## 1 Introduction
Therefore, in security maintenance and system performance evaluations, neural networks are usually used to estimate attack functions to maintain the security of the system [17; 18; 19]. Therefore, the authors propose the NN estimation algorithm for the systems in [17; 18], excluding the strict assumption of parameterization. At the same time, in reference [19], the authors use nonlinear mapping to the cyber-attack function, applying it to a compact set, achieving the estimation requirements for the attack. In addition, an anti-attack estimation algorithm is also designed to suppress the impact of attack signals [19]. Hence, developing new control algorithms against sensor and actuator attacks is of significant importance in both theory and practice. Therefore, the main challenge of adversarial attack recovery algorithms lies in how to achieve complete compensation or suppression of attack signals and ensure convergence to the equilibrium point in any situation (with or without attacks). In this paper, we propose an NN adaptive secure control scheme under the NN estimation algorithm framework. Specifically, we consider the following situations:  
In the above research on secure/resilient control [9; 10; 11; 12; 13; 14; 15; 16], secure/resilient control strategies are designed under the assumption that attacks can be parameterized. If an attack cannot be parameterized, how can we design an attack reconstruction strategy to eliminate the adverse effects of the cyber-attack? Furthermore, compared to [17; 18; 19], how can we utilize this insecure information to restore system security when attacking all channels? The goal of this paper is to obtain the NN attack reconstruction strategies in an insecure network with all information unavailable to cope with the aforementioned situations. The innovations of this paper are as follows:1. We formulate the attack reconstruction algorithm in an insecure network with all information unavailable, where the attack reconstruction strategy involves the final solution of the estimation algorithm. Specifically, by introducing mapping, time is first mapped into a compact set. Then, we use the NN estimation algorithm to solve approximate learning for unknown adversarial attacks, thereby improving the ability to cope with the adversarial attacks. Note that, although [9; 10; 11; 12; 13; 14; 15; 16] also investigates the problems of adversarial attacks, the adversarial attacks model in [9; 10; 11; 12; 13; 14; 15; 16] requires that the adversarial attacks must be parameterized; this article removes this restrictive assumption and is more applicable to general attack models. 2. We study the NN adaptive secure control scheme within the framework of backstepping, where the attack reconstruction algorithm is utilized to solve approximate learning for unknown adversarial attacks. We have demonstrated, based on the Lyapunov stability analysis theory, that the proposed secure control can ensure system performance even in the presence of adversarial attacks. Different from the requirements in the literature [20; 21], and considering that the sensor attacks should be \(s\)-sparse [3; 4; 5; 6; 7; 8; 21; 22], this paper utilizes the attack reconstruction algorithm to solve approximate learning for unknown adversarial attacks, no longer requiring these restrictive assumptions. To ensure clarity and consistency in the text, the abbreviations employed throughout the paper are summarized in Abbreviations prat. The remainder of this paper is organized as follows. Section 2 introduces the system architecture. Section 3 introduces the NN attack reconstruction algorithm to estimate the cyber-attacks. Section 4 discusses the NN adaptive secure control method. In Section 5, a stability analysis was conducted on the designed the NN adaptive secure control method. Section 6 discusses the feasibility of the algorithm through simulation.