Scientific Positioning

PDELie is positioned as a small public research library for empirical Lie-symmetry diagnostics on controlled PDE time-series workflows. Its reports combine residual checks, fitted or supplied generator evidence, configured validation, finite-transform verification, readiness diagnostics, and provenance summaries. These reports are evidence records, not mathematical proofs of symmetry and not a broad data-adapter or benchmark framework.

The Lie-symmetry vocabulary follows the classical differential-equation literature: Olver’s Applications of Lie Groups to Differential Equations, Hydon’s Symmetry Methods for Differential Equations, and Bluman and Kumei’s Symmetries and Differential Equations. PDELie currently uses this theory in a narrow computational setting: scalar 1D uniform periodic fields, polynomial generator families, residual-relative validation, and explicit finite-transform diagnostics.

The numerical differentiation choices are closer to standard spectral-method practice, with Trefethen’s Spectral Methods in MATLAB as the relevant reference point for periodic spectral differentiation. PDELie does not claim general-purpose numerical PDE solving, nonuniform-grid support, or weak-form derivative backends beyond the frozen weak-report slices documented in the API stability policy.

For sparse discovery context, PDELie is adjacent to work such as Brunton, Proctor, and Kutz on sparse identification of nonlinear dynamical systems, and Rudy et al. on data-driven discovery of PDEs. PDELie’s downstream utilities summarize bridge arrays, backend-native results, recovery evidence, and provenance; they do not define a benchmark policy or implement a broad discovery-backend framework.

For symmetry-based augmentation context, PDELie is closer in spirit to the evidence discipline needed before LPSDA-style workflows such as Brandstetter et al.’s Lie point symmetry data augmentation for neural PDE solvers. PDELie does not learn symmetry groups, construct general augmentation orbits, or implement finite multi-generator group actions. It helps users record which empirical Lie-symmetry diagnostics were configured, what passed, what failed, and what remains outside the stable scope.