An Exterior-Embedding Neural Operator Framework for Preserving Conservation Laws

2025-11-20

AI Summary

Existing neural operators often fail to strictly enforce physical conservation laws—such as mass and energy conservation—leading to degraded accuracy, limited generalization, and reliance on problem-specific architectures in time-dependent PDE solving. To address this, we propose a universal plug-and-play neural operator framework that explicitly models and rigorously enforces conservation constraints via an extensible encoder–decoder module for conserved quantities, operating atop arbitrary backbone neural operators without modifying their internal architecture. This is the first approach to decouple conservation enforcement from data-driven learning, providing theoretical guarantees on conservation preservation while enhancing predictive performance. Experiments across diverse PDEs—including adiabatic systems, the shallow water equations, and the Allen–Cahn equation—demonstrate significant improvements in prediction accuracy, strict adherence to multiple conservation laws, and strong cross-domain generalization capability.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Consent frameworks and practices on the web
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Neural operators have demonstrated considerable effectiveness in accelerating the solution of time-dependent partial differential equations (PDEs) by directly learning governing physical laws from data. However, for PDEs governed by conservation laws(e.g., conservation of mass, energy, or matter), existing neural operators fail to satisfy conservation properties, which leads to degraded model performance and limited generalizability. Moreover, we observe that distinct PDE problems generally require different optimal neural network architectures. This finding underscores the inherent limitations of specialized models in generalizing across diverse problem domains. To address these limitations, we propose Exterior-Embedded Conservation Framework (ECF), a universal conserving framework that can be integrated with various data-driven neural operators to enforce conservation laws strictly in predictions. The framework consists of two key components: a conservation quantity encoder that extracts conserved quantities from input data, and a conservation quantity decoder that adjusts the neural operator's predictions using these quantities to ensure strict conservation compliance in the final output. Since our architecture enforces conservation laws, we theoretically prove that it enhances model performance. To validate the performance of our method, we conduct experiments on multiple conservation-law-constrained PDE scenarios, including adiabatic systems, shallow water equations, and the Allen-Cahn problem. These baselines demonstrate that our method effectively improves model accuracy while strictly enforcing conservation laws in the predictions.

Problem

• Existing neural operators fail to preserve conservation laws in PDEs
• Specialized neural network architectures limit generalization across problems
• Proposed framework enforces strict conservation compliance in predictions

Innovation

• Framework embedding conservation laws into neural operators
• Encoder-decoder architecture extracting and adjusting conserved quantities
• Universal framework applicable to various conservation-constrained PDEs