t+2}\mathrm{,}\cdots \mathrm{,}{x}_{t+\tau }\right\}=F_{\text{P}}\left(\left\{x_{t-\omega -1}\mathrm{,}\cdots \mathrm{,}{x}_{t-1}\mathrm{,}{x}_t\right\}\right)\mathrm{.}$ (1) The prediction task involves transferring knowledge from ample simulated data (source domain) to scarce actual data (target domain) in NPPs. The simulated data represents the source domain ${\mathcal{D}}^{\text{S}}=\left\{\left(X_t^{\text{S}}\mathrm{,}{Y}_t^{\text{S}}\right)\right\}$ ($X_t^{\text{S}}\in {\mathcal{X}}^{\text{S}}\mathrm{,}{Y}_t^{\text{S}}\in {\mathcal{Y}}^{\text{S}}$), and the real data represents the target domain ${\mathcal{D}}^{\text{T}}=\left\{\left(X_t^{\text{T}}\mathrm{,}{Y}_t^{\text{T}}\right)\right\}$ ($X_t^{\text{T}}\in {\mathcal{X}}^{\text{T}}\mathrm{,}{Y}_t^{\text{T}}\in {\mathcal{Y}}^{\text{T}}$). They share a common feature subspace, that is, ${\mathcal{X}}^{\text{S}}={\mathcal{X}}^{\text{T}}$ and ${\mathcal{Y}}^{\text{S}}={\mathcal{Y}}^{\text{T}}$. Owing to the different actuation modes and system dynamics, the source and target domains have different marginal probability distributions, that is, $P(X^{\text{S}})\ne P\left(X^{\text{T}}\right)$. Thus, we design a transformation function F_DA to fulfill P(F_DA(X^S))=P(F_DA(X^T)). Function F_DA is combined with function F_P to establish the transferability prediction framework ${\mathcal{F}}_{\text{P}}=F_{\text{DA}}\left(F_{\text{P}}(\cdot )\right)$, which performs forecasting missions in real-world nuclear reactors.

Overview of SRDA

The proposed SRDA framework aims to predict real data in nuclear reactors precisely over time by learning and transferring prior knowledge from simulations to reality. As presented in Fig. 2, the architecture of SRDA resembles that of conventional neural networks and possesses two output modules instead of one. The SRDA model comprises four modules. The blue module is the feature extractor G_f and serves as the backbone network that directly affects the transfer effect. We use a transformer as the feature extractor, which is described in detail in Sect. 3.4. The orange module, representing the parameter predictor G_p, is constructed using an improved logarithmic loss, a fully connected (FC) layer, and a prediction output layer, which forecasts the future variation of operating parameters in the reactors. The purple module represents the domain discriminator G_d and comprises two FC layers and a classification output layer. The purpose of G_d is to classify the training data from each domain and establish an adversarial learning strategy with feature extractor G_f. In addition, the green module represents the multiple kernel maximum mean discrepancy (MK-MMD) used to estimate the discrepancies between the simulated and real reactor data after the feature extraction.

Fig. 2Full-screen View

(Color online) Schematic of the simulation-to-reality domain adaptation (SRDA) framework

Principles of domain adversarial strategy

In the training phase, the samples X_t from ${\mathcal{D}}^{\text{S}}$ and ${\mathcal{D}}^{\text{T}}$ are fed into the feature extractor G_f to extract temporal features. Then, the obtained features $X_t^{\text{f}}$ are forwarded to both the parameter predictor G_p and domain discriminator G_d to forecast the operating parameters ${\tilde{Y}}_t$ and generate the domain label ${\tilde{d}}_t$. This process is expressed as follows: $\{\begin{array}{l}{X}_t^{\text{f}}=G_{\text{f}}\left(X_t;{\theta }_{\text{f}}\right)\hfill \\ {\tilde{Y}}_t=G_{\text{p}}\left(X_t^{\text{f}};{\theta }_{\text{p}}\right)\hfill \\ {\tilde{d}}_t=G_{\text{d}}\left(X_t^{\text{f}};{\theta }_{\text{d}}\right)\mathrm{,}\hfill \end{array}$ (2) where θ_f, θ_p, and θ_d represent the trainable weight matrices in G_f, G_p, and G_d, respectively.

