用对抗学习提升稀疏数据下神经PDE求解器的泛化能力
Adversarial Learning for Neural PDE Solvers with Sparse Data
- 通过系统性增强模型弱点来对抗数据稀缺问题
- 在多种PDE场景中显著降低泛化误差，提升预测精度
- 适合需要高鲁棒性求解器的复杂物理模拟任务
神经网络求解偏微分方程（PDE）取得了显著进展，但仍面临数据稀缺与模型鲁棒性不足的挑战。传统数据增强方法依赖对称性或不变性假设，这些假设在动态复杂的现实应用中常不成立。为此，本文提出一种通用学习策略——系统性模型增强鲁棒训练（SMART），通过聚焦并强化模型的薄弱环节，在数据稀疏条件下有效降低训练过程中的泛化误差，显著提升多种PDE场景下的预测准确率。该方法通过理论分析与大量实验验证了有效性，代码将公开。
原文摘要 · Abstract (English)
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.
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