Tailored Finite Point Operator Networks for Interface problems

TL;DR

Experimental analyses reveal that, in comparison to existing methods such as DeepONet and IONet, TFPONets demonstrate superior learning and generalization capabilities even with limited locations.

Abstract

Interface problems pose significant challenges due to the discontinuity of their solutions, particularly when they involve singular perturbations or high-contrast coefficients, resulting in intricate singularities that complicate resolution. The increasing adoption of deep learning techniques for solving partial differential equations has spurred our exploration of these methods for addressing interface problems. In this study, we introduce Tailored Finite Point Operator Networks (TFPONets) as a novel approach for tackling parameterized interface problems. Leveraging DeepONets and integrating the Tailored Finite Point method (TFPM), TFPONets offer enhanced accuracy in reconstructing solutions without the need for intricate equation manipulation. Experimental analyses conducted in both one- and two-dimensional scenarios reveal that, in comparison to existing methods such as DeepONet and IONet, TFPONets demonstrate superior learning and generalization capabilities even with limited locations.

Figures (5)

• Figure 1: TFPONet integrates a DeepONet with multiple neural networks that receive inputs of the form $(f, x, B_1(x), B_2(x), \dots)$. Specifically, DeepONet is fed with the tuple $(f, x)$, while the $i$-th auxiliary neural network within TFPONet is tasked with processing the local basis $B_i(x)$.
• Figure 2: a): Ground truth and TFPONet's prediction for Eq. \ref{['equ:1d_example']} with $a(x)=0.0001$ on a random $f$ and 257 locations. b), c): Zoomed-in view. d): Test error (MSE) vs. Training resolution $(\# x)$ for TFPOnet and DeepONet.
• Figure 3: a): Ground truth and TFPONet's prediction for Eq. \ref{['equ:1d_example']}, where $a(x) = 1$ for $x \in [0, 0.5)$ and $a(x) = 0.0001$ for $x \in [0.5, 1]$, across 257 locations using a random $f$. b), c): Zoomed-in view. d): Test error (MSE) vs. Training resolution $(\# x)$ for TFPOnet and DeepONet.
• Figure 4: a): Ground truth and TFPONet's prediction for Example 2. b): Test error (MSE) vs. Training resolution $(\# x)$ for TFPONets and IONet.
• Figure 5: Performance of TFPONets and IONet on Eq.\ref{['equ:2d_example']}. a), b): Ground truth. c): Predictions of TFPONets (above) and IONet (below). d): Absolute Point-wise error of TFPONets (above) and IONet (below).