The design of optimizer parameters affects model performance and is widely applied in fields such as image analysis, autonomous driving, and security monitoring. However, the interpretability and generalizability of optimizers are insufficient, limiting their practical applications. To address these challenges, we introduce a novel approach using transfer function and phase trajectory methods to design the parameters and critical conditions for Stochastic Gradient Descent with Momentum (SGD-M) and Nesterov Accelerated Gradient (NAG). The proposed theory is verified through numerical examples and image recognition experiments. First, using the phase trajectory method, a qualitative analysis of the responses of SGD-M and NAG to initial states is conducted, revealing the influence of parameters on the phase trajectory. Then, through the transfer function method, a quantitative analysis of the unit step response of SGD-M and NAG is performed to explain the impact of parameters on system response. Finally, numerical examples and image recognition experiments verify the significant impact of the momentum control parameter $g(\mu )$ and momentum parameter $\alpha$ on optimizer performance, stability, and time-domain characteristics. Experimental results show that adjusting $g(\mu )$ or $\alpha$ improves image classification accuracy on the Modified National Institute of Standards and Technology (MNIST) and Canadian Institute for Advanced Research (CIFAR-10) datasets. It reduces the loss value, validating the effectiveness of the proposed theory.

The development of medical big data analysis technology has enabled vast imaging datasets to play an important role in identifying complex pathologies, supporting individualized diagnoses, and enhancing diagnostic and treatment precision. Pediatric pneumonia, a leading cause of mortality in children, results from infections and is characterized by inflammation of the terminal airways, alveoli, and pulmonary interstitium. Accurate and timely diagnosis is essential for successful intervention. However, most current studies are constrained by simplistic classification tasks, and existing models struggle to effectively extract multi-scale features from Chest X-Ray (CXR) images while eliminating redundant information. Moreover, the datasets used in existing studies lack the diversity and scale necessary to meet the demands of clinical practice. To address these challenges, this paper presents the a multi-branch dual residual attention Convolutional Neural Network (CNN) model for pediatric pneumonia classification model (namely PP-ConvNeXt), designed to enable automatic diagnosis of pediatric pneumonia. First, the model incorporates a novel multi-branch dual-residual structure, enhancing its capacity to extract multi-scale features. Second, the Coordinate Attentional Feature Fusion (CAFF) module is introduced to integrate cross-branch features, allowing for comprehensive acquisition of lesion information. Finally, this research pioneers the automatic five-class diagnosis of pediatric pneumonia, offering clinicians more accurate diagnostic insights. Additionally, a comprehensive dataset comprising 5632 CXR images from Hainan Women and Children’s Medical Center (HWCMC) is presented. Experimental results indicate that the PP-ConvNeXt model surpasses state-of-the-art models and expert human diagnosticians in the five-class diagnosis of pediatric pneumonia, achieving an Area Under the Curve (AUC) score of 89.01%. Additionally, key performance metrics, including accuracy and recall, further validate the model’s superior performance.