ADVANCED APPLICATIONS OF PHYSICS-INFORMED NEURAL NETWORKS (PINNS) IN R FOR SOLVING DIFFERENTIAL EQUATIONS
Abstract
Deep learning, a powerful machine learning technique leveraging artificial neural networks, excels in identifying complex patterns and relationships within data. Among its innovations is the emergence of Physics-Informed Neural Networks (PINNs), which have revolutionized the field of applied mathematics by enabling the solution and discovery of differential equations through neural networks. PINNs address two key challenges: data-driven solutions, where the model approximates the hidden solutions of differential equations with fixed parameters, and data-driven discovery, where the network learns parameters that best describe observed data. This study explores the implementation of PINNs within the R programming environment to solve two differential equations: one with boundary conditions y^'-y=0 with y(0)=0 and y(e)=1 boundaries and the Burgers’ Equation. The research utilizes R libraries, including reticulate for Python integration and torch for neural network operations, to demonstrate the versatility and efficacy of PINNs in addressing both data-centric solutions and parameter discovery. The results showcase the ability of PINNs to handle complex, high-dimensional problems, offering a promising alternative to traditional numerical methods for solving differential equations.
Keywords
Physics-Informed Neural Networks (PINNs), Differential Equations, R-programming language, Burgers’ Equation
Thanks
I would like to thanks to Brown University, Department of Applied Math and Prof.Dr. George Karniadakis.
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