This framework mitigates domain differences and makes precise parameter predictions by jointly training G_f, G_p, and G_d. More specifically, training has two goals: (1) minimizing the prediction loss for G_p and (2) maximizing the domain loss for G_d simultaneously, such that the domain discriminator cannot distinguish the domain from which the obtained features originate [35]. Feature extractor G_f and domain discriminator G_d are trained adversarially to ensure that G_f maps the simulated and real data into a common subspace and generates domain-invariant features. Consequently, the training convergence learns deep domain-invariant features in the feature extractor, which refers to temporal dependencies or generic patterns that do not significantly change between the simulated and real data. To perform adversarial training, the feature extractor G_f and domain discriminator G_d are interconnected by a gradient-reversal layer to achieve optimal results. For efficient backpropagation, the trade-off loss function (${\mathcal{L}}_{\text{total}}$) in the framework is built and formalized as follows: $\begin{array}{ll}{\mathcal{L}}_{\text{total}}\hfill & =\frac{1}{n^{\text{S}}}{\displaystyle \sum _{t=1}^{n^{\text{S}}}}{\mathcal{L}}_{\text{p}}\left(\left\{G_{\text{p}}\left[G_{\text{f}}\left(X_t;{\theta }_{\text{f}}\right);{\theta }_{\text{p}}\right]\right\}\mathrm{,}{Y}_t\right)\hfill \\ \hfill & -\frac{\lambda }{n^{\text{S}}+n^{\text{T}}}{\displaystyle \sum _{t=1}^{n^{\text{S}}+n^{\text{T}}}}{\mathcal{L}}_{\text{d}}\left(\left\{G_{\text{d}}\left[G_{\text{f}}\left(X_t;{\theta }_{\text{f}}\right);{\theta }_{\text{d}}\right]\right\}\mathrm{,}{d}_t\right)\hfill \\ \hfill & +{\mathcal{L}}_{\text{MK}-\text{MMD}}\hfill \\ \hfill & =\frac{1}{n^{\text{S}}}{\displaystyle \sum _{t=1}^{n^{\text{S}}}}{\mathcal{L}}_{\text{p}}\left({\tilde{Y}}_t\mathrm{,}{Y}_t\right)-\frac{\lambda }{n^{\text{S}}+n^{\text{T}}}{\displaystyle \sum _{t=1}^{n^{\text{S}}+n^{\text{T}}}}{\mathcal{L}}_{\text{d}}\left({\tilde{d}}_t\mathrm{,}{d}_t\right)\hfill \\ \hfill & +{\mathcal{L}}_{\text{MK}-\text{MMD}}\mathrm{,}\hfill \end{array}$ (3) where λ denotes the weight coefficient that adjusts the trade-off between the predictor, discriminator, and MK-MMD losses. n^S and n^T are the numbers of samples for training from ${\mathcal{D}}^{\text{S}}$ and ${\mathcal{D}}^{\text{T}}$, respectively. ${\mathcal{L}}_{\text{p}}$, ${\mathcal{L}}_{\text{d}}$, and ${\mathcal{L}}_{\text{MK}-\text{MMD}}$ denote the loss functions of the predictor, discriminator, and MK-MMD. When the reactor operates at low power levels, the magnitudes of certain operating parameters vary considerably; thus, the model cannot accurately fit smaller values. The improved logarithmic function is specifically designed as a predictor loss function ${\mathcal{L}}_{\text{p}}$, and its formula is defined by Eq. (4). The cross-entropy function is used as the discriminator loss function ${\mathcal{L}}_{\text{d}}$. ${\mathcal{L}}_{\text{MK}-\text{MMD}}$ is described in detail in Sect. 3.5. ${\mathcal{L}}_{\text{p}}=\frac{1}{n^{\text{S}}}{\displaystyle \sum _{t=1}^{n^{\text{S}}}}\left|\log \left({\tilde{Y}}_t+ϵ\right)-\log \left(Y_t+ϵ\right)\right|\mathrm{,}$ (4) where ϵ is a small constant that ensures that the predictions inside the logarithmic function are always positive and is set to 0.01.

