Physics-informed Discretization-independent Deep Compositional Operator Network

TL;DR

This research introduces a novel physics-informed model architecture which can generalize to various discrete representations of PDE parameters and irregular domain shapes and involves a discretization-independent learning of parameter embedding repeatedly, inspired by deep operator neural networks.

Abstract

Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE parameter inputs}, have been successfully used. However, the training of neural operators typically demands large training datasets, the acquisition of which can be prohibitively expensive. To address this challenge, physics-informed training can offer a cost-effective strategy. However, current physics-informed neural operators face limitations, either in handling irregular domain shapes or in in generalizing to various discrete representations of PDE parameters. In this research, we introduce a novel physics-informed model architecture which can generalize to various discrete representations of PDE parameters and irregular domain shapes. Particularly, inspired by deep operator neural networks, our model involves a discretization-independent learning of parameter embedding repeatedly, and this parameter embedding is integrated with the response embeddings through multiple compositional layers, for more expressivity. Numerical results demonstrate the accuracy and efficiency of the proposed method. All the codes and data related to this work are available on GitHub: https://github.com/WeihengZ/PI-DCON.

Figures (10)

• Figure 1: The connection between our model architecture and DeepONet is shown. The architecture on the left is DeepONet and the one on the right is our proposed model. Comparing two model architectures, we can observe that our model is a compositional version of the DeepONet.
• Figure 2: The architecture of the physics-informed Deep Compositional Operator Network is shown. The grey blocks represent the coordinates and the blue blocks represent the function values. The red blocks represent the hidden embeddings. The value of $b'_i$ is the boundary condition value evaluated on $(x'_i, y'_i)$. The value of $u$ is the solution value evaluated on $(x, y)$. It should be noted that no activation function is included in the last operator layer.
• Figure 3: The boundary conditions of our Darcy flow experiment setting, and the coarsest and finest meshes used as discretizations to solve the PDE are shown. The samples of Gaussian processes as the boundary conditions $g(\bm{x})$ are also shown above the meshes. The dots on the curve are considered as the representation of the boundary condition values.
• Figure 4: Three representative FEM responses (bottom row) calculated for three realizations of pressure values of the boundary (top row). The dashed line represents the zero pressure.
• Figure 5: The boundary conditions of 2D plate experiment, and the coarsest and finest meshes used as discretizations to solve the PDE are shown. The samples of Gaussian processes as the prescribed displacement on leftmost edges and rightmost edges are also shown. The dots on the curve are considered as the representation of the prescribed displacements.