LSTM-PINN: An Hybrid Method for Prediction of Steady-State Electrohydrodynamic Flow

TL;DR

This work addresses the convergence and stability challenges of physics-informed neural networks (PINNs) when solving 2D steady electrohydrodynamic flow by introducing LSTM-PINN, which embeds spatial dependencies via pseudo-sequential inputs and leverages gated memory to capture long-range correlations. Compared to a baseline MLP-PINN, the LSTM-PINN achieves faster and more robust convergence, maintains physical fidelity (e.g., diagonal symmetry), and tolerates higher learning rates, albeit with increased computational cost. Comprehensive ablations show that memory-based gating (LSTM) outperforms ungated RNNs and that equal-capacity variants fail to surpass the baseline, underscoring the architectural advantages for coupled Navier–Stokes–electrostatics problems. The results demonstrate a mesh-free, physics-consistent surrogate framework with strong potential for micro/nanofluidics design and optimization where boundary conditions and gradient constraints are challenging for traditional solvers.

Abstract

Physics-Informed Neural Networks (PINNs) have demonstrated considerable success in solving complex fluid dynamics problems. However, their performance often deteriorates in regimes characterized by steep gradients, intricate boundary conditions, and stringent physical constraints, leading to convergence failures and numerical instabilities. To overcome these limitations, we propose a hybrid framework that integrates Long Short-Term Memory (LSTM) networks into the PINN architecture, enhancing its ability to capture spatial correlations in the steady-state velocity field of a two-dimensional charged fluid under an external electric field. Our results demonstrate that the LSTM-enhanced PINN model significantly outperforms conventional Multilayer Perceptron (MLP)-based PINNs in terms of convergence rate, numerical stability, and predictive accuracy. This innovative approach offers improved computational efficiency and reliability for modeling electrohydrodynamic flows, providing new insights and strategies for applications in microfluidics and nanofluidics.

Figures (19)

• Figure 1: Schematic of the baseline MLP-based PINN architecture: the network takes spatial coordinates $(x,y)$ as input and outputs velocity components $(u,v)$. Within the PINN, we compute the residuals of the steady electrohydrodynamic momentum equations and the incompressibility constraint (Eqs. \ref{['eq:momentum_x']}–\ref{['eq:continuity']}) via automatic differentiation and assemble a composite loss $\mathcal{L}_{\text{total}}=\mathcal{L}_x+\mathcal{L}_y+\mathcal{L}_c+\mathcal{L}_b$, where the boundary term $\mathcal{L}_b=\mathcal{L}_l+\mathcal{L}_r+\mathcal{L}_u+\mathcal{L}_d$ enforces homogeneous Neumann conditions (Eq. \ref{['eq:bc']}) by penalizing normal velocity gradients on the four sides. The figure highlights the baseline fully connected MLP's feed-forward mapping from coordinates to velocity; training optimizes the learnable parameters $\theta=[W,b]$ with stochastic gradient descent (SGD) over $m$ iterations, serving as a reference for Figure \ref{['fig:lstm']} where an LSTM-PINN replaces the MLP to enhance modeling of spatial correlations.
• Figure 2: LSTM-PINN architecture: we replace the baseline MLP backbone with stacked LSTM layers and construct pseudo-sequential inputs from spatial coordinates so that the hidden state propagates along space and captures long-range spatial correlations. The network still maps $(x,y)$ to $(u,v)$; during training, we evaluate the residuals of the steady electrohydrodynamic momentum equations and the incompressibility constraint (Eqs. \ref{['eq:momentum_x']}–\ref{['eq:continuity']})) via automatic differentiation and assemble the composite loss $\mathcal{L}_{\text{total}}=\mathcal{L}_x+\mathcal{L}_y+\mathcal{L}_c+\mathcal{L}_b$ (Eq. \ref{['eq:total_loss']}), where $\mathcal{L}_b$ enforces homogeneous Neumann boundary conditions (Eq. \ref{['eq:bc']}). Compared with Figure \ref{['fig:mlp']}, this figure highlights how pseudo-sequential spatial encoding enables the LSTM's gating and memory to enhance modeling of complex spatial patterns and improve training stability for steady-state EHD flows.
• Figure 3: LSTM computational flow: the gates regulate information exchange between the cell state $c_t$ and hidden state $h_t$; the candidate memory $\tilde{c}_t$ arises from the current input $x_t$ and the previous hidden state $h_{t-1}$; the update obeys $c_t=f_t\odot c_{t-1}+i_t\odot \tilde{c}_t$ and $h_t=o_t\odot\tanh(c_t)$ (Eqs. \ref{['eq:cell_update']}--\ref{['eq:hidden_update']}), with $\tilde{c}_t=\tanh(W_c x_t+U_c h_{t-1}+b_c)$ (Eq. \ref{['eq:candidate']}) and gate definitions $i_t=\sigma(W_i x_t+U_i h_{t-1}+b_i)$, $f_t=\sigma(W_f x_t+U_f h_{t-1}+b_f)$, $o_t=\sigma(W_o x_t+U_o h_{t-1}+b_o)$ (Eqs. \ref{['eq:input_gate']}--\ref{['eq:output_gate']}). The compact matrix form in Eq. \ref{['eq:matrix_form']} consolidates parameters into $W$ and $b$. The diagram highlights how specific gate settings (e.g., $f_t\!\approx\!0, i_t\!\approx\!1$ to overwrite; $f_t\!\approx\!1, i_t\!\approx\!0$ to retain) control memory updates and, under pseudo-sequential spatial inputs, support long-range correlation modeling in the LSTM-PINN.
• Figure 4: MLP-PINN loss trajectories at learning rates $\lambda\in\{0.002,0.004,0.006,0.008\}$, using a common axis range for side-by-side comparison. The curves reveal strong learning-rate sensitivity: oscillations intensify at $\lambda=0.004$, large-amplitude fluctuations emerge at $\lambda=0.006$, and optimization becomes chaotic at $\lambda=0.008$, which halts training.
• Figure 5: LSTM-PINN loss trajectories at learning rates $\lambda\in\{0.002,0.004,0.006,0.008\}$, displayed on a common axis range for side-by-side comparison. The curves exhibit robust, smooth convergence across the intermediate rates ($0.002$--$0.005$) and show only minor terminal variations at the higher rates ($0.006$--$0.008$), reflecting stable training dynamics in the LSTM-based PINN.