Page 21 - IJAMD-2-1

International Journal of AI for
Materials and Design
Predicting thermal conductivity of sintered Ag

Bayesian optimization has been widely recognized for its Table 1. Bayesian optimization algorithm calculation process
convenience and accuracy.
Bayesian optimization algorithm
Bayesian optimization is derived from the famous For n=1, 2, …, do
“Bayes theorem”: 39
a. Obtain the next evaluation point x by maximizing the
n+1
( pD f )( )p f acquisition function α
( p fD = (XI) b. Get the objective function value of the evaluation point y ;
)
()
pD n+1
c. Augment data D ={D , (x , y )};
n+1
n
n+1
n+1
where f, D ={(x ,y ),(x ,y ),…,(x ,y )}, x , and y d. Update probabilistic proxy model end for
1
2
1
n
n
n
n
2
represent the unknown objective function, the set of
observed sampling points, the decision vector, and the assesses points near the possible minimum value and in
observed value of the sampling point, respectively; p (D| f) regions that have not been sampled. This approach helps avoid
and p(f) represents the likelihood distribution of y and the entrapment in the local optima, improves proxy function
prior probability distribution of f (that is, the assumption approximation of the true objective function, and facilitates
about the unknown objective function state); and p(D) finding the minimum value of the objective function. In
represents the marginal likelihood distribution of f. The simple terms, the Bayesian optimization algorithm selects
function p(D) is usually difficult to calculate because it the next sampling point to maximize the return.
involves the product and integral of the probability density
function. However, since it does not depend on f, it is For the tuning method, the type and range of tuning
treated as a normalized constant in Bayesian optimization. parameters need to be defined. In the ANN model, the
In addition, p (f |D) represents the posterior probability learning rate, number of hidden layers, and number of
distribution of f, which describes the confidence of the neurons in each layer are selected as the tuning parameters.
unknown objective function after modifying the prior Among them, the learning rate in the optimizer is
function from the observed data set. an important hyperparameter in the neural network,
The Bayesian optimization algorithm consists of two regulating the step size of each parameter update and
core parts: the probabilistic agent model and the acquisition directly affecting the convergence speed and performance
function. The probabilistic agent model includes the of the model. The number of hidden layers and neurons
prior probability model and the observation model. The in each layer determines the structure of the model. To
former is p(f), while the latter describes the mechanism streamline parameter adjustment, the number of neurons
by which the observed data are generated, the likelihood in each layer is set to be the same to ensure that the number
distribution p(D| f). The posterior probability distribution of hidden layers and neurons in each layer can be used as
p (f |D), containing the observations of the latest evaluation two independent parameters for parameter adjustment.
points, is obtained using the Bayesian formula to update Figure 10 presents the results of Bayesian optimization
the probabilistic agent model. According to the posterior of hyperparameters in the ANN model. The learning rate
probability distribution, the next most “potential” ranges from 1e to 0.1 and is distributed exponentially to
-4
evaluation point is selected by maximizing the collection improve optimization efficiency. The number of hidden
function, and an effective collection function can ensure layers is presented as an integer, ranging from 1 to 5, while
that the selected evaluation point sequence minimizes the the number of neurons is presented as an integer, ranging
total loss value: from 1 to 140. The evaluation index MSE (Equation IV)

Loss = ∑ n i= 1 y * y− i (XII) of the model is reflected by the color of the data points,
and the corresponding value range is referred to the color
scale on the right. Sampling occurs across the entire
where y* represents the optimal solution of the current
evaluation point. hyperparameter space, avoiding the local optima. The
points around the final optimal result are relatively dense,
The specific calculation process of the Bayesian indicating that the model undergoes fine-tuning and
optimization algorithm involves an iterative process of optimization at the later stages. The final hyperparameters
parameter updates, and its specific algorithm framework is include two hidden layers, each with 32 neurons, and a
presented in Table 1. 40 learning rate of 9.29e .
-4
Figure 9 displays the principle of the Bayesian
optimization algorithm. Each repeat sampling generates a 3. Results and discussion
minimum value for the objective function. After the first The established ANN model was used to map
random sampling of the function, the second sampling the relationship between the input characteristics

Volume 2 Issue 1 (2025) 15 doi: 10.36922/ijamd.5744