# Stochastic Memristor Modeling Framework Based on Physics-Informed Neural Networks
## 1 Introduction  
The architecture proposed by [PERSON] has led to exponential advancements in computing technology. However, technical limitations, such as the memory bottleneck in the [PERSON] architecture, have increased the need for research into new technologies [1]. Additionally, emerging technologies like machine learning, deep learning, and artificial intelligence are rapidly evolving, which is driven by vast amounts of data and high-performance computing hardware. These technologies require low-power, high-efficiency processing systems with parallel structures, such as neuromorphic systems [2]. The emergence of neuromorphic systems has enabled the processing of vast amounts of information that is essential for artificial intelligence (AI). Neuromorphic architectures, inspired by the synaptic structures of the human brain, enable advanced parallel computing technologies [3]. These technologies can be optimized through crossbar arrays, where the selection of elements that serve as synapses between metal substrates is considered crucial [4, 5]. In neuromorphic systems, the role of synapses in the human brain can be performed by memristors [6]. The memristor, known for its high-speed operation and low power consumption, is well suited for use in neuromorphic systems [7]. Therefore, the memristors are considered the next generation of devices for these neuromorphic systems [8]. The memristor is the fourth passive component, along with resistors (R), capacitors (C) and inductors (L) proposed by [PERSON] in 1971 [9]. It has nonlinear characteristics due to a variety of mechanisms and their physical structure [10]. Moreover, the memristor equation of state can be described in various forms depending on the material structure [11]. The memristors have already found industrial applications demonstrating their potential in various fields [12]. Furthermore, it has a wide range of applications such as digital and analog circuits, neuromorphic and in-memory computing and high-speed non-volatile memory [13, 14, 15]. However, these technologies are still evolving, and significant challenges remain. Recently, there has been progress in research utilizing neural networks to integratea number of physical models [16; 17]. Although the memristor has a stochastic nature in the time domain, the models do not include the variability of memristors. The stochastic characteristics of memristor systems refer to the inherent fluctuations arising from structural and chemical properties [18]. The stochastic behavior in the memristor dynamic system is caused by the movement of oxygen vacancies [19]. The research of stochastic dynamics and variability is an interesting topic that must be addressed for device reliability. The occurrence of noise is stochastic, and noise under certain conditions has a significant effect on the change in conductance of a memristor [20]. The stochastic memristor models should be considered to explain phenomena such as noise and nonlinear relaxation in conventional memristor dynamic systems [21]. The stochastic process of a memristor under noise can be represented in the form of the Fokker-Planck equation (FPE) by Brownian motion [22; 23; 24]. In addition, the form of the FPE with noise is represented as a time-varying partial differential equation (PDE). The solution of this is expressed as a probability distribution of the memristor state with noise. The PDEs are very complex and time consuming to solve with traditional numerical methods. Physics-informed neural networks (PINNs) are well suited to solving high-dimensional differential equations [25]. They do not require large amounts of data, which can simplify algorithms and reduce computation costs [17; 25; 26; 27]. PINNs can incur a computational burden to train because they need to calculate weights and biases. However, once the weights and biases are determined, it is possible to implement a simple circuit simulator using them [28]. Thus, we use PINNs to solve them.