${\mathcal{L}}_{\text{p}}$, ${\mathcal{L}}_{\text{d}}$, and ${\mathcal{L}}_{\text{MK}-\text{MMD}}$ have different scales corresponding to the losses in prediction, classification, and statistics, respectively. Thus, a trade-off learning strategy is developed for joint training, in which the weight λ is adjusted dynamically and gradually increase from an initial small weight during training. A formal definition of dynamic λi is expressed as follows: $p_i=\frac{i}{E}$ (5) ${\lambda }_i=\frac{2}{1+\exp \left(-10\times p_i\right)}-\mathrm{1,}$ (6) where pi represents the learning progress, which increases linearly from zero to one. i and E denote the i-th epoch being processed and the maximum number of epochs, respectively. This strategy ensures that the domain discrimination is less affected by noisy data in the initial stages of training.

In the testing phase, the trained SRDA model utilizes a feature extractor (i.e., the transformer) to capture temporal characteristics from real data, which are then fed into the predictor to forecast the operating-parameter variations in a real-world reactor. The domain discriminator and MK-MMD modules are not involved in the testing phase because their purpose is solely to assist the feature extractor in learning domain-invariant features during training.

Principles of transformer

A transformer [36] has an excellent capacity for handling time series and is utilized as the feature extractor within the SRDA model. A standard transformer has a sequence-to-sequence structure that incorporates an encoder and a decoder. To capture the dynamic characteristics and temporal dependencies from the simulated and real data, the encoder in the transformer is utilized to map the inputs into a high-dimensional domain-invariant feature matrix.

As presented in Fig. 3, the encoder comprises positional coding, multi-head attention, layer normalization, and a feed-forward neural network. The positional information is calculated using sine and cosine functions [37]. Multi-head attention aims to capture the dynamic characteristics of special events that can enhance the sensitivity of the model to critical moments or transient scenarios in reactors. As shown in Fig. 4, multi-head attention, as the basic module in the transformer, first expands the input X_t into a new embedding ${X^{\prime }}_t$ by an FC layer, which is described as follows: $X_t^{\text{'}}=X_t{W}^{\text{I}}\mathrm{,}{X}_t^{\text{'}}\in {ℝ}^{k\times d\times 3}\mathrm{,}$ (7) where W^I is the weight of the input FC layer. k and d are the head numbers of the attention mechanism and feature dimension, respectively.

Embedding ${X^{\prime }}_t$ is further propagated to multiple heads where the weights are not shared among them. Each head has three FC layers and a scaled dot-product attention. The FC layers are employed to map ${X^{\prime }}_t$ into a query ($Q\in {ℝ}^{k\times d}$), key ($K\in {ℝ}^{k\times d}$), and value ($V\in {ℝ}^{k\times d}$), which are expressed as follows: $Q\mathrm{,}K\mathrm{,}V=X_t^{\text{'}}{W}^{\text{Q}}\mathrm{,}{X}_t^{\text{'}}{W}^{\text{K}}\mathrm{,}{X}_t^{\text{'}}{W}^{\text{V}}\mathrm{,}$ (8) where W^Q, W^K, and W^V are the weights of the FC layers.

Fig. 3Full-screen View

Structure of transformer

Fig. 4Full-screen View

Structure of multi-head attention

The scaled dot-product attention can calculate the correlation between Q and K to produce an attention map, which is employed as the weight of V; the calculation of which is described formulas follows: $F_S\left(Q\mathrm{,}K\mathrm{,}V\right)=\sigma \left(\frac{Q{K}^{\text{T}}}{\sqrt{d}}\right)V\mathrm{,}$ (9) where FS is the mapping function of the scaled dot-product attention. σ(·) denotes a Softmax activation function.

The output of each head is concatenated and calculated using an output FC layer. This process can be simplified as follows: $S_i=F_S\left(X_t^{\text{'}}{W}_i^{\text{Q}}\mathrm{,}{X}_t^{\text{'}}{W}_i^{\text{K}}\mathrm{,}{X}_t^{\text{'}}{W}_i^{\text{V}}\right)\mathrm{,}i\in (\mathrm{0,}k]$ (10) $F_{\text{A}}\left(Q\mathrm{,}K\mathrm{,}V\right)=\gamma \left(S_1\mathrm{,}\dots \mathrm{,}{S}_k\right)W^O\mathrm{,}$ (11) where F_A denotes the mapping function of multi-head attention. $W_i^{\text{Q}}$, $W_i^{\text{K}}$, and $W_i^{\text{V}}$ represent the weights of the FC layers in the i-th head, whereas W^O represents the weight of the final FC layer. γ denotes the concatenated operation.

The feed-forward neural network, consisting of two FC layers and a rectified linear unit activation function, primarily boosts the nonlinear fitting capability of feature extraction, which is utilized separately for each position.

Principles of MK-MMD

A standalone domain adversarial strategy may have suboptimal effects or instability in simulation-to-reality knowledge transfer. Combining the adversarial strategy with MK-MMD can compensate for these shortcomings while promoting the stability and robustness of the model. In the SRDA framework, the dimensionality of the feature matrix X_f obtained by the feature extractor is reduced to eigenvectors $X_t^{f^{\text{S}}}$ and $X_t^{f^{\text{T}}}$, whose MK-MMD is then calculated using Eq. (12). ${\mathcal{L}}_{\text{MK}-\text{MMD}}$ can quantify the distribution discrepancies between simulated and real data using sophisticated high-dimensional mapping. Compared to traditional MMD, MK-MMD employs a set of kernel functions to fully analyze data across different scales and dimensions, which enhances the ability to identify distributional discrepancies [38]. ${\mathcal{L}}_{\text{MK}-\text{MMD}}={\Vert \frac{1}{n^{\text{S}}}{\displaystyle \sum _{t=1}^{n^{\text{S}}}}\phi \left(X_t^{f^{\text{S}}}\right)-\frac{1}{n^{\text{T}}}{\displaystyle \sum _{t=1}^{n^{\text{T}}}}\phi \left(X_t^{f^{\text{T}}}\right)\Vert }_{\mathcal{H}}^2\mathrm{.}$ (12) φ(·) denotes the function that maps the feature to the reproducing kernel Hilbert space. A kernel function K, which is a convex combination of m positive semi-definite kernels Ku, is defined to avoid a complicated mapping. $\mathcal{K}=\left\{K={\displaystyle \sum _{u=1}^{\tau }}{\beta }_u{K}_u:{\displaystyle \sum _{u=1}^{\tau }}{\beta }_u=\mathrm{1,}{\beta }_u\ge \mathrm{0,}\forall u\right\}\mathrm{,}$ (13) where Ku and βu represent the u-th kernel function defined by the Gaussian kernel and its coefficient, respectively. u denotes the number of kernels, which is set to five.

Experimental setup

Examining the effects of various feature extractors on the SRDA model can provide valuable insights. Six representative deep-learning networks are applied to explore the generalizability of the proposed framework: autoencoder (AE), CNN, recurrent neural network (RNN), LSTM, gated recurrent unit (GRU), and temporal convolutional network (TCN). The key parameters of each model are listed in Tab. 1.

In addition to comparing different feature extractors, the SRDA model is compared with six advanced domain-adaptation methods in parallel to prove its superiority. Owing to the limited domain-adaptation methods available for forecasting tasks, we modified the existing methods proposed for time-series or visual classification. Comparison methods include deep domain confusion (DDC) [39], correlation alignment via domain adaptation (CA-DA) [40], minimum discrepancy estimation for domain adaptation (MDE-DA) [41], a DIRT-T approach to domain-adversarial adaptation (DIRT-T) [42], an adaptive domain-adversarial neural network (ADANN) [43], and adversarial spectral-kernel matching for domain adaptation (ASKM-DA) [44]. The hyperparameters of the aforementioned approaches are rationally set in accordance with corresponding studies to ensure fairness. In the training phase, the learning rate and batch size in all models are critical parameters adjusted by grid optimization. For the proposed SRDA, the parameter settings are listed in Tab. 2. All the AI models are developed using PyTorch 2.0.1 in Python version 3.8.

