Solving Nonlinear Energy Supply and Demand System Using Physics-Informed Neural Networks

TL;DR

The study addresses solving a chaotic four-dimensional nonlinear energy supply-demand (ESD) system by employing Physics-Informed Neural Networks (PINNs) that enforce the differential equations as residuals in a loss function. A four-output neural network is trained to approximate the time-series $x_1(t)$, $x_2(t)$, $x_3(t)$, and $x_4(t)$ across a continuous time domain, with the total loss combining equation residuals and initial-condition constraints. Results show that the PINN solution matches or slightly surpasses the RK45 reference in accuracy, with $R^2$ values near 1 and very small MAE/MSE/RMSE across all four state variables, while offering continuous-domain predictions and leveraging automatic differentiation. The work demonstrates PINNs as a promising, scalable approach for nonlinear, chaotic ODE systems in energy systems, albeit with higher computational demands and sensitivity to architecture and optimization choices. These findings suggest PINNs can enable real-time, continuous forecasting for complex energy systems, provided further advances in stability and efficiency are pursued.

Abstract

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.

Figures (5)

• Figure 1: Overview of the method, where $\varepsilon$ is the desired value to be achieved when minimizing the loss function, and $max$ is the maximum limit of the number of training epochs
• Figure 2: A chart describing the value of the loss function over the training epochs
• Figure 3: The chart visualizes a comparison of the accuracy of the two methods
• Figure 4: A general graph illustrating the direct comparison results between the RK45 numerical method and the neural network method
• Figure 5: A detailed graph illustrating the direct comparison results between the RK45 numerical method and the neural network method