Comparisons in Numerical Analysis and Neural Networks in Stability and Convergence in the 1D Heat Equation
Kendric Hsu1, A. Gupta2
1Clovis North High School, Fresno, United States of America
2Bellarmine College Preparatory, San Jose, United States of America
Publication date: June 30, 2025
1Clovis North High School, Fresno, United States of America
2Bellarmine College Preparatory, San Jose, United States of America
Publication date: June 30, 2025
DOI: http://doi.org/10.34614/JIYRC2025I05
ABSTRACT
We present a comparative study between Physics-Informed Neural Networks (PINNs) and the Finite Difference Method (FDM) to solve the 1D heat equation, examining their different approaches to handling physical differential equations. Our analysis explores how each method addresses traditional stability constraints, including the Courant-Friedrichs-Lewy (CFL) condition in traditional numerical methods. Our study compares PINN and FDM approaches, focusing on stability, accuracy, and convergence. We observe that PINNs exhibit stability characteristics similar to the backward Euler method, while showing different behavior from the forward Euler approach. We conduct a detailed examination of PINN accuracy across various parameter spaces, identifying regions of high performance and challenges. Additionally, we investigate the order of convergence for PINNs in optimal parameter regions and compare it to known FDM convergence rates. These findings provide insights into how both methods can be applied to differential equations under different constraints, allowing for visualization of complex systems across various parameter spaces and offering perspectives on both forward and inverse problems. Our study uses Python with the PyTorch library as the computing language and MATLAB to model and post-process results.
We present a comparative study between Physics-Informed Neural Networks (PINNs) and the Finite Difference Method (FDM) to solve the 1D heat equation, examining their different approaches to handling physical differential equations. Our analysis explores how each method addresses traditional stability constraints, including the Courant-Friedrichs-Lewy (CFL) condition in traditional numerical methods. Our study compares PINN and FDM approaches, focusing on stability, accuracy, and convergence. We observe that PINNs exhibit stability characteristics similar to the backward Euler method, while showing different behavior from the forward Euler approach. We conduct a detailed examination of PINN accuracy across various parameter spaces, identifying regions of high performance and challenges. Additionally, we investigate the order of convergence for PINNs in optimal parameter regions and compare it to known FDM convergence rates. These findings provide insights into how both methods can be applied to differential equations under different constraints, allowing for visualization of complex systems across various parameter spaces and offering perspectives on both forward and inverse problems. Our study uses Python with the PyTorch library as the computing language and MATLAB to model and post-process results.
