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Adversarial Learning for Neural PDE Solvers with Sparse Data

Yunpeng Gong, Yongjie Hou, Zhenzhong Wang, Zexin Lin, Min Jiang

TL;DR

The paper tackles the fragility of neural PDE solvers under data scarcity by proposing Systematic Model Augmentation for Robust Training (SMART), an adversarial augmentation framework that concentrates training on model weaknesses through carefully crafted input perturbations. The authors formalize a theoretical basis via a coverage measure showing that adversarial training expands the effective learning domain and reduces generalization error, and they validate the approach with Burgers’, Advection, and 2D Navier-Stokes experiments, demonstrating significant accuracy gains over standard data augmentation, especially in sparse-data settings. They also compare SMART to General Covariance Data Augmentation and show complementary gains when combined with Lie Point Symmetry augmentation, across multiple neural PDE solvers. The work offers a practical pathway to more robust, data-efficient neural PDE solvers with potential impact across scientific computing domains.

Abstract

Neural network solvers for partial differential equations (PDEs) have made significant progress, yet they continue to face challenges related to data scarcity and model robustness. Traditional data augmentation methods, which leverage symmetry or invariance, impose strong assumptions on physical systems that often do not hold in dynamic and complex real-world applications. To address this research gap, this study introduces a universal learning strategy for neural network PDEs, named Systematic Model Augmentation for Robust Training (SMART). By focusing on challenging and improving the model's weaknesses, SMART reduces generalization error during training under data-scarce conditions, leading to significant improvements in prediction accuracy across various PDE scenarios. The effectiveness of the proposed method is demonstrated through both theoretical analysis and extensive experimentation. The code will be available.

Adversarial Learning for Neural PDE Solvers with Sparse Data

TL;DR

The paper tackles the fragility of neural PDE solvers under data scarcity by proposing Systematic Model Augmentation for Robust Training (SMART), an adversarial augmentation framework that concentrates training on model weaknesses through carefully crafted input perturbations. The authors formalize a theoretical basis via a coverage measure showing that adversarial training expands the effective learning domain and reduces generalization error, and they validate the approach with Burgers’, Advection, and 2D Navier-Stokes experiments, demonstrating significant accuracy gains over standard data augmentation, especially in sparse-data settings. They also compare SMART to General Covariance Data Augmentation and show complementary gains when combined with Lie Point Symmetry augmentation, across multiple neural PDE solvers. The work offers a practical pathway to more robust, data-efficient neural PDE solvers with potential impact across scientific computing domains.

Abstract

Neural network solvers for partial differential equations (PDEs) have made significant progress, yet they continue to face challenges related to data scarcity and model robustness. Traditional data augmentation methods, which leverage symmetry or invariance, impose strong assumptions on physical systems that often do not hold in dynamic and complex real-world applications. To address this research gap, this study introduces a universal learning strategy for neural network PDEs, named Systematic Model Augmentation for Robust Training (SMART). By focusing on challenging and improving the model's weaknesses, SMART reduces generalization error during training under data-scarce conditions, leading to significant improvements in prediction accuracy across various PDE scenarios. The effectiveness of the proposed method is demonstrated through both theoretical analysis and extensive experimentation. The code will be available.
Paper Structure (16 sections, 27 equations, 1 figure, 3 tables)

This paper contains 16 sections, 27 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. PDE Data Augmentation Examples
  3. General Covariance Data Augmentation
  4. Lie Point Symmetry Data Augmentation
  5. Proposed Methods
  6. One-dimensional Burgers’ Equation Adversarial Sample Generation Example
  7. Adversarial Sample Generation Steps
  8. Final Adversarial Sample Generation.
  9. Two-dimensional Incompressible Navier-Stokes Equation Adversarial Sample Generation Example
  10. Adversarial Sample Generation Steps
  11. Theoretical Analysis
  12. Example Pseudocode
  13. Experiments
  14. Evaluation Criteria
  15. Comparison Experiments
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Displays the time evolution of the 1D Advection equation: (a) exact solution, (b) neural network prediction, (c) prediction with added random noise, (d) prediction after adversarial perturbation. The amplitude of both random and adversarial noise is 8% of the grid size.