Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)

TL;DR

The paper addresses the efficiency and accuracy limitations of physics-informed neural networks (PINNs) under domain decomposition by proposing Initialization-enhanced PINN with Domain Decomposition (IDPINN). IDPINN bootstraps subdomain networks from a single PINN trained on a small dataset (IDPINN-init) and enforces high-order interface continuity using augmented losses ($\mathcal{L}_{\text{inter}}$, $\mathcal{L}_{\nabla}$, $\mathcal{L}_{PDE_g}$). Across Helmholtz, 2D Poisson, Heat, and Burgers equations, IDPINN demonstrates superior interface accuracy and overall prediction quality compared with PINN and XPINN, with notable gains from initialization and interface-smoothness terms. The framework is robust to different domain geometries (straight lines and irregular curves) and offers a practical path to faster, more accurate forward-PDE solutions in large-scale or multi-domain settings.

Abstract

We propose a new physics-informed neural network framework, IDPINN, based on the enhancement of initialization and domain decomposition to improve prediction accuracy. We train a PINN using a small dataset to obtain an initial network structure, including the weighted matrix and bias, which initializes the PINN for each subdomain. Moreover, we leverage the smoothness condition on the interface to enhance the prediction performance. We numerically evaluated it on several forward problems and demonstrated the benefits of IDPINN in terms of accuracy.

Figures (14)

• Figure 1: Slices of the exact (blue) and predicted (red) solution ($u(x,y)$) of the Helmholtz equation for given $y$ values, and their corresponding PDE residuals ($f(x,y,u)$). First and second row: We use the loss functions on the interface: $\mathcal{L}_{residual}(\theta)$\ref{['8']} and $\mathcal{L}_{avg}(\theta)$\ref{['99']}. Third and fourth row: We use IDPINN-3. ($\mathcal{L}_{\nabla}(\theta)$\ref{['9']} and $\mathcal{L}_{PDE_g}(\theta)$\ref{['10']}).
• Figure 2: Structure of IDPINN, where each subdomain uses a neural network to approximate the function on the subdomain, and these neural networks are integrated through interface conditions. The PINN to the subdomains $\Omega_i$ and $\Omega_j$ have different parameters $\theta_i$ and $\theta_j$, respectively. Here, due to the simplicity, we use $\theta$.
• Figure 3: Point selection for the Helmholtz equation with two subdomains. The selected points in each subdomain are represented by red (Subdomain 1) and blue (Subdomain 2), respectively. The boundary points in two subdomains and interface points are represented by green, orange, and purple, respectively.
• Figure 4: First row: exact solution of the Helmholtz equation. Second row: predicted solution and point-wise error for XPINN after 100k total iterations, resulting in a $1.41\times 10^{-1}$ L2 error. Third row: predicted solution and point-wise error for IDPINN-3 after 100k total iterations, resulting in a $1.1\times 10^{-2}$ L2 error. Last row: predicted solution and point-wise error for IDPINN-3 without initialization after 100k total iterations, resulting in a $4.82\times 10^{-2}$ L2 error.
• Figure 5: The L2 error history in 100,000 iterations for the Helmholtz equation for XPINN, IDPINN-3, and IDPINN-3 without initialization, under two different learning rates.