A refined physics-informed neural network framework for solving nonlinear partial differential equations and extracting analytical expressions via symbolic regression

This work presents a novel physics-informed machine learning approach to solve nonlinear Partial Differential Equations (PDEs) that arise in the modeling of arterial blood flow under the influence of a magnetic field. We used a mathematical model to simulate viscoelastic arterial flow using reductive perturbation, leading to nonlinear forced Burger, forced Korteweg-de Vries, and other evolutionary equations. Solutions of the forced Burgers equation at different time scales are obtained using a refined Physics-Informed Neural Network (PINN) approach called Adaptive Residual Enhancement PINNs (ARE-PINNs). The optimal network architecture is determined via Bayesian Hyperparameter Optimization. We also used Symbolic Regression (SR) algorithm to derive approximate analytical expressions representing the solutions for higher order nonlinear PDE. This study also explores the impact of the parsimony factor and operator complexity on the discovery of symbolic expressions. This PINN framework serves as a computationally efficient substitute for conventional discretization techniques in solving higher-order nonlinear PDEs, with its accuracy validated through both residual error analysis and numerical benchmarks. Graphical representations of pulse wave propagation can assist in interpreting cardiovascular parameters, supporting diagnosis and treatment through digital tools and potentially accelerating the adoption of deep learning in medical applications.