在自相似变量中学习热方程
Learning Heat-based Equations in Self-similar variables
AI总结：
本文提出在自相似变量中学习热方程的方法，通过对比不同坐标系下的模型表现，证明自相似坐标能提升长期动态预测的准确性和稳定性。
摘要：
我们研究了在自相似变量（SSV）中热方程的解学习。我们开发了一个与标准神经算子训练兼容的SSV训练框架。我们在此框架上实例化于二维不可压缩纳维-斯托克斯方程和一维粘性伯格斯方程，并使用两种简单的全连接架构（标准多层感知机和因子化全连接网络）进行受控比较，比较在物理坐标和相应自相似坐标上训练的模型。在两个系统和两种架构中，SSV训练的网络在训练窗口之外的外推上表现得更加准确和稳定，并且能更好地捕捉长期趋势。这些结果表明，自相似坐标为学习热方程的长期动态提供了数学动机的归纳偏差。
英文摘要：
We study solution learning for heat-based equations in self-similar variables (SSV). We develop an SSV training framework compatible with standard neural-operator training. We instantiate this framework on the two-dimensional incompressible Navier-Stokes equations and the one-dimensional viscous Burgers equation, and perform controlled comparisons between models trained in physical coordinates and in the corresponding self-similar coordinates using two simple fully connected architectures (standard multilayer perceptrons and a factorized fully connected network). Across both systems and both architectures, SSV-trained networks consistently deliver substantially more accurate and stable extrapolation beyond the training window and better capture qualitative long-time trends. These results suggest that self-similar coordinates provide a mathematically motivated inductive bias for learning the long-time dynamics of heat-based equations.
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