为图神经网络求解偏微分方程设定消息传递下限，避免盲目调参。
On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL condition
- 根据方程物理特性推导消息传递最少次数
- 低于下限则信息无法有效传播，解的质量差
- 适用于三类基本偏微分方程，可指导模型设计
本文为图神经网络（GNN）求解偏微分方程（PDE）时所需的消息传递迭代次数提出了精确的下界。该下界显著减少了对超参数的穷举式调优需求。针对三类基本PDE（双曲型、抛物型、椭圆型），通过关联问题的物理特性与GNN的消息传递机制，推导出下界。具体涉及控制方程的物理常数、空间与时间离散化方式，以及消息传递结构之间的关系。当消息传递次数低于该下界时，即使使用深层网络，信息也无法有效传播，导致解的质量下降。反之，满足下界后，模型能准确捕捉物理现象，获得足够精度的求解器。文中以四类典型方程为例验证了所提下界的紧性。
原文摘要 · Abstract (English)
This paper proposes sharp lower bounds for the number of message passing iterations required in graph neural networks (GNNs) when solving partial differential equations (PDE). This significantly reduces the need for exhaustive hyperparameter tuning. Bounds are derived for the three fundamental classes of PDEs (hyperbolic, parabolic and elliptic) by relating the physical characteristics of the problem in question to the message-passing requirement of GNNs. In particular, we investigate the relationship between the physical constants of the equations governing the problem, the spatial and temporal discretisation and the message passing mechanisms in GNNs. When the number of message passing iterations is below these proposed limits, information does not propagate efficiently through the network, resulting in poor solutions, even for deep GNN architectures. In contrast, when the suggested lower bound is satisfied, the GNN parameterisation allows the model to accurately capture the underlying phenomenology, resulting in solvers of adequate accuracy. Examples are provided for four different examples of equations that show the sharpness of the proposed lower bounds.
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