Learning Heat-Based Equations in Self-Similar Variables

Shihao Wang, Qipeng Qian, Jingquan Wang

Keywords: PDE Learning, Self-similar Variables, Long-time dynamics, Operator learning, Heat-based Equations

TL;DR: Showing better performance of learning heat-based solutions (u_t - u_xx = ....) in self-similar variables rather than in physical variables.

Abstract: 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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Submission Number: 16