Temporal Spectral Analysis of Late-Time Error in a Physics-Informed Neural Network Solution of the One-Dimensional Advection–Diffusion Equation

Abstract

Persistent late-time variation can remain in physics-informed neural network (PINN) solutions after the governing transient has effectively decayed, while conventional error norms do not reveal whether this variation has a systematic temporal–frequency structure. This study develops an offline temporal–spectral diagnostic and postprocessing workflow for a one-dimensional advection–diffusion benchmark. A high-accuracy analytical reference and three fixed-resolution finite-difference baselines are used to assess a PINN whose architecture is selected by a fully supervised neural architecture search and whose parameters are trained with progressive temporal windowing. Candidate late-time intervals are selected without using the reference solution by applying the Bayesian information criterion (BIC) to a breakpoint model for the inter-reconstruction sensitivity; the selected field is subsequently reconstructed by retaining a prescribed fraction of its temporal spectral energy and is evaluated independently through reference-error and physics-consistency measures. For [tcut,tmax]=[1.8,5], the zero-frequency component contains 0.9999996 of the raw-field energy, so the q=0.95 reconstruction retains only the temporal mean. This projection reduces the final-time spatial error norm from 2.70×10−3 to 1.06×10−3, a factor of approximately 2.5, while changing the discrete governing-equation residual by less than 0.3% over the filtered window. Mean-removed tests for q=0.90,0.95,0.99 show that the discarded fluctuation is dominated by low-frequency approximation error rather than high-frequency noise. The result supports the proposed selection–validation workflow for this controlled benchmark but does not establish a universally transferable filter.