Accelerating MHD shock predictions using deep learning surrogate neural networks trained on finite difference solutions

Abstract

Cylindrically converging magnetohydrodynamic (MHD) shock waves in non-ideal gases exhibit complex interactions between compressibility, magnetic field effects, and real-gas thermodynamics. These flows are governed by a system of coupled nonlinear ordinary differential equations (ODEs) derived through similarity transformation from the conservation laws in cylindrical geometry. In this work, we focus on the specific case where the gas obeys the Van der Waals equation of state (EOS), as studied in \cite{revathi2025similarity}. We first implement a finite difference (FD) scheme to numerically solve the resulting similarity ODEs over a broad range of physical parameters, including the adiabatic index \(\gamma\), magnetic field strength coefficient \(C_0\), and Van der Waals constants \(a_1\) and \(b_1\). The computed solutions serve as the training data for a fully connected deep neural network that learns to map inputs \((\xi, \gamma, C_0, a_1, b_1)\) directly to the shock profile variables \((\rho, u, p, h, E)\). The trained surrogate model achieves excellent predictive accuracy, with \(R^2 > 0.99998\) and RMSE \(< 10^{-3}\) across all output variables. Compared to traditional numerical solvers, the neural network enables orders-of-magnitude faster evaluations while retaining physical fidelity. This work demonstrates a robust and efficient surrogate modeling strategy for accelerating MHD shock predictions in non-ideal gases, enabling rapid parametric studies and real-time deployment in computational MHD applications.