Scale-Consistent Learning for Partial Differential Equations

2025-07-24

AI Summary

Existing PDE-informed machine learning models exhibit limited generalization, particularly across varying physical parameters such as Reynolds number and domain scale. To address this, we propose a scale-consistent learning framework that enforces consistency between full-domain solutions and analytically scaled subdomain solutions—leveraging the inherent scale invariance of PDEs. Our method integrates scale-aware data augmentation, a dedicated scale-consistency loss, and multi-scale boundary/parameter reconstruction into a neural operator architecture. A multi-scale matching loss further refines solution fidelity across scales. We evaluate the framework on Burgers, Darcy flow, Helmholtz, and Navier–Stokes equations. Trained solely at Re = 1000, the model generalizes robustly across Re = 250–10,000, achieving an average 34% error reduction over baseline methods. This demonstrates substantial improvement in cross-scale generalization—marking the first approach to explicitly embed scale consistency as a training constraint in PDE learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Generalizing ML models for PDEs beyond training data scales
• Solving PDEs across varied domains and parameters consistently
• Enhancing neural operators with scale-consistency for multi-scale accuracy

Innovation

• Scale-consistent data augmentation for PDEs
• Scale-informed neural operator design
• Multi-scale generalization with consistency loss