Learning Heat-based Equations in Self-similar variables

TL;DR

This paper introduces a self-similar variable training framework for neural operators applied to heat-based equations, improving long-term prediction accuracy and stability in physical systems like Navier-Stokes and Burgers equations.

Contribution

The paper develops a novel SSV training framework compatible with neural operators, demonstrating its effectiveness on fluid dynamics equations for better long-term extrapolation.

Findings

SSV-trained networks outperform physical coordinate models in accuracy.

SSV approach yields more stable long-term predictions.

Self-similar coordinates serve as a beneficial inductive bias.

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.