A Physics-Informed Neural Network Atlas: Sub-Percent on Smooth Forward PDEs, Orders Worse on Inverse, High-Frequency and Real Data

Abstract

Physics-informed neural networks (PINNs) are promoted as a general differential-equation solver, but the accuracy actually achieved varies by orders of magnitude across problem types, and that variation is rarely laid out in one place. This report is a method atlas: a runnable catalogue in which each of the state-of-the-art PINN families (hard constraints, residual-based adaptive sampling, Fourier features, domain decomposition, operator learning, physics-informed operators, conformal-prediction uncertainty, structure-preserving Hamiltonian learning, and inverse solving) is exercised on at least one differential-equation case, trained by a dedicated engine, validated against an analytic or numerical reference, and scored by relative L2 error. Across the seventeen cases with an archived reference-validated error, the result is a clean stratification. PINNs reach sub-percent accuracy on smooth, well-posed forward PDEs (a one-dimensional wave at 5e-5, a reaction-diffusion Allen-Cahn field at 1.2e-3, several mining and transport reduced models below 1e-3), with ten of seventeen cases under 1% relative error. But they degrade by two to three orders of magnitude on the hard classes: inverse problems (4e-2), high-wavenumber Helmholtz (1.0e-1), the lid-driven cavity Navier-Stokes benchmark (1.7e-1), operator learning (5.6e-2), and real noisy field data (6.9e-2). The finding matches the field's own failure-mode literature: PINNs are an accurate and flexible instrument where the solution is smooth and the problem well-posed, and they are not a drop-in replacement for a good classical solver where it is not. Every case ships its equations, its method, a synthetic provenance label, and a reproducible error; every number and figure regenerates from the archived artifacts. Runnable catalogue and reproducible pipeline (MIT): https://github.com/fsantibanezleal/CAOS_PINNLAB ; catalogue: https://pinnlab.fasl-work.com .