Three precision metrics, namely the root mean square error (δ_RMSE), mean absolute error (δ_MAE), and symmetric mean absolute percentage error (δ_SMAPE), are adopted to evaluate the forecasting performance. The smaller the δ_RMSE, δ_MAE, and δ_SMAPE metrics, the higher the prediction accuracy. These can be calculated as follows: ${\delta }_{\text{RMSE}}=\sqrt{\frac{{\displaystyle {\sum }_{t=1}^n{\left(y_t-{\tilde{y}}_t\right)}^2}}{n}},$ (14) ${\delta }_{\text{MAE}}=\frac{1}{n}{\displaystyle \sum _{t=1}^n}\left|y_t-{\tilde{y}}_t\right|,$ (15) ${\delta }_{\text{SMAPE}}=\frac{1}{n}{\displaystyle \sum _{t=1}^n}\frac{\left|y_t-{\tilde{y}}_t\right|}{\left[y_t+{\tilde{y}}_t\right]/2}\times \mathrm{100,}$ (16) where n is the total number of test samples. yt and ${\tilde{y}}_t$ are the actual and predicted values at time t, resepctively.

Forecasting results

Experiments on forecasting tasks are conducted using source-only, target-only, and source-target models as baselines to validate the effectiveness of the proposed SRDA model for knowledge transfer from simulation to reality in nuclear reactors. The source-only model is trained exclusively on the source-domain training set and directly tested on the target-domain test set. Similarly, the target-only model is trained on the target-domain training set and directly tested on the target-domain test set. The source-target model is trained on the source-domain data using conventional transfer learning, and it fine-tunes the weights of its final layer on the target-domain data. Furthermore, a transformer and improved logarithmic loss are used to construct the three baselines mentioned above. Domain adaptation is not employed in this process. Forecasting of the neutron flux N₁ and hot-leg temperature T₁ are performed as examples. As shown in Fig. 6, the target-only model trained using the first 5% of the steady-operation data exhibits the largest predictive deviation. After learning a sufficient number of simulated samples, the trained source-only model can adapt to the basic variational trend of real neutron fluxes and temperatures, resulting in suboptimal effects attributable to the difference between the simulation and reality. Although the source-target model outperforms the above two models, it exhibits a certain deviation in the local area. Compared with the three baselines, the predictive trend obtained by the SRDA model with domain adaptation is generally closer to the real curves and free from the interference of operational noise in complex nuclear systems.

Fig. 6Full-screen View

(Color online) Forecast curves of the SRDA model compared with various baselines. (a) Forecast curves for neutron flux N₁ in shutdown; (b) Forecast curves for outlet temperature T₁ in shutdown; (c) Forecast curves for neutron flux N₁ in power variation; (d) Forecast curves for outlet temperature T₁ in power variation

Table 3 provides the specific average and standard deviation of the errors (${\overline{\delta }}_{\text{RMSE}}$, ${\overline{\delta }}_{\text{MAE}}$, and ${\overline{\delta }}_{\text{SMAPE}}$) for all the models in the two test sets. For neutron-flux forecasts, the SRDA model demonstrates superior performance during shutdown, with a remarkably low ${\overline{\delta }}_{\text{RMSE}}$ of 0.010 V and ${\overline{\delta }}_{\text{MAE}}$ of 0.012 V. Moreover, ${\overline{\delta }}_{\text{SMAPE}}$ is an order of magnitude lower than that of its counterparts, at 1.248%. This precision is also reflected in the power-variation scenarios, where the SRDA model achieves an ${\overline{\delta }}_{\text{RMSE}}$ of 0.008 V and ${\overline{\delta }}_{\text{MAE}}$ of 0.009 V, along with a notably low ${\overline{\delta }}_{\text{SMAPE}}$ of 0.636%. For the inlet and outlet temperature forecasts, the SRDA model achieves excellent results, with the lowest ${\overline{\delta }}_{\text{RMSE}}$, ${\overline{\delta }}_{\text{MAE}}$, and ${\overline{\delta }}_{\text{SMAPE}}$ under the shutdown and power-variation conditions. In addition, the low standard deviation further highlights the stability of the SRDA model. Case experiments demonstrate the effectiveness of domain adaptation in transferring knowledge from simulation to reality in a practical reactor.

