用生成对抗网络指导残差，提升物理信息Transformer在复杂方程中的精度与因果性。
A Residual Guided strategy with Generative Adversarial Networks in training Physics-Informed Transformer Networks
- 引入残差感知GAN动态识别高误差区域，引导模型重点训练。
- 在三类典型方程上实现相对均方误差降低三个数量级。
- 适合需要高精度时空建模的多尺度物理系统研究者。
非线性偏微分方程（PDEs）在复杂物理系统建模中至关重要，但传统物理信息神经网络（PINNs）常在关键时空区域存在未解决的残差并违反时间因果性。为此，我们提出一种基于生成对抗网络（GAN）的残差引导训练策略，用于物理信息Transformer。该框架采用仅解码器的Transformer结构，通过自回归处理自然捕捉时间相关性；同时引入残差感知GAN，动态识别并优先处理高残差区域。通过添加因果性惩罚项和自适应采样机制，方法在保证时间因果性的前提下，显著提升问题区域的精度。在Allen-Cahn、Klein-Gordon和Navier-Stokes方程上的大量数值实验表明，相比基线方法，相对均方误差最高降低三个数量级。本工作弥合了深度学习与物理驱动建模之间的差距，为多尺度、时变的PDE系统提供了一种鲁棒解决方案。
原文摘要 · Abstract (English)
Nonlinear partial differential equations (PDEs) are pivotal in modeling complex physical systems, yet traditional Physics-Informed Neural Networks (PINNs) often struggle with unresolved residuals in critical spatiotemporal regions and violations of temporal causality. To address these limitations, we propose a novel Residual Guided Training strategy for Physics-Informed Transformer via Generative Adversarial Networks (GAN). Our framework integrates a decoder-only Transformer to inherently capture temporal correlations through autoregressive processing, coupled with a residual-aware GAN that dynamically identifies and prioritizes high-residual regions. By introducing a causal penalty term and an adaptive sampling mechanism, the method enforces temporal causality while refining accuracy in problematic domains. Extensive numerical experiments on the Allen-Cahn, Klein-Gordon, and Navier-Stokes equations demonstrate significant improvements, achieving relative MSE reductions of up to three orders of magnitude compared to baseline methods. This work bridges the gap between deep learning and physics-driven modeling, offering a robust solution for multiscale and time-dependent PDE systems.
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