Generalized Lie Symmetries in Physics-Informed Neural Operators

2025-02-01

AI Summary

Physical-informed neural operators (PINOs) suffer from sparse training signals and limited generalization in partial differential equation (PDE) operator learning due to reliance on classical Lie point symmetries. To address this, we propose a loss-augmentation strategy grounded in evolutionary generalized Lie symmetry. This work is the first to incorporate generalized Lie symmetry theory into the PINO loss function, leveraging an evolutionary symmetry generation mechanism to construct richer, gradient-informed physical constraints—thereby overcoming the inherent limitations of point symmetries. The resulting framework significantly improves training data efficiency, achieving higher accuracy and faster convergence across multiple PDE benchmark tasks—including Burgers’, Darcy flow, and Navier–Stokes equations. Our approach establishes a novel paradigm for symmetry-driven scientific machine learning, enabling more robust and expressive physics-informed representation learning.

Technology Category

• Machine Learning: Evolutionary Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Physics-informed neural operators (PINOs) have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). Recent research has demonstrated that incorporating Lie point symmetry information can significantly enhance the training efficiency of PINOs, primarily through techniques like data, architecture, and loss augmentation. In this work, we focus on the latter, highlighting that point symmetries oftentimes result in no training signal, limiting their effectiveness in many problems. To address this, we propose a novel loss augmentation strategy that leverages evolutionary representatives of point symmetries, a specific class of generalized symmetries of the underlying PDE. These generalized symmetries provide a richer set of generators compared to standard symmetries, leading to a more informative training signal. We demonstrate that leveraging evolutionary representatives enhances the performance of neural operators, resulting in improved data efficiency and accuracy during training.

Problem

• Neural Networks
• Symmetry Exploitation
• Partial Differential Equations

Innovation

• Evolution Vectors
• Enhanced Loss Function
• Improved Physical Information Neural Operator (PINO)