HYBRID NEWTON-RAPHSON WITH PHYSICS-INFORMED NEURAL NETWORKS FOR SOLVING HIGH-DIMENSIONAL NONLINEAR EQUATIONS IN REAL-TIME SYSTEMS

Abstract

Nonlinear systems of equations with high dimensions are crucial for power systems, robotics, optimization, control, and many real-time engineering applications; however, conventional approaches like Newton-Raphson have the problems of being potentially slow, sensitive to initial guesses, instable because of Jacobian, and computationally expensive. In the current study, a novel HNR-PINN (Hybrid Newton-Raphson with Physics-Informed Neural Networks) approach is proposed that combines the prediction ability of PINNs with the numeric capability of N-R refinement. The physics-informed PINN produces an educated guess and Jacobian approximation based on learning of nonlinear structure and physical laws, followed by iterative refinement provided by the Newton-Raphson. The new method was tested on 1,000,000 nonlinear problems of various difficulties and dimensionalities. It was proven that HNR-PINN converges with 67% fewer iterations and does not diverge in 89% of cases in comparison with the conventional approach in ill-conditioned problems. Moreover, HNR-PINN was applied to high-dimensional problem (1,000 variables) related to power flow analysis to estimate the computational performance in real-time applications. The execution time is decreased from 2.40 s to 0.31 s; that is, an 87.1% reduction or 7.74x acceleration rate is achieved. The hybrid framework keeps the necessary numerical refinement but decreases the computational load. Hence, it is proved that the combination of physics-informed learning and traditional numerical approaches allows finding a balance between the computational speed, convergence stability, scalability, and accuracy. HNR-PINN appears to be promising tool for real-time solving of high-dimensional nonlinear systems of equations.