## 4 Result
### Domain Knowledge Embedding
Therefore, we redesigned the neural network model, as illustrated in \[\tau=\tau_{y}+\mu_{p}\hat{\gamma} \tag{10}\]1 In this new architecture, a module named \"Flowability Characterization Neurons\" was introduced. This module incorporates the temperature-corrected consistency coefficient and flow behavior index as independent neurons, which are embedded into the model structure and directly influence the output layer. This enhancement significantly improves the model's ability to represent the actual downhole rheological behavior of cement slurry. Conventional neural networks typically adopt the linear transformation form wx+b as the fundamental structure of neurons. In this study, a physics-informed embedding approach is proposed, wherein the traditional linear mapping is replaced by temperature-corrected rheological equations, as defined in Equations (13) and (14). During backpropagation, the nodes governed by these physical equations are automatically differentiated using the chain rule, and their corresponding gradient expressions are given as follows:  
\[\tau=\tau_{y}+\mu_{p}\hat{\gamma} \tag{10}\]2 \[\tau=\tau_{y}+\mu_{p}\hat{\gamma} \tag{10}\]3  
To evaluate the impact of embedding temperature-corrected rheological equations into neural networks, a comparative experiment was designed with two structurally consistent models:  
Model A adopts the proposed Wide and Deep (WD) architecture, as illustrated in \[\tau=\tau_{y}+\mu_{p}\hat{\gamma} \tag{10}\]4 The relationship between downhole temperature and the rheological parameters--consistency coefficient \(k\) and flow behavior index \(n\)--is explicitly defined by Equations (13) and (14), and these temperature-corrected values are embedded as intermediate neurons into the network. Model B shares the same overall structure and input features, including temperature. However, instead of using physically derived equations, the intermediate neurons \(k\) and \(n\) are generated via simple learnable linear transformations of temperature, i.e., y = wx + b. Both models were trained for 200 epochs, during which the average gradients of the \(k\) and \(n\) neurons with respect to the downstream fully connected layer were recorded. \[\tau=\tau_{y}+\mu_{p}\hat{\gamma} \tag{10}\]5 Improved wide and deep neural network architecture. This gradient-based analysis enables a direct comparison of model sensitivity and feature utilization under the two embedding strategies. The results are presented in \[\tau=\tau_{y}+\mu_{p}\hat{\gamma} \tag{10}\]6  
As shown in the figure, the average gradients of the \(k\) and \(n\) neurons in Model A remain consistently high throughout the training process, indicating that the model continuously attends to these physical features--the consistency coefficient \(k\) and the flow behavior index \(n\). In contrast, in Model B, where \(k\) and \(n\) neurons are generated through simple linear mappings (designed to maintain structural consistency but lacking physical significance), their gradients rapidly decay to near zero within the first 50 epochs. This suggests that the Model B fails to effectively utilize these features. These results demonstrate that embedding physical equations enhances the model's sensitivity to critical variables, thereby improving both learning efficiency and interpretability. \[\tau=\tau_{y}+\mu_{p}\hat{\gamma} \tag{10}\]7 Gradient of neurons during model training: (**a**) Neuron k. (**b**) Neuron n.