LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries

A new method called LieSolver promises to solve initial-boundary value problems more efficiently by embedding physical laws directly into the model, according to a preprint posted on arXiv [1]. The approach uses Lie symmetries to enforce partial differential equations exactly, learning solutions solely from initial and boundary data [2]. Initial-boundary value problems, or IBVPs, are a standard framework for modeling phenomena across physics and engineering [2]. The preprint, authored by Ivan Timofeev and submitted to the machine learning section of the open-access repository, describes a technique that leverages symmetry transformations to build the governing PDE into the model's architecture [1][2]. Because the physical law is satisfied by construction, the model's boundary loss provides a direct measure of domain-wide error, enabling rigorous error estimates for well-posed problems [2]. The paper reports that LieSolver outperforms physics-informed neural networks, known as PINNs, in both speed and accuracy while producing more compact models [1][2]. The work focuses on linear homogeneous PDEs, and the authors state the approach significantly improves the efficiency and reliability of predictions for PDE-constrained problems [2]. The preprint appeared on arXiv, a repository that hosts electronic preprints across mathematics, physics, computer science, and related fields, and which does not conduct peer review before posting [6]. As of late 2024, the site was receiving roughly 24,000 new articles per month [6]. The LieSolver manuscript was first submitted in October 2025 and revised in June 2026 [1].