用物理约束+频域注意力，少数据也能精准模拟复杂系统演化。
PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems
- 融合物理规律与频域注意力，捕捉局部微分算子和全局依赖。
- 在有限数据下长期预测误差显著低于现有方法。
- 适合物理知识不全、观测数据稀少的科学建模场景。
准确模拟由偏微分方程（PDE）控制的复杂系统在众多科学与工程领域至关重要。然而，传统数值方法在真实场景中因物理定律不完整或未知而受限；机器学习方法则常因观测数据稀缺且难以同时捕捉局部与全局特征而泛化能力不足。为此，我们提出物理编码谱注意力网络（PeSANet），通过整合局部与全局信息，在数据有限且物理先验不全的情况下实现复杂系统的精准预测。模型包含两个核心组件：物理编码块利用硬约束从有限数据中近似局部微分算子；谱增强块在频域中捕捉长程全局依赖，引入新颖的谱注意力机制以建模频谱间关系并学习长距离空间特征。实验结果表明，PeSANet在各项指标上均优于现有方法，尤其在长期预测精度方面表现突出，为数据少、物理不全的复杂系统模拟提供了有效解决方案。
原文摘要 · Abstract (English)
Accurately modeling and forecasting complex systems governed by partial differential equations (PDEs) is crucial in various scientific and engineering domains. However, traditional numerical methods struggle in real-world scenarios due to incomplete or unknown physical laws. Meanwhile, machine learning approaches often fail to generalize effectively when faced with scarce observational data and the challenge of capturing local and global features. To this end, we propose the Physics-encoded Spectral Attention Network (PeSANet), which integrates local and global information to forecast complex systems with limited data and incomplete physical priors. The model consists of two key components: a physics-encoded block that uses hard constraints to approximate local differential operators from limited data, and a spectral-enhanced block that captures long-range global dependencies in the frequency domain. Specifically, we introduce a novel spectral attention mechanism to model inter-spectrum relationships and learn long-range spatial features. Experimental results demonstrate that PeSANet outperforms existing methods across all metrics, particularly in long-term forecasting accuracy, providing a promising solution for simulating complex systems with limited data and incomplete physics.
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