LieSolver: A PDE-constrained solver for IBVPs using Lie symmetries
René P. Klausen, Ivan Timofeev, Johannes Frank, Jonas Naujoks, Thomas Wiegand, Sebastian Lapuschkin, Wojciech Samek
TL;DR
LieSolver addresses solving IBVPs by embedding Lie symmetry transformations directly into the model, ensuring the PDE is satisfied exactly by construction. It represents solutions as linear combinations of symmetry-generated base functions built from seed solutions, reducing learning to fitting initial and boundary data using a two-stage optimization that combines greedy basis selection, variable projection, and nonlinear least squares. Across 1D heat and wave problems, LieSolver achieves higher accuracy with far fewer parameters and substantially faster runtimes than PINNs, while providing interpretable, PDE-consistent representations of the solution. The approach offers rigorous error estimates for well-posed IBVPs and demonstrates practical gains in efficiency and reliability, albeit being limited currently to linear homogeneous PDEs and dependent on the choice of base solutions. Overall, LieSolver presents a symmetry-informed alternative to physics-informed learning that yields compact, interpretable solvers with strong theoretical and empirical performance advantages.
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.