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Derivative-enhanced Deep Operator Network

Yuan Qiu, Nolan Bridges, Peng Chen

TL;DR

A derivative-enhanced deep operator network (DE-DeepONet) is proposed, which leverages derivative information to enhance the solution prediction accuracy and provides a more accurate approximation of solution-to-parameter derivatives, especially when training data are limited.

Abstract

The deep operator networks (DeepONet), a class of neural operators that learn mappings between function spaces, have recently been developed as surrogate models for parametric partial differential equations (PDEs). In this work we propose a derivative-enhanced deep operator network (DE-DeepONet), which leverages derivative information to enhance the solution prediction accuracy and provides a more accurate approximation of solution-to-parameter derivatives, especially when training data are limited. DE-DeepONet explicitly incorporates linear dimension reduction of high dimensional parameter input into DeepONet to reduce training cost and adds derivative loss in the loss function to reduce the number of required parameter-solution pairs. We further demonstrate that the use of derivative loss can be extended to enhance other neural operators, such as the Fourier neural operator (FNO). Numerical experiments validate the effectiveness of our approach.

Derivative-enhanced Deep Operator Network

TL;DR

A derivative-enhanced deep operator network (DE-DeepONet) is proposed, which leverages derivative information to enhance the solution prediction accuracy and provides a more accurate approximation of solution-to-parameter derivatives, especially when training data are limited.

Abstract

The deep operator networks (DeepONet), a class of neural operators that learn mappings between function spaces, have recently been developed as surrogate models for parametric partial differential equations (PDEs). In this work we propose a derivative-enhanced deep operator network (DE-DeepONet), which leverages derivative information to enhance the solution prediction accuracy and provides a more accurate approximation of solution-to-parameter derivatives, especially when training data are limited. DE-DeepONet explicitly incorporates linear dimension reduction of high dimensional parameter input into DeepONet to reduce training cost and adds derivative loss in the loss function to reduce the number of required parameter-solution pairs. We further demonstrate that the use of derivative loss can be extended to enhance other neural operators, such as the Fourier neural operator (FNO). Numerical experiments validate the effectiveness of our approach.
Paper Structure (42 sections, 1 theorem, 69 equations, 20 figures, 11 tables)

This paper contains 42 sections, 1 theorem, 69 equations, 20 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem setup
  4. High fidelity approximation
  5. DeepONet
  6. DE-DeepONet
  7. Model architecture
  8. Loss function
  9. Dimension reduction
  10. Karhunen–Loève Expansion (KLE) basis.
  11. Active Subspace Method (ASM) basis.
  12. Experiments
  13. Input probability measure
  14. Governing equations
  15. Evaluation metrics
  16. ...and 27 more sections

Key Result

Theorem 1

Suppose the PDE in the general form of eq:PDE is well-posed with a unique solution map from the input function $m\in V^{\text{in}}$ to the output function $u\in V^{\text{out}}$ with dual $(V^{\text{out}})^{\prime}$. Suppose the PDE operator $\mathcal{R}:V^{\text{in}}\times V^{\text{out}}\to (V^{\tex

Figures (20)

  • Figure 1: Mean relative errors ($\pm$ standard deviation) over 5 random seeds of neural network training for a varying number of training samples for the [top: hyperelasticity; bottom: Navier--Stokes] equation using different methods. Relative errors in the $L^2(\Omega)$ norm (left) and $H^1({\Omega})$ norm (middle) for the prediction of $u=(u_1,u_2)$. Right: Relative error in the Frobenius (Fro) norm for the prediction of $du(m;\omega)=(du_1(m;\omega), du_2(m;\omega))$.
  • Figure 2: Mean relative errors ($\pm$ standard deviation) over 5 random seeds versus model training time for the Navier--Stokes equations when the number of training samples is [top: 16; bottom: 256].
  • Figure 3: Visualization of one parameter-solution pair of hyperelasticity equation. The color of the output indicates the magnitude of the displacement $u$ (which maps from domain $\Omega$ to $\mathbb{R}^2$) instead of its componentwise function $u_1$ or $u_2$. The skewed square shows locations of any domain point after deformation $X\to x$. See \ref{['fig:truth_prediction_hyperelasticity_u1', 'fig:truth_prediction_hyperelasticity_u2']} for $u_1$ and $u_2$.
  • Figure 4: Visualization of one parameter-solution pair of Navier--Stokes equations.
  • Figure 5: Visualization of the $10$ by $10$ unit square mesh. Left: diagonal='right'; Right: diagonal='crossed'
  • ...and 15 more figures

Theorems & Definitions (4)

  • Remark
  • Theorem 1
  • Remark
  • Remark