Physics-Informed Geometry-Aware Neural Operator

TL;DR

A novel method, the Physics-Informed Geometry-Aware Neural Operator (PI-GANO), designed to simultaneously generalize across both PDE parameters and domain geometries is introduced.

Abstract

Engineering design problems often involve solving parametric Partial Differential Equations (PDEs) under variable PDE parameters and domain geometry. Recently, neural operators have shown promise in learning PDE operators and quickly predicting the PDE solutions. However, training these neural operators typically requires large datasets, the acquisition of which can be prohibitively expensive. To overcome this, physics-informed training offers an alternative way of building neural operators, eliminating the high computational costs associated with Finite Element generation of training data. Nevertheless, current physics-informed neural operators struggle with limitations, either in handling varying domain geometries or varying PDE parameters. In this research, we introduce a novel method, the Physics-Informed Geometry-Aware Neural Operator (PI-GANO), designed to simultaneously generalize across both PDE parameters and domain geometries. We adopt a geometry encoder to capture the domain geometry features, and design a novel pipeline to integrate this component within the existing DCON architecture. 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/Physics-informed-Neural-Foundation-Operator.

Figures (12)

• Figure 1: The architecture of the DeepONet is shown. The input to the branch net is the sampled discrete representations of variable parameters $k(\bm x)$ and $g(\bm x)$, with $l$ and $m$ discretization points, respectively.
• Figure 2: A schematic of how information on the domain geometry can be integrated with the location and parameter embeddings. The variable inputs include $l$ and $m$ discrete values of parameter functions, together with the coordinates of $n$ points on the domain boundary.
• Figure 3: The architecture of the physics-informed Geometry aware Neural Operator 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 4: The differences between our model architecture and PI-PointNet are shown. The black solid arrows are forward computation path for PDE solution prediction, while the red dashed arrows are backward computation path for derivative calculation of the approximated function.
• Figure 5: The architecture of the modified physics-informed PointNet (PI-PointNet*).