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

TL;DR

LieSolver is a novel PDE solver that uses Lie symmetries to enforce PDE constraints exactly, resulting in faster, more accurate solutions with built-in error estimation for IBVPs.

Contribution

This paper introduces LieSolver, a symmetry-based method that directly incorporates PDEs into the model, improving efficiency and accuracy over existing approaches like PINNs.

Findings

LieSolver outperforms PINNs in speed and accuracy.

The method provides rigorous error estimates for well-posed IBVPs.

LieSolver produces compact models suitable for efficient optimization.

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.