LieSolver: A PDE-constrained solver for IBVPs using Lie symmetries

2025-10-29

AI Summary

This work addresses the challenge of rigorously satisfying partial differential equations (PDEs) at the construction level when solving initial-boundary value problems (IBVPs). We propose a novel solution framework grounded in Lie symmetry theory. By performing explicit Lie symmetry analysis, the method intrinsically embeds the PDE’s invariance structure into the model architecture, thereby internalizing physical laws as structural priors and ensuring that learned solutions automatically satisfy the governing equations. Combining symmetry reduction with PDE-constrained optimization, the framework directly learns symmetry-invariant solutions from initial and boundary data. Compared to physics-informed neural networks (PINNs), our approach achieves higher accuracy, faster convergence, and more compact model representations on diverse linear homogeneous PDEs. Moreover, it enables rigorous a posteriori error estimation, significantly enhancing predictive reliability and interpretability.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Learning to Search
• 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
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We introduce a method for efficiently solving initial-boundary value problems (IBVPs) that uses Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construction. By leveraging symmetry transformations, the model inherently incorporates the physical laws and learns solutions from initial and boundary data. As a result, the loss directly measures the model's accuracy, leading to improved convergence. Moreover, for well-posed IBVPs, our method enables rigorous error estimation. The approach yields compact models, facilitating an efficient optimization. We implement LieSolver and demonstrate its application to linear homogeneous PDEs with a range of initial conditions, showing that it is faster and more accurate than physics-informed neural networks (PINNs). Overall, our method improves both computational efficiency and the reliability of predictions for PDE-constrained problems.

Problem

• Solving initial-boundary value problems using Lie symmetries
• Enforcing PDE constraints exactly through symmetry transformations
• Improving computational efficiency and accuracy over PINNs

Innovation

• Uses Lie symmetries to enforce PDE exactly
• Leverages symmetry transformations for physical laws
• Enables rigorous error estimation for well-posed IBVPs