用自相似变量训练神经算子，显著提升热方程长期动态预测精度。
Learning Heat-based Equations in Self-similar variables
- 在自相似坐标下训练神经算子，引入数学驱动的归纳偏置。
- 相比物理坐标训练，外推误差更小，长期趋势捕捉更准确。
- 适用于需要稳定预测长期行为的流体动力学问题。
我们研究了在自相似变量（SSV）框架下学习热基方程的解。提出一种与标准神经算子训练兼容的SSV训练框架，并在二维不可压缩纳维-斯托克斯方程和一维黏性伯格斯方程上进行验证。采用两种简单全连接架构（标准多层感知机与因子化全连接网络），对比物理坐标与自相似坐标训练模型。在两类系统及两种架构下，SSV训练模型均表现出更优的外推性能和稳定性，且能更好刻画长期定性演化趋势。结果表明，自相似坐标为学习热基方程的长期动态提供了数学上合理的归纳偏置。
原文摘要 · Abstract (English)
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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