Scale-Consistent Learning for Partial Differential Equations

• URL: http://arxiv.org/abs/2507.18813v1
• Date: Thu, 24 Jul 2025 21:29:52 GMT
• Title: Scale-Consistent Learning for Partial Differential Equations
• Authors: Zongyi Li, Samuel Lanthaler, Catherine Deng, Michael Chen, Yixuan Wang, Kamyar Azizzadenesheli, Anima Anandkumar,
• Abstract summary: We propose a data augmentation scheme based on scale-consistency properties of PDEs.<n>We then design a scale-informed neural operator that can model a wide range of scales.<n>With scale-consistency, the model trained on $Re$ of 1000 can generalize to $Re$ ranging from 250 to 10000.
• Score: 79.48661503591943
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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