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Resolution invariant deep operator network for PDEs with complex geometries

Jianguo Huang, Yue Qiu

TL;DR

The paper tackles the limitation of existing neural operators that require identical input and output domains by introducing the resolution-invariant deep operator (RDO). RDO replaces the branch net of DeepONet with a neural-operator–based integral transform, forming a composition $\mathcal{G}(a)=\langle \mathcal{T}\circ\mathcal{G}_0(a), \mathbf{f}(\cdot) \rangle$ which preserves resolution-invariance and supports functional inputs/outputs on different domains. A universal approximation theorem for RDO is established, leveraging Fourier integral operators (FIO) and attention integral operators (AIO) to realize the branch, along with a learnable readout $\mathbf{f}_\theta$. Numerical experiments on SVBP, Darcy flow with irregular domains, and Burgers’ equation demonstrate RDO’s effectiveness, including zero-shot super-resolution, and show improvements over DeepONet and FNO in accuracy and flexibility for complex geometries and time-dependent problems.

Abstract

Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other deep learning methods. However, the spatial domain of its input function needs to be identical to its output, which limits its applicability. For instance, the widely used Fourier neural operator (FNO) fails to approximate the operator that maps the boundary condition to the PDE solution. To address this issue, we propose a novel framework called resolution-invariant deep operator (RDO) that decouples the spatial domain of the input and output. RDO is motivated by the Deep operator network (DeepONet) and it does not require retraining the network when the input/output is changed compared with DeepONet. RDO takes functional input and its output is also functional so that it keeps the resolution invariant property of NO. It can also resolve PDEs with complex geometries whereas NO fail. Various numerical experiments demonstrate the advantage of our method over DeepONet and FNO.

Resolution invariant deep operator network for PDEs with complex geometries

TL;DR

The paper tackles the limitation of existing neural operators that require identical input and output domains by introducing the resolution-invariant deep operator (RDO). RDO replaces the branch net of DeepONet with a neural-operator–based integral transform, forming a composition which preserves resolution-invariance and supports functional inputs/outputs on different domains. A universal approximation theorem for RDO is established, leveraging Fourier integral operators (FIO) and attention integral operators (AIO) to realize the branch, along with a learnable readout . Numerical experiments on SVBP, Darcy flow with irregular domains, and Burgers’ equation demonstrate RDO’s effectiveness, including zero-shot super-resolution, and show improvements over DeepONet and FNO in accuracy and flexibility for complex geometries and time-dependent problems.

Abstract

Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other deep learning methods. However, the spatial domain of its input function needs to be identical to its output, which limits its applicability. For instance, the widely used Fourier neural operator (FNO) fails to approximate the operator that maps the boundary condition to the PDE solution. To address this issue, we propose a novel framework called resolution-invariant deep operator (RDO) that decouples the spatial domain of the input and output. RDO is motivated by the Deep operator network (DeepONet) and it does not require retraining the network when the input/output is changed compared with DeepONet. RDO takes functional input and its output is also functional so that it keeps the resolution invariant property of NO. It can also resolve PDEs with complex geometries whereas NO fail. Various numerical experiments demonstrate the advantage of our method over DeepONet and FNO.
Paper Structure (18 sections, 7 theorems, 52 equations, 8 figures, 7 tables)

This paper contains 18 sections, 7 theorems, 52 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Operator regression
  3. Problem settings
  4. DeepONet
  5. Proposed Architecture
  6. RDO
  7. Approximation theorem
  8. Parameterization and implementation of RDO
  9. Fourier integral operator
  10. Attention integral operator
  11. Architecture of RDO
  12. Numerical experiments
  13. Stochastic boundary value problem (SVBP)
  14. Darcy flow
  15. Darcy problem in a triangular domain
  16. ...and 3 more sections

Key Result

Lemma 1

Let $f \in C(\mathcal{D})$ be a continuous function defined on the domain $\mathcal{D}$ and $x_1, \dots, x_m$ be $m$ fixed points in $\mathcal{D}$. The vector $f_m = (f(x_1), \dots, f(x_m))^T \in \mathbb{R}^m$ is obtained by evaluating the function $f$ at these fixed points. Then, the mapping $\math

Figures (8)

  • Figure 1: DeepONet architecture.
  • Figure 2: FIO architecture.
  • Figure 3: AIO architecture.
  • Figure 4: RDO architecture.
  • Figure 5: Computational results for $\ell_x=1$. (a) gives the input function $a(\cdot,\cdot)$. (b), (c) and (d) shows the prediction results for the resolution 33, 65, 129, respectively.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 1
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 2 more