# Non-Vertical Well Trajectory Design Based on Multi-Objective Optimization
## 4 Information Entropy Guided Multi-Reference Vector NSGA-III Optimization Algorithm
### Information Entropy Guided Search Direction Adaptation  
The core of this mechanism is to combine the Gaussian mixture model (GMM) clustering and Shannon entropy calculation to identify high uncertainty regions in the target space, and guide the search direction through a dynamic reference vector adjustment strategy to balance the contradiction between global exploration and local development. Firstly, in the target space entropy calculation stage, the algorithm performs GMM clustering on the normalized target value matrix \(F_{norm}\), divides it into \(c\) components, and calculates the mean value of each component, \(\mu_{1}\), \(\mu_{2}\ldots\), \(\mu_{c}\). Then, for each component i, the local uncertainty is quantified by Shannon entropy:  
\[H_{i}=-\sum_{k=1}^{n_{i}}p_{ik}\log p_{ik} \tag{39}\]  
For example, if the sample probability distribution in a cluster is [0.4, 0.3, 0.3], the entropy H = 1.57 bits, identifying it as a high-uncertainty region. The algorithm then increases the density of reference vectors in this region to enhance exploration. \(p_{ik}\) represents the probability that the \(k\) sample belongs to component i, and \(n_{i}\) is the number of samples of component i. This formula is derived from the definition of Shannon entropy in information theory:  
\[H(x)=-\sum_{x\in X}P(x)\log P(x) \tag{40}\]The core idea is to measure uncertainty by the degree of dispersion of the probability distribution. For example, in the classification task mentioned in the knowledge base, if the sample distribution in a component is uniform (such as category probability \(p_{1}\) = 0.5, \(p_{2}\) = 0.3, \(p_{3}\) = 0.2), the entropy value is higher (about 1.49 bits), indicating that there is a large uncertainty in the region; on the contrary, if the sample is highly concentrated in a certain category (e.g., \(p_{1}\) = 0.9, \(p_{2}\) = 0.05, \(p_{3}\) = 0.05), the entropy value is low (about 0.67 bits), indicating that the region has high certainty. By calculating the local entropy \(H_{i}\) of all components, the algorithm can identify high-uncertainty (high-entropy) regions in the target space, which usually correspond to complex or underexplored parts of the Pareto front.  
Based on the entropy information, the algorithm further designs a dynamic reference vector adjustment strategy to guide the search direction to the high entropy region. Specifically, define the mixing coefficient \(\alpha_{t}\) as follows:  
\[\alpha_{t}=\beta\left(1-\frac{t}{T}\right) \tag{41}\]  
where \(\beta\in[0,1]\) is the entropy weight factor, t is the current number of iterations, and T is the maximum number of iterations. The formula shows that with the increase in the number of iterations t, \(\alpha_{t}\) gradually decreases, that is, the algorithm relies more on the uniform reference vector \(u\) in the early stage (\(t\to 0\)) to achieve global exploration; in the later stage (\(t\to T\)), the weight of the high entropy regional center FFFF is gradually increased to focus on local development. Finally, the generation formula of the reference vector GGGGG is as follows:  
\[w=\alpha_{t}\cdot u+(1-\alpha_{t})\cdot\mu_{i}. \tag{42}\]  
\(\mu_{i}\) is the center point of the high entropy region. This strategy dynamically adjusts the distribution of the reference vector, so that the algorithm preferentially covers the macro structure of the target space (low entropy region) at the initial stage of iteration, and gradually concentrates to the high entropy region at the later stage, thus improving the convergence accuracy while ensuring diversity.