## 1 Introduction  
With the development of aerospace technology, the demand for landings to explore the moon and other exoplanets has increased dramatically. The future mission requires the vehicle to achieve a high-precision landing to explore regions of scientific value efficiently [1]. In recent years, rocket vertical recovery technology developed from planetary soft-landing technology has attracted the attention of many researchers. The purpose of the technology is to guide the rocket core stage to achieve vertical recovery, which significantly reduces the cost of space transportation [2]. The above scenarios both lead to new requirements for the powered landing guidance algorithm. Due to the complex dynamic environment, the initial state of the powered landing phase cannot be accurately predicted in advance, so the tracking guidance method based on offline calculation cannot be adopted. Therefore, the real-time optimal guidance algorithm has become the key technology for the powered landing problem. The primary purpose of the work is to design a high-precision landing guidance algorithm based on convex optimization fusing the deep neural network (DNN) predictor, which satisfies optimality and real-time performance. The early guidance algorithms mainly adopted analytical algorithms, including the gravity turn [3, 4] and Apollo polynomial guidance algorithm [5]. The gravity turn guidance algorithm cannot guarantee the landing accuracy. The Apollo guidance algorithm can achieve a high-precision lunar landing. However, the algorithm cannot deal with process constraints or guarantee the optimality of fuel consumption [6]. After the 1960s, many scholars have proposed various indirect guidance methods based on [PERSON]'s minimum principle. These methods transform the optimal control problem into a two-point boundary value problem (TPBVP), and different methods were used to solve the TPBVP [7, 8, 9, 10]. However, the convergence of the indirect method is sensitive to the costate variables, and how to select the initial costate variables needs further research. To sum up, the traditional guidance algorithms lack real-time and optimality performance, and it is difficult to achieve an accurate landing in a complex dynamic environment. In recent years, many scholars have proposed some potential real-time guidance algorithms, which can be divided into two categories. The first category is based on convex optimization (CVX) [11; 12; 13; 14; 15; 16; 17]. [PERSON] [11] and [PERSON] [12] first used variable substitution and lossless convex optimization (LCvx) to deal with the nonconvex dynamic equations and constraints, and the second-order cone programming (SOCP) was built to solve the Mars landing problem. [PERSON] and [PERSON] [13; 14] proposed the principle of successive convex optimization (SCvx). By linearizing the constraint and adding the trust-region constraint, the nonlinear optimal control problem was transformed into a sequence of sub-SOCPs. This method can deal with more complex dynamic constraints and has been successfully applied to various aerospace guidance problems, including powered landing. The most remarkable advantage of the CVX-based algorithm is that it has strict convergence proof, and the algorithm can obtain the optimal solution in polynomial time. However, due to the limitation of the current onboard CPU, it is still hard to achieve real-time guidance. The second category is ML-based algorithms [18; 19; 20; 21; 22]. [PERSON] [19] proposed a supervised learning guidance algorithm based on DNN. Firstly, the indirect method generated abundant training samples, and then the DNN was used to learn the relationship between state and control variables. The control signals can be mapped directly from the state variables when running onboard. The algorithm is very efficient, but it cannot guarantee the correctness of the mapped control signals. The landing accuracy and optimality are lower than the CVX-based algorithm. To sum up, CVX and ML algorithms are the two main ways to realize real-time guidance.