Dilated convolution neural operator for multiscale partial differential equations

TL;DR

The paper tackles learning the operator $\mathcal{S}:\bm a\mapsto \bm u$ for multiscale PDEs where high-frequency content is crucial and standard neural operators suffer spectral bias. It introduces the Dilated Convolution Neural Operator (DCNO), an Encode-Process-Decode architecture that combines Fourier layers for global structure with dilated convolution layers for local detail. Through experiments on multiscale elliptic equations, Navier–Stokes, Helmholtz, and inverse coefficient identification, DCNO achieves superior accuracy with favorable cost versus baselines and includes spectral-bias analyses and ablations. The work provides a scalable, data-driven tool for multiscale operator learning with potential impact on physics-based modeling and engineering simulations.

Abstract

This paper introduces a data-driven operator learning method for multiscale partial differential equations, with a particular emphasis on preserving high-frequency information. Drawing inspiration from the representation of multiscale parameterized solutions as a combination of low-rank global bases (such as low-frequency Fourier modes) and localized bases over coarse patches (analogous to dilated convolution), we propose the Dilated Convolutional Neural Operator (DCNO). The DCNO architecture effectively captures both high-frequency and low-frequency features while maintaining a low computational cost through a combination of convolution and Fourier layers. We conduct experiments to evaluate the performance of DCNO on various datasets, including the multiscale elliptic equation, its inverse problem, Navier-Stokes equation, and Helmholtz equation. We show that DCNO strikes an optimal balance between accuracy and computational cost and offers a promising solution for multiscale operator learning.

Figures (11)

• Figure 1.1: We demonstrate the effectiveness of the DCNO scheme using a challenging multiscale trigonometric benchmark. The coefficient and corresponding solution derivative are presented in (a) and (b), respectively (refer to Section \ref{['sec:mul']}) for a detailed description. We observe that DCNO accurately captures the solution derivatives. In (c) and (d), we analyze the (testing) dynamics for high-frequency ($> 10\pi$) and low-frequency ($\leq 10\pi$) errors, respectively. It is evident that DCNO achieves outstanding performance in terms of both high-frequency and low-frequency errors.
• Figure 3.1: The architecture of the DCNO neural operator.
• Figure 3.2: An example of a two-layer dilated convolution with dilation rates (1, 3).
• Figure 4.1: (a) multiscale trigonometric coefficient, (b)comparison of predicted solutions on the slice $x=0$, (c) comparison of predicted derivative $\frac{\partial u}{\partial y}$ on the slice $x=0$.
• Figure 4.2: Top: Darcy rough example, (a) coefficient, (b) reference solution, (c) DCNO, absolute (abs.) error, (d) FNO, abs. error; Bottom: multiscale trignometric example, (a) coefficient (in $\log_{10}$ scale), (b) reference solution, (c) DCNO, abs. error (in $\log_{10}$ scale), (d) FNO, abs. error (in $\log_{10}$ scale).