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Abstract This work explores an interpretable approach based on physics-informed neural networks (PINNs) combined with symbolic regression (SR) to determine mathematical expressions for the predicted solutions of nonlinear partial differential equations that describe arterial blood flow influenced by external magnetic fields. PINNs are excellent at capturing the underlying physics, but can be computationally intensive and hard to interpret. The predictive power of PINN has been combined with evolutionary symbolic regression using PySR, an open-source Python module that employs genetic programming and customizable mathematical operators, including trigonometric, exponential, and arithmetic functions. This hybrid approach enables the derivation of concise, transparent mathematical expressions that closely replicate the behavior of these complex systems. This blend of PINNs and symbolic regression helps us to better understand how pulsatile blood flow and magnetic fields interact in the viscoelastic arterial circulation. The comparison graphs of the SR model and the PINN-predicted data at different time scales signify a better fit for the discovered mathematical expressions with data. As illustrated by the low mean squared error and statistical validation on residual losses, the symbolic expressions are extremely accurate and quick enough for real-time execution. Additionally, the solutions provided by the PINN are validated numerically to demonstrate the effectiveness of the proposed method. Our results demonstrate that combining symbolic regression with PINNs provides practical and interpretable solutions in biofluid mechanics, offering a more transparent and reliable alternative to traditional methods.
Das et al. (Sun,) studied this question.