Forecasting comparison and analysis

Temporal feature extraction within the SRDA model is interlinked with the operating-parameter prediction performance. To analyze the impact of various feature extractors, experiments are conducted to replace the transformer in the SRDA framework with the six representative neural networks specified in Sect. 4.1: AE, CNN, RNN, LSTM, GRU, and TCN. As depicted in Fig. 7, the SRDA framework consistently achieves favorable outcomes across various feature extractors, demonstrating its broad versatility. However, different backbone networks moderately affect the prediction accuracy for neutron fluxes and temperatures during the shutdown and power-variation phases. Owing to the lack of inherent architecture in AE and CNN for recording temporal dependencies, the SRDA (AE) and SRDA (CNN) models exhibit insufficient feature-extraction capabilities. The SRDA (RNN) makes unstable predictions, as reflected in its δ_SMAPE, owing to the absence of effective memory mechanisms. The SRDA (LSTM) and SRDA (GRU) models incorporate memory cells and gating mechanisms to mitigate issues, such as vanishing gradients, thereby bolstering the temporal feature-extraction process. SRDA (TCN), which incorporates dilated causal convolutions with a larger receptive field to capture long-term characteristics, exhibits precision comparable to that of SRDA (Trans) and is a robust contender. The multi-head attention block of the proposed SRDA (Trans) allows it to capture both subtle long- and short-term dependencies, which makes it superior to other feature extractors in adapting to complex variations. The experimental results are demonstrated by the steady ${\overline{\delta }}_{\text{SMAPE}}$ in the forecasting under reactor-shutdown and power-variation conditions.

Fig. 7Full-screen View

(Color online) Comparison of δ_SMAPE by various feature extractors in the SRDA model. (a) Prediction errors in shutdown; (b) Prediction errors in power variation

Table 4 presents a detailed comparison of the predictive precision across various feature extractors for the two test sets. In summary, the models intricately designed for time-series analysis, such as TCN and the transformer within the SRDA framework, possess advanced temporal feature-extraction capabilities. This facilitates more effective domain adaptation, resulting in enhanced predictive performance. Although SRDA (TCN) yields formidable and competitive results, SRDA (Trans) demonstrates unparalleled performance for both neutron fluxes and temperatures. For example, in inlet- and outlet-temperature prediction, compared with the average errors (${\overline{\delta }}_{\text{RMSE}}$, ${\overline{\delta }}_{\text{MAE}}$, and ${\overline{\delta }}_{\text{SMAPE}}$) of SRDA (TCN), SRDA (Trans) is improved by 36.158%, 34.807%, and 36.207% in shutdown, as well as by 12.205%, 14.444%, and 12.048% during power variation, respectively, further showcasing the superiority of the transformer for capturing temporal dependencies in complex, dynamic nuclear-reactor systems.

