Comparison sub-equation neural network method and neural network predictions method for the nonlinear systems in fluid and nuclear physics.

Abstract

This study aims to compare the effectiveness of the sub-equation neural network method (SENNM) and the neural network predictions method (NNPM) for solving nonlinear systems commonly encountered in fluid dynamics and nuclear physics. The comparison focuses on accuracy, convergence behavior, training performance, and the preservation of key physical properties. The SENNM integrates analytical sub-equation structures with neural network learning to efficiently capture localized waveforms and soliton-like behaviors. In contrast, the NNPM employs a fully data-driven or physics-informed neural architecture to approximate global solutions without relying on predefined functional assumptions. Both methods are implemented on representative nonlinear differential systems and evaluated using numerical simulations, training-loss analysis, and prediction-performance metrics. Numerical results demonstrate that SENNM achieves higher accuracy in modeling systems with strong nonlinear interactions and supports stable learning dynamics. NNPM exhibits superior robustness and generalization across varying boundary and initial conditions. A detailed comparison of neural network predictions and training losses shows that SENNM converges faster with lower residual error for localized structures, while NNPM maintains consistent prediction quality across the full domain. Figures depicting neural network predictions and training-loss curves further illustrate the relative performance, highlighting SENNM’s sharper solution reconstruction and NNPM’s smoother global approximation behavior. The findings reveal that SENNM and NNPM provide complementary advantages for solving nonlinear physical systems. SENNM excels in accuracy and convergence for highly nonlinear or localized patterns, whereas NNPM offers broader predictive stability and adaptability. Together, these insights support the advancement of hybrid analytical–neural and prediction-based neural frameworks for modeling complex fluid and nuclear physics phenomena.