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A Unified Benchmark Study of Shock-Like Problems in Two-Dimensional Steady Electrohydrodynamic Flow Based on LSTM-PINN

Chao Lin, Ze Tao, Fujun Liu

Abstract

Accurately resolving steady electrohydrodynamic (EHD) flows presents a formidable computational challenge due to the strong nonlinear coupling between charged-particle density, velocity fields, and electric potential. These interactions frequently induce sharp transition layers, crossing fronts, and multiscale spatial structures, which notoriously degrade the predictive accuracy of standard mesh-free solvers like Physics-Informed Neural Networks (PINNs). To systematically address this bottleneck, we formulate a unified four-variable operator framework and develop a comprehensive benchmark suite for two-dimensional steady EHD shock-like problems. The benchmark comprises eight rigorously designed cases featuring diverse front geometries, such as oblique, curved, and intersecting layers, alongside complex multiscale patterns. Under strictly identical configurations, including governing equations, source terms, sampling strategies, and loss formulations, we evaluate a Standard MLP-based PINN, a Residual Attention PINN (ResAtt-PINN), and an LSTM-PINN that leverages pseudo-sequential spatial encoding. Extensive numerical experiments demonstrate that the LSTM-PINN consistently achieves the highest predictive accuracy across all eight cases. It successfully reconstructs sharp gradients and intricate multiscale structures where other architectures fail or over-smooth. Furthermore, the LSTM backbone efficiently captures long-range spatial correlations while maintaining an exceptionally low computational overhead and GPU memory footprint. These findings not only establish the LSTM-PINN as a robust and efficient solver for strongly coupled PDEs with shock-like features, but also provide the computational physics community with a standardized, reproducible benchmark for future algorithmic evaluations.

A Unified Benchmark Study of Shock-Like Problems in Two-Dimensional Steady Electrohydrodynamic Flow Based on LSTM-PINN

Abstract

Accurately resolving steady electrohydrodynamic (EHD) flows presents a formidable computational challenge due to the strong nonlinear coupling between charged-particle density, velocity fields, and electric potential. These interactions frequently induce sharp transition layers, crossing fronts, and multiscale spatial structures, which notoriously degrade the predictive accuracy of standard mesh-free solvers like Physics-Informed Neural Networks (PINNs). To systematically address this bottleneck, we formulate a unified four-variable operator framework and develop a comprehensive benchmark suite for two-dimensional steady EHD shock-like problems. The benchmark comprises eight rigorously designed cases featuring diverse front geometries, such as oblique, curved, and intersecting layers, alongside complex multiscale patterns. Under strictly identical configurations, including governing equations, source terms, sampling strategies, and loss formulations, we evaluate a Standard MLP-based PINN, a Residual Attention PINN (ResAtt-PINN), and an LSTM-PINN that leverages pseudo-sequential spatial encoding. Extensive numerical experiments demonstrate that the LSTM-PINN consistently achieves the highest predictive accuracy across all eight cases. It successfully reconstructs sharp gradients and intricate multiscale structures where other architectures fail or over-smooth. Furthermore, the LSTM backbone efficiently captures long-range spatial correlations while maintaining an exceptionally low computational overhead and GPU memory footprint. These findings not only establish the LSTM-PINN as a robust and efficient solver for strongly coupled PDEs with shock-like features, but also provide the computational physics community with a standardized, reproducible benchmark for future algorithmic evaluations.
Paper Structure (22 sections, 19 equations, 38 figures, 13 tables)

This paper contains 22 sections, 19 equations, 38 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Unified Governing Equations
  4. Unified Physical Interpretation of the Benchmark Cases
  5. Methodology
  6. LSTM-PINN Architecture
  7. Unified Residual Construction and Loss Function
  8. Results and Analysis
  9. Case 1: Vertical Internal Shock Layer
  10. Case 2: Horizontal Shock Layer
  11. Case 3: Single Oblique Shock
  12. Case 4: Crossing Oblique Layers
  13. Case 5: Curved Radial Front
  14. Case 6: Electro-Shear Layer with Localized Density Pocket
  15. Case 7: Double Front with Secondary Transverse Layer
  16. ...and 7 more sections

Figures (38)

  • Figure 1: Conceptual architecture of the LSTM-PINN workflow adapted from Tao et al., showing the pseudo-sequential encoding, stacked recurrent backbone, output head, and physics-informed loss construction. In the present study, the same design idea is extended from the original two-output formulation to the four-variable field $(n,u_x,u_y,\phi)$.
  • Figure 2: Gate-level structure of an LSTM unit, illustrating the coordinated roles of the input gate, forget gate, output gate, hidden state, and cell state in information updating and propagation.
  • Figure 3: 3$\times$3 reconstruction comparison of the $n$-field for Case 01. The first row shows the reference solution, the second row shows the predictions of the three models, and the third row shows the corresponding absolute errors.
  • Figure 4: 3$\times$3 reconstruction comparison of the $u_x$-field for Case 01.
  • Figure 5: 3$\times$3 reconstruction comparison of the $u_y$-field for Case 01.
  • ...and 33 more figures