Scale-Consistent Learning for Partial Differential Equations

TL;DR

This paper introduces a scale-consistent learning framework for PDEs that enhances neural operators' ability to generalize across different scales and parameters, significantly improving accuracy and robustness in solving various PDEs.

Contribution

The authors propose a novel scale-consistency loss and a scale-informed neural operator that enable PDE models to generalize across multiple scales and parameters, addressing limitations of previous ML approaches.

Findings

Model trained on Re=1000 generalizes from Re=250 to 10000.

Reduces error by 34% on average across datasets.

Effective on Burgers', Darcy Flow, Helmholtz, and Navier-Stokes equations.

Abstract

Machine learning (ML) models have emerged as a promising approach for solving partial differential equations (PDEs) in science and engineering. Previous ML models typically cannot generalize outside the training data; for example, a trained ML model for the Navier-Stokes equations only works for a fixed Reynolds number ($Re$) on a pre-defined domain. To overcome these limitations, we propose a data augmentation scheme based on scale-consistency properties of PDEs and design a scale-informed neural operator that can model a wide range of scales. Our formulation leverages the facts: (i) PDEs can be rescaled, or more concretely, a given domain can be re-scaled to unit size, and the parameters and the boundary conditions of the PDE can be appropriately adjusted to represent the original solution, and (ii) the solution operators on a given domain are consistent on the sub-domains. We leverage these facts to create a scale-consistency loss that encourages matching the solutions evaluated on a given domain and the solution obtained on its sub-domain from the rescaled PDE. Since neural operators can fit to multiple scales and resolutions, they are the natural choice for incorporating scale-consistency loss during training of neural PDE solvers. We experiment with scale-consistency loss and the scale-informed neural operator model on the Burgers' equation, Darcy Flow, Helmholtz equation, and Navier-Stokes equations. With scale-consistency, the model trained on $Re$ of 1000 can generalize to $Re$ ranging from 250 to 10000, and reduces the error by 34% on average of all datasets compared to baselines.