LieSolver: A PDE-constrained solver for IBVPs using Lie symmetries
This addresses PDE-constrained problems in computational physics and engineering, offering improved efficiency and reliability, though it appears incremental as an enhancement over existing PINN methods.
The authors tackled the problem of solving initial-boundary value problems (IBVPs) by introducing LieSolver, a method that uses Lie symmetries to enforce PDE constraints exactly, resulting in faster and more accurate solutions compared to physics-informed neural networks (PINNs).
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.