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Numerical Approximation of Electrohydrodynamics Model: A Comparative Study of PINNs and FEM

Mara Martinez, B. Veena S. N. Rao, S. M. Mallikarjunaiah

TL;DR

This study compares physics-informed neural networks (PINNs) to the finite element method (FEM) for solving a nonlinear one-dimensional Electrohydrodynamic boundary-value problem. By embedding the governing ODE residual and boundary conditions into a total $L^2$-type loss, the PINN learns the solution without requiring an exact analytical form, enabling a mesh-free, data-efficient approach. A comprehensive hyperparameter sweep identifies an optimal PINN configuration (100 collocation points, two hidden layers with 16 neurons, Sigmoid activation, Adamax with $\eta=0.001$) that achieves an $L^2$-error of the order $10^{-4}$, and in a direct comparison, it attains a slightly lower error with far fewer degrees of freedom than a high-resolution FEM run (1024 cells). The results demonstrate the competitive accuracy and parameter efficiency of PINNs, while also highlighting the enduring rigor and convergence guarantees of FEM. The work points to the potential of PINNs for scalable, mesh-free simulations in higher dimensions and more complex multiphysics problems, with future work focusing on theoretical convergence and scalability analyses.

Abstract

The accurate representation of numerous physical, chemical, and biological processes relies heavily on differential equations (DEs), particularly nonlinear differential equations (NDEs). While understanding these complex systems necessitates obtaining solutions to their governing equations, the derivation of precise approximations for NDEs remains a formidable task in computational mathematics. Although established techniques such as the finite element method (FEM) have long been foundational, remarkable promise for approximating continuous functions with high efficacy has recently been demonstrated by advancements in physics-informed deep-learning feedforward neural networks. In this work, a novel application of PINNs is presented for the approximation of the challenging Electrohydrodynamic (EHD) problem. A specific $L^2$-type \textit{total loss function} is employed, notably without reliance on any prior knowledge of the exact solution. A comprehensive comparative study is conducted, juxtaposing the approximation capabilities of the proposed neural network with those of the conventional FEM. The PINN training regimen is composed of two critical steps: forward propagation for adjustments to gradient and curvature, and backpropagation for the refinement of hyperparameters. The critical challenge of identifying optimal neural network architectures and hyperparameter configurations for efficient optimization is meticulously investigated. Excellent performance is shown to be delivered by the neural network even with a limited training dataset. Simultaneously, it is demonstrated that the accuracy of the FEM can be substantially enhanced through the judicious selection of smaller mesh sizes.

Numerical Approximation of Electrohydrodynamics Model: A Comparative Study of PINNs and FEM

TL;DR

This study compares physics-informed neural networks (PINNs) to the finite element method (FEM) for solving a nonlinear one-dimensional Electrohydrodynamic boundary-value problem. By embedding the governing ODE residual and boundary conditions into a total -type loss, the PINN learns the solution without requiring an exact analytical form, enabling a mesh-free, data-efficient approach. A comprehensive hyperparameter sweep identifies an optimal PINN configuration (100 collocation points, two hidden layers with 16 neurons, Sigmoid activation, Adamax with ) that achieves an -error of the order , and in a direct comparison, it attains a slightly lower error with far fewer degrees of freedom than a high-resolution FEM run (1024 cells). The results demonstrate the competitive accuracy and parameter efficiency of PINNs, while also highlighting the enduring rigor and convergence guarantees of FEM. The work points to the potential of PINNs for scalable, mesh-free simulations in higher dimensions and more complex multiphysics problems, with future work focusing on theoretical convergence and scalability analyses.

Abstract

The accurate representation of numerous physical, chemical, and biological processes relies heavily on differential equations (DEs), particularly nonlinear differential equations (NDEs). While understanding these complex systems necessitates obtaining solutions to their governing equations, the derivation of precise approximations for NDEs remains a formidable task in computational mathematics. Although established techniques such as the finite element method (FEM) have long been foundational, remarkable promise for approximating continuous functions with high efficacy has recently been demonstrated by advancements in physics-informed deep-learning feedforward neural networks. In this work, a novel application of PINNs is presented for the approximation of the challenging Electrohydrodynamic (EHD) problem. A specific -type \textit{total loss function} is employed, notably without reliance on any prior knowledge of the exact solution. A comprehensive comparative study is conducted, juxtaposing the approximation capabilities of the proposed neural network with those of the conventional FEM. The PINN training regimen is composed of two critical steps: forward propagation for adjustments to gradient and curvature, and backpropagation for the refinement of hyperparameters. The critical challenge of identifying optimal neural network architectures and hyperparameter configurations for efficient optimization is meticulously investigated. Excellent performance is shown to be delivered by the neural network even with a limited training dataset. Simultaneously, it is demonstrated that the accuracy of the FEM can be substantially enhanced through the judicious selection of smaller mesh sizes.
Paper Structure (16 sections, 1 theorem, 11 equations, 6 figures, 19 tables, 1 algorithm)

This paper contains 16 sections, 1 theorem, 11 equations, 6 figures, 19 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Fully Connected Artificial Neural Network Architecture and Training
  4. Neural Network-Based Solution Methodology
  5. Impact of Data Point Density on Network Performance
  6. Evaluation of Activation Functions
  7. Approximation by Deep Neural Networks: Impact of Network Depth
  8. Single hidden-layer with different number of neurons
  9. Impact of Learning Rate on Network Optimization
  10. Summary of Optimal Hyperparameters
  11. Finite Element Discretization of the EHD Boundary Value Problem
  12. Variational Formulation and Linearization
  13. Galerkin Discretization and Solution Algorithm
  14. Numerical Validation and Convergence Study
  15. Comparative Analysis of ANN and FEM Approaches
  16. ...and 1 more sections

Key Result

Theorem 1

For any $\alpha > 0$ and $Ha^2 \neq 0$, there exists a solution to the BVP EHD_DE-EHD_Bcs, and such a solution is monotonically decreasing and satisfies $0 < v(\tau) < 1/(\alpha + 1)$ for all $\tau \in (0, 1)$. Furthermore, the solution is unique.

Figures (6)

  • Figure 1: Configuration showing EHD fluid flow in a circular cylindrical conduit.
  • Figure 2: Evolution of the total loss function across epochs for a single-hidden-layer network trained with varying numbers of data points.
  • Figure 3: Evolution of the total loss function across epochs for a single-hidden-layer network utilizing different activation functions.
  • Figure 4: Evolution of the total loss function across epochs for different numbers of hidden layers in the neural network.
  • Figure 5: The total loss value as a function of several epochs for a network with a single hidden layer with a different number of neurons.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Remark 1
  • Theorem 1