# Stochastic Memristor Modeling Framework Based on Physics-Informed Neural Networks
## 3 PINN for Memristor Noise Modeling  
A memristor stochastic system with a noise term has the form of a complex differential equation. A PINN is an artificial neural network that allows the numerical analysis of physical laws described by a nonlinear PDE. To construct system equations for nonlinear systems using the PINN model, the physical laws that govern the memristor's dynamic behavior, such as voltage-dependent conductance changes, are incorporated into the loss function. For example, the voltage applied to the memristor over time causes state changes in the memristor [22; 24; 30; 31]. These state changes can be represented by differential equa  
Figure 1: The basic equivalent circuit of a TiO\({}_{2}\) twoterminal memristor based on dopant concentration. \(R_{ON}\) and \(R_{OFF}\) represent the regions with high and low dopant concentrations, respectively. tions that capture the nonlinear I-V characteristics. By incorporating the time-dependent voltage variable into the system equations, the PINN model can accurately predict the memristor's dynamic behavior under various operating conditions, such as sinusoidal or pulsed voltages. Dynamic analysis is performed by modeling the memristor's response to different voltage conditions. For instance, when a sinusoidal or pulsed voltage is applied, the memristor's state evolves over time due to the influence of stochastic factors such as white noise or cycle-to-cycle variations [22; 23; 24]. The PINN model allows for the integration of these time-dependent fluctuations into the system's differential equations, enabling accurate predictions of the memristor's dynamic behavior. The residual terms in the differential equations capture both the deterministic and stochastic aspects of the system, making the model suitable for a wide range of practical scenarios. Once the PINN model is trained, the extracted weights and biases can be applied to circuit simulators, enabling real-time dynamic simulations of the memristor's behavior under varying voltage conditions. The PINN requires more time than numerical methods to achieve high accuracy during training. However, while numerical methods face challenges in simulating the stochastic behavior of memristors in circuit simulators, the PINN offers a simpler implementation using weights and biases. It is a model with a simple structure and a higher accuracy than traditional deep neural networks (DNNs) and recurrent neural networks (RNNs). DNN has the ability to learn from large amounts of data, and RNN excels at processing time-series and sequential data. In order to incorporate physical laws, these neural networks require additional components, such as an LSTM [16; 28; 33]. The PINN is designed to minimize not only data-driven errors but also errors derived from physical laws by incorporating the residual terms of the governing equations and the initial and boundary conditions into the loss function. The inclusion of residual terms allows for accurate predictions in nonlinear systems even when data are scarce. This network transforms the problem into an optimization task that minimizes residual terms using automatic differentiation based on input state and time coordinates [25; 26; 27; 34]. This approach provides analytical solutions to differential equations, offering an advantage over traditional data-driven learning methods such as DNN and RNN. The characteristics of conventional memristors are derived in the form of differential equations over time. When noise is incorporated into these characteristics, the equations become more complex. PINN is well suited for learning such equations, as it does not require vast amounts of data to solve high-dimensional partial differential equations. The residual term \(u(x,t)\) of the PDE is given by [23; 24; 32]  
\[u(x,t)\colon\frac{\partial P(x,t)}{\partial t}+\frac{\partial}{\partial x}[r(x,t)P(x,t)]-\frac{\Gamma}{2}\frac{\partial^{2}P(x,t)}{\partial x^{2}}=0 \tag{7}\]  
\(u(x,t)\) defined in Equation (7) is minimized using a PINN.