XI-DeepONet: An operator learning method for elliptic interface problems

TL;DR

XI-DeepONet tackles parametric elliptic interface problems with discontinuous coefficients by learning the solution operator in a mesh-free fashion. It extends DeepONet with two key innovations: (i) encoding the interface geometry as a level-set input $\Phi$ and (ii) using an extension function to enforce a homogeneous jump condition, enabling the operator to depend continuously on the interface. Training can be data-driven or physics-informed, reducing the need for paired training data. Across 1D, 2D, 3D, high-contrast, irregular, and even high-dimensional cases, XI-DeepONet achieves high accuracy and robust generalization to varying interfaces, offering a fast, mesh-free alternative for parametric elliptic interface problems.

Abstract

Scientific computing has been an indispensable tool in applied sciences and engineering, where traditional numerical methods are often employed due to their superior accuracy guarantees. However, these methods often encounter challenges when dealing with problems involving complex geometries. Machine learning-based methods, on the other hand, are mesh-free, thus providing a promising alternative. In particular, operator learning methods have been proposed to learn the mapping from the input space to the solution space, enabling rapid inference of solutions to partial differential equations (PDEs) once trained. In this work, we address the parametric elliptic interface problem. Building upon the deep operator network (DeepONet), we propose an extended interface deep operator network (XI-DeepONet). XI-DeepONet exhibits three unique features: (1) The interface geometry is incorporated into the neural network as an additional input, enabling the network to infer solutions for new interface geometries once trained; (2) The level set function associated with the interface geometry is treated as the input, on which the solution mapping is continuous and can be effectively approximated by the deep operator network; (3) The network can be trained without any input-output data pairs, thus completely avoiding the need for meshes of any kind, directly or indirectly. We conduct a comprehensive series of numerical experiments to demonstrate the accuracy and robustness of the proposed method.

Figures (8)

• Figure 1: The network architecture of Extended Interface DeepONet.
• Figure 2: Four representative curve of solution from the test sample data (distinguished by different colors). The reference solutions obtained by MIB and the predicted solutions computed by NNs are depicted by solid and dashed lines respectively. Left: The approximate solution acquired by DD-XI-DeepONets. Right: The approximate solutions acquired by PI-XI-DeepONets.
• Figure 3: Top: $l_1=l_2=0.3$. Bottom: $l_1=l_2=0.15$. The solid and dashed lines are the reference solution and the predicted solutions obtained by MIB and NNs, and the training data is generated by $l_1=0.2, l_2=0.1$.
• Figure 4: Example \ref{['exa2']}: The sensors distribution. Blue dots represent the location of sensors, and solid lines are interface $\Gamma$ and $\partial \Omega$ in the case where $r_0=0.5$.
• Figure 5: Example \ref{['exa2']}: The profile of NN solutions $u_{\mathcal{S}}$ and absolute point-wise errors $\left|u - u_{\mathcal{S}}\right|$ for the considered problem with different interface position and size. Top: $r_0=0.5$. Middle: $r_0=0.6$. Bottom: $r_0=0.7$.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Example 1
• Example 2
• Example 3
• Example 4
• Example 5