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Learning Heat-based Equations in Self-similar variables

Shihao Wang, Qipeng Qian, Jingquan Wang

TL;DR

The paper addresses the challenge of fragile long-time extrapolation in time-dependent PDEs by introducing a self-similar-variable (SSV) training framework that leverages the parabolic scaling of heat-based dynamics. By reformulating the 2D Navier–Stokes vorticity and 1D viscous Burgers equations in variables $(ξ,τ)$, and training with a windowed $L^2$ loss on a fixed self-similar disk, the authors demonstrate that learning in SSV coordinates yields substantially more accurate and stable long-horizon predictions than learning in physical coordinates, across standard MLP and FCN architectures. Key contributions include a precise Monte Carlo sampling scheme on the SSV window, a fair comparison protocol, and evidence that SSV training better captures qualitative long-time features such as merging in NS and diffusion waves in Burgers. The findings suggest a physically motivated inductive bias that can improve operator-learning methods for heat-based PDEs and potentially extend to other systems with self-similar asymptotics. The work provides practical guidelines and code for reproducing the experiments, highlighting the value of aligning representations with intrinsic scaling laws for improved rollout stability and generalization.

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.

Learning Heat-based Equations in Self-similar variables

TL;DR

The paper addresses the challenge of fragile long-time extrapolation in time-dependent PDEs by introducing a self-similar-variable (SSV) training framework that leverages the parabolic scaling of heat-based dynamics. By reformulating the 2D Navier–Stokes vorticity and 1D viscous Burgers equations in variables , and training with a windowed loss on a fixed self-similar disk, the authors demonstrate that learning in SSV coordinates yields substantially more accurate and stable long-horizon predictions than learning in physical coordinates, across standard MLP and FCN architectures. Key contributions include a precise Monte Carlo sampling scheme on the SSV window, a fair comparison protocol, and evidence that SSV training better captures qualitative long-time features such as merging in NS and diffusion waves in Burgers. The findings suggest a physically motivated inductive bias that can improve operator-learning methods for heat-based PDEs and potentially extend to other systems with self-similar asymptotics. The work provides practical guidelines and code for reproducing the experiments, highlighting the value of aligning representations with intrinsic scaling laws for improved rollout stability and generalization.

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.
Paper Structure (16 sections, 2 theorems, 29 equations, 6 figures)

This paper contains 16 sections, 2 theorems, 29 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Networks
  3. Learning the vorticity of the 2D Navier-Stokes equation
  4. System and self-similar variables
  5. Training in the self-similar framework
  6. Monte Carlo estimator (uniform sampling on the SSV window).
  7. Training objective.
  8. Physical-space window (expanding disk)
  9. Expanding physical window and correspondence with SSV.
  10. Fair comparison protocol
  11. Experiments
  12. Viscous Burgers
  13. Equation Setup
  14. Sampling windows and training loss
  15. Experiments
  16. ...and 1 more sections

Key Result

Proposition 1

Fix $\mu\in(0,\tfrac{1}{2})$. There exist positive constants $r_2$ and $C$ such that, for any initial data $\Omega_0$ with $\|\Omega_0\|_2\leq r_2$, the solution $\Omega(\cdot,\tau)$ of eq:ssv-eq satisfies:

Figures (6)

  • Figure 1: Relative MSE over $t\in[0.3,5]$ under two architectures.
  • Figure 2: Example of extrapolation to $t=0.5,1,1.5$ under the FCN: truth (left), physical head (middle), and self-similar head mapped to physical space (right).
  • Figure 3: Example of extrapolation to $t=0.5,1,1.5$ under the MLP.
  • Figure 4: Relative MSE over $t\in[0.5,5]$ under two architectures.
  • Figure 5: Comparison between the physical FCN and the SSV FCN at t=1, 1.5, 2, 2.5.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 1: Long-time Behavior
  • Proposition 2: Long-time Behavior. kim_tzavaras_2001