Conservation-preserved Fourier Neural Operator through Adaptive Correction

2025-05-30

AI Summary

Standard Fourier Neural Operators (FNOs) struggle to strictly satisfy physical conservation laws—such as mass and momentum conservation—limiting their reliability in physics-driven PDE solving. To address this, we propose a learnable adaptive correction matrix, embedding a parameterized linear correction module at the FNO output layer. This module is jointly trained with conservation constraints, unifying spectral-domain modeling and physical priors. Crucially, conservation is enforced dynamically and exactly during training—without post-hoc correction—yielding theoretically guaranteed fidelity: our method’s data-fitting error is provably no worse than that of the optimal conserving FNO. Evaluated across diverse canonical PDE benchmarks—including Burgers’, Navier–Stokes, and Darcy flow equations—our approach consistently outperforms existing conservation-enhanced methods, achieving both high predictive accuracy and strict, end-to-end conservation compliance.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Fourier Neural Operators (FNOs) have recently emerged as a promising and efficient approach for learning the numerical solutions to partial differential equations (PDEs) from data. However, standard FNO often fails to preserve key conservation laws, such as mass conservation, momentum conservation, norm conservation, etc., which are crucial for accurately modeling physical systems. Existing methods for incorporating these conservation laws into Fourier neural operators are achieved by designing related loss function or incorporating post-processing method at the training time. None of them can both exactly and adaptively correct the outputs to satisfy conservation laws, and our experiments show that these methods can lead to inferior performance while preserving conservation laws. In this work, we propose a novel adaptive correction approach to ensure the conservation of fundamental quantities. Our method introduces a learnable matrix to adaptively adjust the solution to satisfy the conservation law during training. It ensures that the outputs exactly satisfy the goal conservation law and allow for more flexibility and adaptivity for the model to correct the outputs. We theoretically show that applying our adaptive correction to an unconstrained FNO yields a solution with data loss no worse than that of the best conservation-satisfying FNO. We compare our approach with existing methods on a range of representative PDEs. Experiment results show that our method consistently outperform other methods.

Problem

• FNOs fail to preserve key conservation laws in PDEs
• Existing methods lack exact and adaptive correction for conservation laws
• Proposing adaptive correction to ensure exact conservation in FNOs

Innovation

• Adaptive correction for conservation laws
• Learnable matrix adjusts solutions adaptively
• Ensures exact conservation law satisfaction