mlsanityData to 7 Oct 2026

Adversarial Learning for Neural PDE Solvers with Sparse Data

arXiv 2409.02431 ↗First posted 4 Sep 2024cs.LGcited by 16

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Abstract (arXiv)

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.

Key facts. Cited by 16 papers in the map (data through 7 Oct 2026). Builds on 18 of its 29 references that are in the map. First posted on arXiv 4 Sep 2024.

Builds on

18 of 29 references.

References · 18

  1. 2014Explaining and Harnessing Adversarial Examplescited by 7,287
  2. 2017Towards Deep Learning Models Resistant to Adversarial Attackscited by 6,326
  3. 2013Intriguing properties of neural networkscited by 5,114
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  5. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  6. 2016Adversarial examples in the physical worldcited by 1,762
  7. 2015Multi-Scale Context Aggregation by Dilated Convolutionscited by 1,128
  8. 2019Hamiltonian Neural Networkscited by 702
  9. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 396
  10. 2020Time Series Data Augmentation for Deep Learning: A Surveycited by 225
  11. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 123
  12. 2021Learned Coarse Models for Efficient Turbulence Simulationcited by 90
6 more
  1. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 76
  2. 2021Person Re-identification Method Based on Color Attack and Joint Defencecited by 56
  3. 2021Eliminate Deviation with Deviation for Data Augmentation and a General Multi-modal Data Learning Methodcited by 42
  4. 2024Exploring Color Invariance through Image-Level Ensemble Learningcited by 23
  5. 2024Beyond Augmentation: Empowering Model Robustness under Extreme Capture Environmentscited by 23
  6. 2024Beyond Dropout: Robust Convolutional Neural Networks Based on Local Feature Maskingcited by 19

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 10; 2026: 4.

16 papers.

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  1. 2024Cross-Modality Attack Boosted by Gradient-Evolutionary Multiform Optimizationcited by 11
  2. 2025StruSR: Structure-Aware Symbolic Regression with Physics-Informed Taylor Guidancecited by 5
  3. 2025Diffusion-Based Image Editing: An Unforeseen Adversary to Robust Invisible Watermarkscited by 2
  4. 2026Vanishing Watermarks: Diffusion-Based Image Editing Undermines Robust Invisible Watermarkingcited by 2
  5. 2025Geometrically Regularized Transfer Learning with On-Manifold and Off-Manifold Perturbationcited by 1
  6. 2026Editing on the Generative Manifold: A Theoretical and Empirical Study of General Diffusion-Based Image Editing Trade-offscited by 1
  7. 2024Fusing Physics-Driven Strategies and Cross-Modal Adversarial Learning: Toward Multi-Domain Applicationscited by 0
  8. 2025On the Mechanisms of Adversarial Data Augmentation for Robust and Adaptive Transfer Learningcited by 0
  9. 2025GAMA: Geometry-Aware Manifold Alignment via Structured Adversarial Perturbations for Robust Domain Adaptationcited by 0
  10. 2025Disentangled Geometric Alignment with Adaptive Contrastive Perturbation for Reliable Domain Transfercited by 0
  11. 2025Dynamic Epsilon Scheduling: A Multi-Factor Adaptive Perturbation Budget for Adversarial Trainingcited by 0
  12. 2025Dynamic Modality Scheduling for Multimodal Large Models via Confidence, Uncertainty, and Semantic Consistencycited by 0
4 more
  1. 2025FADE: Adversarial Concept Erasure in Flow Modelscited by 0
  2. 2025Robust Concept Erasure in Diffusion Models: A Theoretical Perspective on Security and Robustnesscited by 0
  3. 2026When Denoising Becomes Unsigning: Theoretical and Empirical Analysis of Watermark Fragility Under Diffusion-Based Image Editingcited by 0
  4. 2026Editing Away the Evidence: Diffusion-Based Image Manipulation and the Failure Modes of Robust Watermarkingcited by 0