## 1 Introduction  
The wireless communication connection between physical components and network computing layers in cyber-physical systems (CPSs) introduces vulnerabilities in network attacks [1, 2]. Consequently, there is a crucial need to thoroughly study and address the security issues associated with CPSs to enhance the security of CPSs and safeguard CPSs against cyber-attacks. Thus, many researchers focus on improving the means of keeping CPSs secure. The secure state estimation (SSE) is the problem of estimating the system state from the compromised data packets and the noisy [3]. To ensure an SSE under cyber-attacks, it is crucial to identify the information transmission channel affected when a portion of the collected information is tampered with. In the context of an SSE amidst cyber-attacks, the challenge lies in identifying the precise set of compromised channels among numerous possibilities, rendering the problem an instance of combinatorial optimization and falling within the realm of NP-hard problems [4, 5, 6, 7, 8]. To solve such difficulties, the author devotes the \(L_{1}/L_{r}\) relaxation technique in reference [4]. The author applies the ideas from reference [4] to the event-triggered projected gradient descent algorithm [5]. Note that the aforementioned algorithms [4, 5] can only guarantee the correctness of the estimates under certain restrictive assumptions about the system structure. Reference [6] investigates the impact of cyber-attacks, establishing that the stability of the observation error system is stable as long as the 2s-sparse observable condition of the original system is satisfied. The existing work provides fundamental limits for estimation against integrity attacks, specifically proving the necessity of \(2s\)-sparse observable for SSE against \(s\)-sparse attacks. For example, reference [7] analyzes the convergence of observation errors under \(s\)-sparse attacks using the separation principle and Lyapunov stability theory via a system based on \(2s\)-sparse observable. However, it is worth noting that control systems are often nonlinear. In such complex systems, accurate SSE may require the use of complex models and algorithms, increasing computational complexity and cost. Existing studies suggest that increasing secure/resilient control or prior knowledge can alleviate the damage caused by cyber-attacks. Through further research on the assumption that attacks can be parameterized, which is state-dependent, reference [9; 10] relaxed the requirement for attacks to be bounded, allowing attacks to be unbounded. Reference [11] introduces the coordinate transformations in controller design, which can overcome the negative impact caused by cyber-attacks. References [12; 13] propose an attack compensator that effectively suppresses the impact of attacks by removing extreme values, thereby ensuring that normal nodes correctly estimate the system state under malicious node interference and avoid traversal search methods. Meanwhile, reference [14] proposes a novel attack compensator to suppress the effect of the attack signal on state information by compressing the amplitude of abnormal measurement signals. Due to the challenge of unknown time-varying gains in the design of control systems after the parameterization of attacks, to solve it, the author proposes a Nussbaum gain technique [15]. In addition, the authors propose an observer with attack compensation in [15], which successfully solved the problem of unknown control gain caused by unknown time-varying coefficients. However, the assumption that attacks can be parameterized is too strict, and it is difficult to find conditions that satisfy this assumption in real physical systems. As deep learning continues to be studied, the performance of NN models is constantly improving. Researchers have proposed many prediction methods [16], making the application of NN more reliable and understandable.