To demonstrate the knowledge-transfer superiority of the proposed method from simulation to reality, the SRDA model is compared with the six advanced domain-adaptation methods specified in Sect. 4.1, namely the DDC, CA-DA, MDE-DA, DIRT-T, ADANN, and ASKM-DA models. To ensure a fair evaluation, the transformer and improved logarithm losses are applied in the aforementioned methods. As presented in Fig. 8, the DDC and CA-DA models are statistic-based domain adaptations, and are designed to mitigate the distribution discrepancies between the simulated and real features. Their knowledge-transfer ability for neutron fluxes and temperatures during shutdown is inadequate, leading to larger errors. MDE-DA is a composite method that improves the predictive stability on the two test sets by fusing second-order statistics in CA-DA with MMD in DDC. With the incorporation of conditional entropy and a teacher model, the adversarial-based DIRT-T approach effectively forces the feature extractor to align domain-invariant features. This methodology demonstrates moderate performance in terms of inlet- and outlet-temperature predictions. In contrast, ADANN and ASKM-DA, specially designed for time-series-analysis tasks, exhibit more refined accuracy because of their marginally smaller errors compared to DIRT-T. Based on Fig. 8, adversarial-based methods tend to perform consistently better, thereby enhancing the generalization from simulation to reality. The finesse of the SRDA model involves adopting a domain adversarial strategy to extract deep domain-invariant features between the simulated and real data, in addition to utilizing MK-MMD to mitigate their distribution discrepancies. This dual approach guarantees that the model maintains the essential characteristics of the simulation while adjusting to the distribution and fluctuations that exist in a real-world reactor, which considerably improves its forecast precision.

Fig. 8Full-screen View

(Color online) Comparison of forecast errors by various domain-adaptation methods. (a) Forecast errors for neutron fluxes in shutdown; (b) Forecast errors for inlet and outlet temperatures in shutdown; (c) Forecast errors for neutron fluxes in power variation; (d) Forecast errors for inlet and outlet temperatures in power variation

Model interpretation

The proposed trade-off loss function includes the predictor ${\mathcal{L}}_{\text{p}}$, discriminator ${\mathcal{L}}_{\text{d}}$, and MK-MMD losses ${\mathcal{L}}_{\text{MK}-\text{MMD}}$. Because ${\mathcal{L}}_{\text{d}}$ and ${\mathcal{L}}_{\text{MK}-\text{MMD}}$ have been proven to enhance performance in Sect. 4.3, we focus on investigating the contribution of ${\mathcal{L}}_{\text{p}}$ to the prediction precision. ${\mathcal{L}}_{\text{p}}$ acts as an improved logarithmic loss and is responsible for adapting the predictor losses under different power levels in nuclear reactors. We compared the logarithmic loss with conventional mean square error (MSE) loss. An example (only six layers of neutron flux in one channel are shown) of the results for the neutron-flux forecasting task is shown in Fig. 9. The SRDA model using the MSE loss achieves qualified predictions for the SRDA model, with only a marginal divergence from the actual values observed at lower reactor-power levels. This deviation can be attributed to situations in which the MSE loss changes significantly with data scaling, making the model less sensitive at low powers, and the value is close to zero. The proposed SRDA model exhibits superior predictive accuracy for different power operations by balancing different numerical scales. Enhanced precision is critical for fine-tuned applications in nuclear-reactor operations, where minute deviations can have significant implications.

Fig. 9Full-screen View

(Color online) Forecast curves of different predictor loss functions in the SRDA model. (a) Forecast curves of the proposed SRDA model. (b) Forecast curves of the SRDA model using MSE loss

To present the transferability effect of the domain adaptation intuitively, feature distributions are visualized using t-distributed stochastic embedding (t-SNE). t-SNE is a nonlinear dimensionality-reduction algorithm that can transform a high-dimensional feature matrix into a two-dimensional eigenvector for visualization. As depicted in Fig. 10, the two domains are color-coded, with red denoting the simulated data and blue denoting the real data. In detail, Fig. 10a and c show the feature distributions without domain adaptation for both datasets. The features of the two domains only overlap locally, illustrating the similarities and discrepancies that commonly exist between simulation and reality. Thus, directly applying a model trained on simulated data to real data results in unsatisfactory forecasting results owing to a domain shift. In contrast, Fig. 10b and d show the feature distribution after the feature extraction in the SRDA model. Notably, after the domain adaptation, the distributions of the extracted features from the simulation and reality are uniformly mixed, illustrating that the SRDA model can mitigate the domain discrepancies effectively to enhance the operating-parameter prediction in reactors.

Fig. 10Full-screen View

(Color online) Feature visualization of simulation-to-reality knowledge transfer. (a) Without domain adaptation in shutdown; (b) Domain adaptation in shutdown; (c) Without domain adaptation in power variation; (d) Domain adaptation in power variation