# Adversarial Learning for Neural PDE Solvers with Sparse Data

arXiv: https://arxiv.org/abs/2409.02431 · HTML: https://mlsanity.com/paper/2409.02431/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 29 references that are in the map.
- First posted on arXiv 4 Sep 2024.

## Builds on (most cited first, 18 of 18)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Adversarial examples in the physical world](https://mlsanity.com/paper/1607.02533/) (2016)
- [Multi-Scale Context Aggregation by Dilated Convolutions](https://mlsanity.com/paper/1511.07122/) (2015)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Time Series Data Augmentation for Deep Learning: A Survey](https://mlsanity.com/paper/2002.12478/) (2020)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Person Re-identification Method Based on Color Attack and Joint Defence](https://mlsanity.com/paper/2111.09571/) (2021)
- [Eliminate Deviation with Deviation for Data Augmentation and a General Multi-modal Data Learning Method](https://mlsanity.com/paper/2101.08533/) (2021)
- [Exploring Color Invariance through Image-Level Ensemble Learning](https://mlsanity.com/paper/2401.10512/) (2024)
- [Beyond Augmentation: Empowering Model Robustness under Extreme Capture Environments](https://mlsanity.com/paper/2407.13640/) (2024)
- [Beyond Dropout: Robust Convolutional Neural Networks Based on Local Feature Masking](https://mlsanity.com/paper/2407.13646/) (2024)

## Built on by (most cited first, 16 of 16)

- [Cross-Modality Attack Boosted by Gradient-Evolutionary Multiform Optimization](https://mlsanity.com/paper/2409.17977/) (2024)
- [StruSR: Structure-Aware Symbolic Regression with Physics-Informed Taylor Guidance](https://mlsanity.com/paper/2510.06635/) (2025)
- [Diffusion-Based Image Editing: An Unforeseen Adversary to Robust Invisible Watermarks](https://mlsanity.com/paper/2511.05598/) (2025)
- [Vanishing Watermarks: Diffusion-Based Image Editing Undermines Robust Invisible Watermarking](https://mlsanity.com/paper/2602.20680/) (2026)
- [Geometrically Regularized Transfer Learning with On-Manifold and Off-Manifold Perturbation](https://mlsanity.com/paper/2505.15191/) (2025)
- [Editing on the Generative Manifold: A Theoretical and Empirical Study of General Diffusion-Based Image Editing Trade-offs](https://mlsanity.com/paper/2603.29736/) (2026)
- [Fusing Physics-Driven Strategies and Cross-Modal Adversarial Learning: Toward Multi-Domain Applications](https://mlsanity.com/paper/2412.00341/) (2024)
- [On the Mechanisms of Adversarial Data Augmentation for Robust and Adaptive Transfer Learning](https://mlsanity.com/paper/2505.12681/) (2025)
- [GAMA: Geometry-Aware Manifold Alignment via Structured Adversarial Perturbations for Robust Domain Adaptation](https://mlsanity.com/paper/2505.15194/) (2025)
- [Disentangled Geometric Alignment with Adaptive Contrastive Perturbation for Reliable Domain Transfer](https://mlsanity.com/paper/2505.15241/) (2025)
- [Dynamic Epsilon Scheduling: A Multi-Factor Adaptive Perturbation Budget for Adversarial Training](https://mlsanity.com/paper/2506.04263/) (2025)
- [Dynamic Modality Scheduling for Multimodal Large Models via Confidence, Uncertainty, and Semantic Consistency](https://mlsanity.com/paper/2506.12724/) (2025)
- [FADE: Adversarial Concept Erasure in Flow Models](https://mlsanity.com/paper/2507.12283/) (2025)
- [Robust Concept Erasure in Diffusion Models: A Theoretical Perspective on Security and Robustness](https://mlsanity.com/paper/2509.12024/) (2025)
- [When Denoising Becomes Unsigning: Theoretical and Empirical Analysis of Watermark Fragility Under Diffusion-Based Image Editing](https://mlsanity.com/paper/2603.04696/) (2026)
- [Editing Away the Evidence: Diffusion-Based Image Manipulation and the Failure Modes of Robust Watermarking](https://mlsanity.com/paper/2603.12949/) (2026)

## 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.