###### Abstract  
This manuscript presents an analysis of machine learning (ML) usage in the Frequency Symbolic Method (FSM) to enhance the diagnosis of faults in parametric circuit analysis and optimization, with a particular focus on Linear Periodically Time-Variable (LPTV) systems. We put forth a few ML-based approaches for fault diagnosis (including anomaly detection), invisible feature detection, and the prediction of FSM output. These methodologies concentrate on identifying and diagnosing faults by evaluating particular ML techniques, extracting pertinent features, and determining the desired diagnostic outputs. The use cases of ML application considered in this paper demonstrate that machine learning can enhance fault detection and diagnosis, reduce human errors and identify previously unnoticed anomalies within the FSM framework. ML has never been used in FSM before, so the key aim of this paper is to consider possible use cases of AI application in FSM. Additionally, feature extraction, required as an input stage for the ML model, is proposed based on FSM peculiarities. This work can be considered a study of ML application in FSM. machine learning: frequency symbolic method; linear periodically time-variable circuits +
Footnote †: journal: _aciences_  
0000-0001-9993-3227-3274][PERSON]. [PERSON], [PERSON]. It is a well-known fact in circuit theory that there are two basic ways to analyze linear and time-invariant (LTI) systems. The first method is to analyze systems in the time domain, either by solving differential equations describing the systems or by using convolution techniques. The second group of methods, called frequency analysis, is based on the use of Laplace or Fourier transforms. LTV systems represent a direct generalization of classical linear and time-invariant (LTI) systems. Accordingly, an analysis of their dynamic and frequency properties can be performed in both the time and frequency domains. Furthermore, the concepts of impulse response, frequency response, and transmission, familiar from the theory of stationary systems, can be generalized to non-stationary systems. In the time domain, parametric systems can be described using two different methods. The first method involves representing systems in the time domain using a parametric linear differential Equation (1) [7; 9; 14]:  
\[a_{n}(t)y^{(n)}(t)+a_{n-1}(t)y^{(n-1)}(t)+\ldots+a_{1}(t)y^{\prime}+a_{0}(t)y(t )=x(t), \tag{1}\]  
where \(x(t)\), \(y(t)\), are the input and output signals, \(t\) is the independent variable (time), and \(a_{i}(t)\) are the time-varying coefficients of the differential equation, which are called parametric functions. Except for a few special cases, there are no well-established analytical methods for solving this equation. The authors' current research focuses on a class of parametric systems with periodically varying coefficients known as LPTV, or Linear Time Periodic Systems [15]. The alternative method of describing an LTV system in the time domain, as opposed to the parametric differential equation approach, is a parametric convolution using the impulse response \(h(t,\tau)\) as the kernel:  
\[y(t)=\int\limits_{0}^{t}h(t,\tau)x(\tau)d\tau. \tag{2}\]  
It is important to note that the impulse response of LTV systems differs from that of time-invariant systems in that it depends not only on the time \(t\), but also on the instant \(\tau\) at which an excitation is applied to the input of the circuit. In the frequency domain, two approaches may be used to describe a parametric circuit. The first is identical to that of LTI systems and is defined as the ratio of the complex output signal \(y(t)\) to the complex input signal, which is the monoharmonic function \(\mathrm{e}^{\mathrm{i}\omega t}\).