Solving Viscous Burgers’ Equation: Hybrid Approach Combining Boundary Layer Theory and Physics-Informed Neural Networks

Rubén Darío Ortiz Ortiz (), Oscar Martínez Núñez and Ana Magnolia Marín Ramírez
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Rubén Darío Ortiz Ortiz: Grupo Ondas, Instituto de Matemáticas Aplicadas, Universidad de Cartagena, Cartagena de Indias 130014, Colombia
Oscar Martínez Núñez: Grupo Ondas, Instituto de Matemáticas Aplicadas, Universidad de Cartagena, Cartagena de Indias 130014, Colombia
Ana Magnolia Marín Ramírez: Grupo Ondas, Instituto de Matemáticas Aplicadas, Universidad de Cartagena, Cartagena de Indias 130014, Colombia

Mathematics, 2024, vol. 12, issue 21, 1-30

Abstract: In this paper, we develop a hybrid approach to solve the viscous Burgers’ equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers’ equation, such as shock wave formation and the smoothing effects of viscosity, and show how the combination of these methods provides a powerful tool for solving nonlinear partial differential equations.

Keywords: boundary layer theory; Physics-Informed Neural Networks (PINNs); nonlinear partial differential equations; Burgers’ equation; shock waves; traveling waves (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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