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

• URL: http://arxiv.org/abs/2510.25731v1
• Date: Wed, 29 Oct 2025 17:37:27 GMT
• Title: LieSolver: A PDE-constrained solver for IBVPs using Lie symmetries
• Authors: René P. Klausen, Ivan Timofeev, Johannes Frank, Jonas Naujoks, Thomas Wiegand, Sebastian Lapuschkin, Wojciech Samek,
• Abstract summary: We introduce a method for solving initial-boundary value problems (IBVPs) that uses Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construction.<n>By leveraging symmetry transformations, the model inherently incorporates the physical laws and learns solutions from initial and boundary data.
• Score: 20.541443888815873
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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