Nonlinear differential equations and systems play a crucial role in modeling systems where time-dependent factors exhibit nonlinear characteristics. Due to their nonlinear nature, solving such systems often presents significant difficulties and challenges. In this study, we propose a method utilizing Physics-Informed Neural Networks (PINNs) to solve the nonlinear energy supply–demand (ESD) system. We design a neural network with four outputs, where each output approximates a function that corresponds to one of the unknown functions in the nonlinear system of differential equations describing the four-dimensional ESD problem. The neural network model is then trained, and the parameters are identified and optimized to achieve a more accurate solution. The solutions obtained from the neural network for this problem are equivalent when we compare and evaluate them against the Runge–Kutta numerical method of order 5(4) (RK45). However, the method utilizing neural networks is considered a modern and promising approach, as it effectively exploits the superior computational power of advanced computer systems, especially in solving complex problems. Another advantage is that the neural network model, after being trained, can solve the nonlinear system of differential equations across a continuous domain. In other words, neural networks are not only trained to approximate the solution functions for the nonlinear ESD system but can also represent the complex dynamic relationships between the system’s components. However, this approach requires significant time and computational power due to the need for model training. Furthermore, as this method is evaluated based on experimental results, ensuring the stability and convergence speed of the model poses a significant challenge. The key factors influencing this include the manner in which the neural network architecture is designed, such as the selection of hyperparameters and appropriate optimization functions. This is a critical and highly complex task, requiring experimentation and fine-tuning, which demand substantial expertise and time.

To monitor temperature as a function of varying inductance and resistance, we propose a data-driven digital twin approach for the rapid and efficient real-time estimation of the rotor temperature in an induction motor. By integrating differential equations with online signal processing, the proposed data-driven digital twin approach is structured into three key stages: (1) transforming the nonlinear differential equations into discrete algebraic equations by substituting the differential operator with the difference quotient based on the sampled voltage and current; (2) deriving approximate analytical solutions for rotor resistance and stator inductance, which can be utilized to estimate the rotor temperature; and (3) developing a general procedure for obtaining approximate analytical solutions to nonlinear differential equations. The feasibility and validity of the proposed method were demonstrated by comparing the test results with a 1.5 kW AC motor. The experimental results indicate that our method achieves a minimum estimation error that falls within the standards set by IEC 60034-2-1. This work provides a valuable reference for the overheating protection of induction motors where direct temperature measurement is challenging.

The paper focuses on the stationary self-consistent problem of magnetic insulation for a vacuum diode with space-charge limitation, described by a singularly perturbed Vlasov-Maxwell system of dimension 1.5. The case of insulated diode when the electrons are deflected back towards the cathode at the point 𝑥* is considered. First, the initial VM system is reduced to the nonlinear singular limit system of ODEs for the potentials of electric and magnetic fields. The second step deals with the limit system’s reduction to the new nonlinear singular ODE equation for effective potential 𝜃(𝑥). The existence of non-negative solutions is proved for the last equation on the interval [0, 𝑥* ) where 𝜃(𝑥) > 0. The most interesting and unexplored case takes place if 𝜃(𝑥) < 0 on the interval (𝑥*, 1] and such case corresponds to the case of an insulated diode. For the first time, a numerical analysis of complex bifurcation of solutions in insulated diode is considered for 𝜃(𝑥) < 0 depending on parameters and boundary conditions. Bifurcation diagrams of the dependence of solution 𝜃(𝑥) on a free point (free boundary) 𝑥* were constructed. Insulated diode spacing is found. These results could lead to more efficient and effective magnetically insulated diodes in future power conversion systems.

An approach for saving energy consumption of the electric submersible motor (ESM) has been proposed based on combining the thermal field model with electrical control. For decreasing the high energy consumption that has become increasingly serious in oil fields, one of our works is to increase the identification accuracy of the ESM temperature by overcoming the influence of the long cable. The thermal field model of the long cable has been derived and used to correct the equivalent circuit of the ESM first; then the procedure that can be used to identify the rotor temperature is given based on improving the calculation accuracy and efficiency of the rotor slot harmonics. The control rule for modulating the inverter frequency to match the actual oil production was simulated by summarizing the experiment results. Theory analysis and experiments denote that the average identification error of the rotor temperature has been controlled within 1° C and the average power consumption of the ESM may be decreased for about 12% by the proposed approach. This work can effectively reduce the power consumption and provide overheat protection of the ESM.