Learning Heat-based Equations in Self-similar variables

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.

💡 Research Summary

The paper investigates whether learning the solution operators of heat‑type partial differential equations (PDEs) in self‑similar variables can improve long‑time extrapolation compared with the conventional approach of training directly in physical space. Heat‑based equations possess a characteristic parabolic scaling: spatial length grows like √t while amplitudes decay like t⁻¹/². By introducing the change of variables ξ = x/√(t+1) and τ = log(t+1), the authors map the original PDEs onto a new coordinate system where the linear heat kernel becomes stationary. In this self‑similar frame the equations acquire an additional drift term (½ ξ·∇_ξ) and an identity operator, but crucially the long‑time attractor (Gaussian for Navier‑Stokes vorticity, diffusion wave for Burgers) becomes a time‑independent, rapidly decaying profile in ξ. This observation suggests that a fixed sampling window in ξ captures the essential dynamics even for very large physical times.

To test the hypothesis, the authors consider two benchmark heat‑type systems: (i) the two‑dimensional incompressible Navier‑Stokes equations written in vorticity form, and (ii) the one‑dimensional viscous Burgers equation. For each system they train two neural‑operator architectures: a standard multilayer perceptron (MLP) that directly maps the concatenated coordinates (z = (x,t) or (ξ,τ)) to the solution value, and a factorized fully‑connected network (FCN) inspired by branch‑trunk designs, where a “branch” processes the time coordinate and a “trunk” processes the full space‑time coordinate, and the output is a bilinear combination of their latent vectors. Importantly, the same depth, width, optimizer, learning‑rate schedule, batch size, and total training steps are used for the physical‑coordinate and self‑similar models, ensuring a fair comparison.

The training protocol enforces a one‑to‑one correspondence between samples in the two coordinate systems. In the self‑similar setting, points (ξ,τ) are drawn uniformly from a fixed disk D_C = { |ξ| ≤ C } and τ ∈