# Parameter-Matching Multi-Objective Optimization for Diesel Engine Torsional Dampers
## 2 Research Methodology
### Neural Network Training of Torsional Vibration Damper Parameters  
The nonlinear relationships between torsional damper parameters (the moment of inertia, stiffness, and damping) and system responses (the maximum torsional angle and stress amplitudes) necessitate advanced modeling techniques. Neural networks excel at mapping these input-output correlations, yet their predictive accuracy critically depends on hyperparameter tuning. Bayesian optimization algorithms outperform conventional genetic algorithms in both precision and computational efficiency for hyperparameter selection [24; 25; 26], making them ideal for this high-dimensional problem. Figure 4 presents the Bayesian optimization framework proposed in this study, which automates the adjustment of a fully connected neural network's hidden layer architecture (depth/width) and learning rate. The framework integrates damper input features to constrain the hyperparameter search space. A Gaussian process surrogate model efficiently evaluates hyperparameter candidates, combined with cosine annealing learning rate scheduling and early stopping to prevent overfitting. This method significantly reduces prediction errors for vibration and stress metrics, resulting in a compact, high-fidelity surrogate model that supports multi-objective damper optimization under tight tolerance constraints. The relationship between torsional damper parameters and crankshaft responses is represented through a hyperparameter-optimized neural network. This optimized network enables the more accurate prediction of the crankshaft system's dynamic responses and maximum component stress compared to conventional approaches. The Bayesian optimization algorithm, functioning as a global optimization framework, relies on probabilistic surrogate models to approximate the probability distribution of the objective function. Through iterative cycles, the algorithm refines its search for optimal solutions. Central to this process is the construction of Gaussian process-based surrogate models, where parameter combinations are systematically evaluated to balance the exploration of underexamined regions and the exploitation of known high-performance domains--a mechanism critical for avoiding local optima traps in high-dimensional design spaces [27; 28]. \[\left\{ \begin{aligned} f(x)&\sim GP(m(x),k(x,x^{\prime}))\\ m(x)&=E[f(x)]\\ k(x,x^{\prime})&=E[(f(x)-m(x))(f(x^{\prime})-m(x^{ \prime}))]\end{aligned} \right. \tag{21}\]  
In the equation, \(m(x)\) is the mean function and \(k(x,x^{\prime})\) is a covariance function. The probabilistic surrogate model iteratively approximates the objective function by initializing prior information and progressively refining estimates through Bayesian updates. At each iteration, an acquisition function (Equation (22)) strategically selects high-potential evaluation points to balance the exploration of uncertain regions and the exploitation of known optima [29]. \[x_{n+1}=\max_{x\in X}\alpha_{n}(x;D_{1:n}) \tag{22}\]  
Figure 5 demonstrates the predictive performance of the Bayesian-hyperparameter-optimized neural network, achieving a coefficient of determination (R\({}^{2}\)) of 0.9683 on the training set and 0.9495 on the testing set. Corresponding mean squared error (MSE) values reached 0.0643 (training) and 0.076 (testing), confirming the model's robust predictive capabilities. These metrics validate the network's effectiveness in capturing nonlinear relationships between damper parameters and system responses. Figure 4: Bayesian hyperparameter optimization neural network framework. The SHAP analysis quantifies the contributions of the torsional stiffness, moment of inertia, and damping coefficients to twist angle and stress, uncovering inherent contradictions in multi-objective optimization and parameter control. As shown in Figure 6, high torsional stiffness reduces twist deformation but amplifies stress concentration, demonstrating a vibration stress trade off. The bidirectional fluctuations of the moment of inertia confirm its nonlinear effects.