Strong chromospheric absorption lines such as H
α 6562.8 A and Ca II 8542.1 A provide vital diagnostics of plasma dynamics and thermal structure in the solar chromosphere. Multilayer spectral inversion (MLSI) offers a physically interpretable framework for modeling these lines using a finite number of radiative-transfer layers, but conventional MLSI relies on pixel-by-pixel nonlinear least-squares fitting, making it computationally expensive for large imaging spectroscopic data sets. Here, we introduce a physics-informed neural-network (PINN) framework to accelerate MLSI while preserving its analytic radiative-transfer formulation. The network predicts MLSI parameters directly from observed line profiles and passes them through a differentiable MLSI forward model to synthesize spectra. Training follows a two-stage approach: an initial stage optimized solely via spectral reconstruction loss, followed by fine-tuning that combines spectral consistency with parameter-space supervision from conventional MLSI results on a single reference image. This strategy eliminates the need for large precomputed training sets while maintaining physical interpretability. Applied to Fast Imaging Solar Spectrograph (FISS) observations from the Goode Solar Telescope (GST) targeting both quiet-Sun and active-region regions, MLSI-PINN parameter maps reproduce the primary spatial structures of direct inversions, achieving an arithmetic mean pixel-wise Pearson correlation coefficient of 0.933 across all evaluated parameters. The reconstructed spectra closely match both observed profiles and conventional MLSI fits. Post-training, MLSI-PINN processes a raster in approximately 5-15 seconds compared to 3-5 minutes for conventional MLSI, delivering an inference speedup of about 12-60 times without substantial loss in reconstruction quality, enabling efficient MLSI analysis on large chromospheric data sets.