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R-adaptive DeepONet: Learning Solution Operators for PDEs with Discontinuous Solutions Using an R-adaptive Strategy

Yameng Zhu, Jingrun Chen, Weibing Deng

TL;DR

The paper tackles learning solution operators for PDEs with discontinuities, where vanilla DeepONet suffers from a fundamental linear-reconstruction limitation. It introduces R-adaptive DeepONet, which decouples the operator into an adaptive coordinate transform and an adaptive solution operator learned by two DeepONets, with post-processing by interpolation guided by an equidistribution-based mesh. The authors prove that this RA framework reduces the reconstruction-error upper bound and obtain superlinear convergence for representative PDEs (linear advection and inviscid Burgers') in contrast to the linear decay seen in vanilla DeepONet. Numerical experiments on linear advection, viscous Burgers', and Euler shock-tube problems demonstrate that RA-DeepONet outperforms vanilla DeepONet and remains competitive with Shift-DeepONet, while requiring less data in some settings and better handling of sharp gradients and discontinuities.

Abstract

DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with discontinuous solutions, DeepONet poses a foundational approximation lower bound due to its linear reconstruction property. Inspired by the moving mesh (R-adaptive) method, we propose an R-adaptive DeepONet method, which contains the following components: (1) the output data representation is transformed from the physical domain to the computational domain using the equidistribution principle; (2) the maps from input parameters to the solution and the coordinate transformation function over the computational domain are learned using DeepONets separately; (3) the solution over the physical domain is obtained via post-processing methods such as the (linear) interpolation method. Additionally, we introduce a solution-dependent weighting strategy in the training process to reduce the final error. We establish an upper bound for the reconstruction error based on piecewise linear interpolation and show that the introduced R-adaptive DeepONet can reduce this bound. Moreover, for two prototypical PDEs with sharp gradients or discontinuities, we prove that the approximation error decays at a superlinear rate with respect to the trunk basis size, unlike the linear decay observed in vanilla DeepONets. Therefore, the R-adaptive DeepONet overcomes the limitations of DeepONet, and can reduce the approximation error for problems with discontinuous solutions. Numerical experiments on PDEs with discontinuous solutions, including the linear advection equation, the Burgers' equation with low viscosity, and the compressible Euler equations of gas dynamics, are conducted to verify the advantages of the R-adaptive DeepONet over available variants of DeepONet.

R-adaptive DeepONet: Learning Solution Operators for PDEs with Discontinuous Solutions Using an R-adaptive Strategy

TL;DR

The paper tackles learning solution operators for PDEs with discontinuities, where vanilla DeepONet suffers from a fundamental linear-reconstruction limitation. It introduces R-adaptive DeepONet, which decouples the operator into an adaptive coordinate transform and an adaptive solution operator learned by two DeepONets, with post-processing by interpolation guided by an equidistribution-based mesh. The authors prove that this RA framework reduces the reconstruction-error upper bound and obtain superlinear convergence for representative PDEs (linear advection and inviscid Burgers') in contrast to the linear decay seen in vanilla DeepONet. Numerical experiments on linear advection, viscous Burgers', and Euler shock-tube problems demonstrate that RA-DeepONet outperforms vanilla DeepONet and remains competitive with Shift-DeepONet, while requiring less data in some settings and better handling of sharp gradients and discontinuities.

Abstract

DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with discontinuous solutions, DeepONet poses a foundational approximation lower bound due to its linear reconstruction property. Inspired by the moving mesh (R-adaptive) method, we propose an R-adaptive DeepONet method, which contains the following components: (1) the output data representation is transformed from the physical domain to the computational domain using the equidistribution principle; (2) the maps from input parameters to the solution and the coordinate transformation function over the computational domain are learned using DeepONets separately; (3) the solution over the physical domain is obtained via post-processing methods such as the (linear) interpolation method. Additionally, we introduce a solution-dependent weighting strategy in the training process to reduce the final error. We establish an upper bound for the reconstruction error based on piecewise linear interpolation and show that the introduced R-adaptive DeepONet can reduce this bound. Moreover, for two prototypical PDEs with sharp gradients or discontinuities, we prove that the approximation error decays at a superlinear rate with respect to the trunk basis size, unlike the linear decay observed in vanilla DeepONets. Therefore, the R-adaptive DeepONet overcomes the limitations of DeepONet, and can reduce the approximation error for problems with discontinuous solutions. Numerical experiments on PDEs with discontinuous solutions, including the linear advection equation, the Burgers' equation with low viscosity, and the compressible Euler equations of gas dynamics, are conducted to verify the advantages of the R-adaptive DeepONet over available variants of DeepONet.
Paper Structure (25 sections, 9 theorems, 76 equations, 9 figures, 5 tables, 1 algorithm)

This paper contains 25 sections, 9 theorems, 76 equations, 9 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Operator Learning and DeepONet
  3. Problem setting
  4. A brief introduction to DeepONet
  5. Limitation of DeepONet
  6. Variant Models of DeepONet
  7. Proposed Methodology: R-adaptive DeepONet
  8. R-adaptive DeepONet
  9. Training settings
  10. Theoretical Analysis
  11. Reconstruction error of DeepONets
  12. Bounds of the reconstruction error
  13. R-adaptive to lessen the upper bound of the reconstruction error
  14. Approximation properties for concrete examples
  15. Linear Advection Equation
  16. ...and 10 more sections

Key Result

Theorem 2.1

\newlabelthm:lowerbound0 Let $\mathcal{X}$ be a separable Banach space, $\mathcal{Y}$ a separable Hilbert space, and let $\mu$ be a probability measure on $\mathcal{X}$. Let $\mathcal{G}: \mathcal{X} \to \mathcal{Y}$ be a Borel measurable operator with $\mathbb{E}_{a\sim \mu} [\|\mathcal{G}(a)\|_{ where the optimal error $\mathcal{E}_{opt}$ is written in terms of the eigenvalues $\lambda_1 \geq \

Figures (9)

  • Figure 1: Illustration of an example of processed data for advection equation.
  • Figure 1: An example of the processed data for Spatial-temporal Burgers' equation.
  • Figure 2: Left: the validation error of different models during training; Right: eigenvalues of the covariance operators of different data sets.
  • Figure 2: Illustration of an example of outputs for spatial-temporal Burgers' equation.
  • Figure 3: An example of the prediction results of the three models for linear advection equation.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Theorem 2.1: Lanthaler et al. 2022, Thm. 3.4
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Remark 4.4
  • Theorem 4.5
  • Theorem 4.6
  • Theorem 4.7
  • Theorem 4.8
  • Theorem 4.9