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Sobolev Training for Operator Learning

Namkyeong Cho, Junseung Ryu, Hyung Ju Hwang

TL;DR

This work introduces Sobolev Training for operator learning, integrating derivative information into the training loss to improve convergence and generalization when learning mappings between function spaces. The authors propose a derivative-approximation pipeline on irregular meshes using moving least squares within a locally PCA-derived frame, and provide a convergence analysis for a one-layer ReLU operator with an integral kernel, showing faster convergence when derivatives are included. They also employ PCGrad to stabilize multi-task objectives that combine $L^2$ and derivative losses, and demonstrate via experiments on six datasets and multiple baselines that the method consistently reduces relative $L^2$ errors, sometimes by over 30%, and improves robustness to noise. Theoretical and empirical results collectively support Sobolev Training as a principled, effective enhancement to neural-operator models, with practical impact for faster, more reliable operator learning in irregular-meshes and nonlocal PDE settings.

Abstract

This study investigates the impact of Sobolev Training on operator learning frameworks for improving model performance. Our research reveals that integrating derivative information into the loss function enhances the training process, and we propose a novel framework to approximate derivatives on irregular meshes in operator learning. Our findings are supported by both experimental evidence and theoretical analysis. This demonstrates the effectiveness of Sobolev Training in approximating the solution operators between infinite-dimensional spaces.

Sobolev Training for Operator Learning

TL;DR

This work introduces Sobolev Training for operator learning, integrating derivative information into the training loss to improve convergence and generalization when learning mappings between function spaces. The authors propose a derivative-approximation pipeline on irregular meshes using moving least squares within a locally PCA-derived frame, and provide a convergence analysis for a one-layer ReLU operator with an integral kernel, showing faster convergence when derivatives are included. They also employ PCGrad to stabilize multi-task objectives that combine and derivative losses, and demonstrate via experiments on six datasets and multiple baselines that the method consistently reduces relative errors, sometimes by over 30%, and improves robustness to noise. Theoretical and empirical results collectively support Sobolev Training as a principled, effective enhancement to neural-operator models, with practical impact for faster, more reliable operator learning in irregular-meshes and nonlocal PDE settings.

Abstract

This study investigates the impact of Sobolev Training on operator learning frameworks for improving model performance. Our research reveals that integrating derivative information into the loss function enhances the training process, and we propose a novel framework to approximate derivatives on irregular meshes in operator learning. Our findings are supported by both experimental evidence and theoretical analysis. This demonstrates the effectiveness of Sobolev Training in approximating the solution operators between infinite-dimensional spaces.
Paper Structure (28 sections, 7 theorems, 86 equations, 6 figures, 5 tables, 1 algorithm)

This paper contains 28 sections, 7 theorems, 86 equations, 6 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Sobolev spaces and Sobolev Training
  5. Operator Learning
  6. Results on the convergence analysis
  7. Proposed method
  8. Approximating derivatives in non-uniform meshes
  9. Convergence analysis for the Sobolev Training for operator learning
  10. Sobolev Training with PCGrad
  11. Experiments
  12. Baseline models
  13. Datasets
  14. Error Criteria
  15. Main Results
  16. ...and 13 more sections

Key Result

Lemma 3.1

Suppose $u \in W^{M,2}(D)$ for some $M \in \mathbb{N}$, and consider a set of grid points $\{x_j\}_{j=1}^{J}$ satisfing the Monte-Carlo approximation assumption eq:approx_ for all $D^\alpha_x u$ with $|\alpha| \leq M$ and for some $C_0 > 0$. Define and let $c_j$ be the coefficient vector obtained from Algorithm alg:gradient_est. Then, for all $m \leq M$, there exists a constant $C = C(C_0, N, K,

Figures (6)

  • Figure 1: The relative error for each $K$ from 6 to 30. The y-axis is the relative $L_2$ error; the x-axis is the value of $K$.
  • Figure 2: Visualization of coefficient function $a$ of the Darcy2d-dataset. Due to jump discontinuity, $a \in L^{\infty}((0,1)^{2};\mathbb{R}_{+})$ fails to belong to $W^{M, 2}((0,1)^{2})$ for all $M \in \mathbb{N}$.
  • Figure 3: Visualization of comparison between loss function landscape for fixed $\theta\in [0, \pi]$.
  • Figure 4: Visualization of comparison between loss function landscape for fixed $x\in [0, 3]$.
  • Figure 5: Visualization of loss function landscape for $L^2$ loss and Sobolev loss in 3D.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof