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.

We conduct a numerical study of integral algebraic equations (IAEs) with singular points, which pose significant challenges for standard computational methods. The presence of singular points often renders classical discretization schemes unstable and inaccurate. This work explores the reformulation of such problems using a least squares framework to restore numerical stability. By recasting the singular IAE as a minimization problem, the least squares method effectively handles the non-integrability and ill-conditioning inherent in direct approaches. We provide a numerical analysis of the proposed scheme and present results from several test cases, demonstrating its superior performance in terms of the convergence rate and solution quality compared to conventional methods. Our findings establish the least squares method as a viable and effective tool for solving singular IAEs.

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, 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 4/5 (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.

N this study, we implement and estimate various numerical methods for solving a nonlinear differential equation system modeling energy resources supply-demand dynamics. Both single-step methods (Taylor series, Runge-Kutta) and multi-step methods (Adams – Bashforth, Adams Predictor-Corrector) are employed. In addition to standard fourth-order methods, higher-order techniques such as the fifth-order Runge-Kutta method and the sixth- order Taylor series method are also applied. Furthermore, along with fixed-step numerical methods, we implement and assess adaptive step-size methods, including the explicit Runge- Kutta method of order 5(4) (that is RK45), the explicit Runge-Kutta method of order 8(5,3) (or DOP853), the implicit Runge-Kutta method from the Radau IIA family of order 5 (Radau), the implicit method based on backward differentiation formulas (BDF), and the Adams/BDF method with automatic switching (LSODA). The results indicate that, in the cases we considered, single-step methods are more effective than multi-step ones in capturing and tracking rapid variations of the system, while multi-step methods require less computation time. Adaptive step-size numerical methods demonstrate both flexibility and stability. Through the evaluation and analysis of numerical solutions obtained by various methods, the behaviour and dynamic characteristics of the system are explored.