# Research on Wellbore Trajectory Optimization and Drilling Control Based on the TD3 Algorithm
## 2 Related Work
### TD3 Network
In order to prevent the strategy from being too aggressive, the target strategy action \(a^{\prime}\) adds noise and performs truncation processing. \[a^{\prime}=\pi^{\prime}(s^{\prime};\phi^{\prime})+\varepsilon,\varepsilon \sim N(0,\sigma^{2}),\varepsilon\in[-clip,clip] \tag{23}\]  
The noise \(\varepsilon\) obeys the Gaussian distribution \(N(0,\sigma^{2})\), which is used to smooth the Q value update. Truncation processing \(\varepsilon\in[-clip,clip]\) avoids the extreme value interference strategy. (3) Theoretical properties and convergence. Theorem 3.1 (Stability of TD3). The stability of TD3 is based on the following assumptions:  
A1: [PERSON] continuous strategy;  
A2: Bounded reward function. The policy error of TD3 satisfies the following inequality:  
\[r(s_{t},\ a_{t}) \tag{15}\]0  
where \(\beta\) represents the estimation error of the Q network and \(\eta\) represents the regulatory factor determined by the delayed update step \(d\). This theorem confirms that TD3 stabilizes to the optimal policy as iterations increase, with delay step d regulating convergence speed. Compared to DDPG, TD3 reduces policy variance, making it suitable for high-dimensional drilling decisions. This theorem shows that TD3 can guarantee the stable convergence of the strategy under reasonable conditions. TD3 effectively suppresses the overestimation bias of DDPG through mechanisms such as the double Q network, delayed update, and target strategy smoothing. It also improves training stability and sample utilization, making it an important benchmark algorithm for continuous control tasks in reinforcement learning. As presented in Table 1, the performance comparison highlights significant improvements of the proposed method across all metrics. (4) TD3 algorithm network architecture. The architecture of TD3 is shown in \[r(s_{t},\ a_{t}) \tag{15}\]1 There are two network structures: the actor network (actor) and the critic network (critic). The actor network is responsible for strategy generation. The output is the probability distribution of continuous actions, \begin{table}
\begin{tabular}{c c c c} \hline
**Characteristics** & **DDPG** & **TD3** & **Theoretical Advantages** \\ \hline Q value estimation & Single network, overestimation & Dual network, min suppression & Deviation reduction \(\mathcal{O}(\sigma)\) \\ Strategy update frequency & Real-time update & Update with a delay of \(d\) steps & Variance reduction \(\mathcal{O}(1/d)\) \\ Target stability & Hard update & Soft update (\(\tau=5\times 10^{-3}\)) & Tracking error reduction 40\% \\ Convergnore rate & \(\mathcal{O}(1/\sqrt{k})\) & \(\mathcal{O}(1/\sqrt{k})\) & The sample efficiency is increased by \\ \hline \end{tabular}
\end{table} 
Table 1: Performance comparison table. and the next action is determined by sampling. The critic network is composed of two independent Q networks, which evaluate the value of the actions generated by the actor network and take the smaller value of the two Q values as the target, so as to reduce the risk of overestimation. In addition, TD3 introduces a target strategy smoothing mechanism to add noise to the target action to improve the robustness of the strategy. At the same time, by delaying the strategy update, that is, only the actor network is updated several times after the reviewer network is updated, the instability caused by the frequent update of the strategy network is avoided. In addition, based on the improvement in the training convergence effect and training stability, TD3 adopts the idea of experience playback in the DQN and randomly extracts the historical data stored in the data for learning. In this way, each data point does not generate a time correlation but slows down the training speed. The critic network is mainly used to judge the probability of an action (for an action) in a given state, while the critic network is implemented by a double Q network to avoid the problem of overestimation of the Q value.