用域分解+超网络，高效学习复杂偏微分方程的物理信息算子。
FB-HyDON: Parameter-Efficient Physics-Informed Operator Learning of Complex PDEs via Hypernetwork and Finite Basis Domain Decomposition
- 基于有限基函数与超网络实现域分解，降低训练难度。
- 在高频率振荡器、不同粘性布格尔斯方程上性能超越现有模型。
- 适合需要少数据、强物理约束的科学计算场景。
深度算子网络（DeepONet）和神经算子因其可映射无限维函数空间并实现零样本超分辨率而备受关注。然而，这些模型通常需要大量数据进行有效训练。尽管物理信息算子提供了无数据学习方法，但在高度非线性系统中仍存在额外训练复杂性和收敛难题。为此，我们提出有限基物理信息超深度算子网络（FB-HyDON），一种具备内在域分解的先进算子架构。通过超网络与有限基函数的结合，FB-HyDON有效缓解了现有物理信息算子学习方法的训练局限。我们在高频谐振子、不同粘性水平下的布格尔斯方程及艾伦-蔡恩方程上验证了该方法，结果表明其在多项指标上显著优于其他算子学习模型。
原文摘要 · Abstract (English)
Deep operator networks (DeepONet) and neural operators have gained significant attention for their ability to map infinite-dimensional function spaces and perform zero-shot super-resolution. However, these models often require large datasets for effective training. While physics-informed operators offer a data-agnostic learning approach, they introduce additional training complexities and convergence issues, especially in highly nonlinear systems. To overcome these challenges, we introduce Finite Basis Physics-Informed HyperDeepONet (FB-HyDON), an advanced operator architecture featuring intrinsic domain decomposition. By leveraging hypernetworks and finite basis functions, FB-HyDON effectively mitigates the training limitations associated with existing physics-informed operator learning methods. We validated our approach on the high-frequency harmonic oscillator, Burgers' equation at different viscosity levels, and Allen-Cahn equation demonstrating substantial improvements over other operator learning models.
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