Physics-Constrained Learning of Dose-Dependent Spectral Degradation in Metal--Organic Frameworks from In Situ Low-Loss EELS

Electron-beam irradiation limits atomic-resolution characterization of beam-sensitive hybrid materials, yet quantitative models that connect $\mathit{\text{in}}\mathit{\text{situ}}$ spectroscopy to dose-dependent degradation remain scarce. Here we use a physics-informed neural network (PINN) to model beam-induced spectral evolution in MIL-101(Fe) from an in situ low-loss electron energy-loss spectroscopy (EELS) dose series. Each spectrum is reduced to fixed-window low-loss descriptors, ${\tilde{n}}_{\mathrm{e}\mathrm{f}\mathrm{f},j}(\mathrm{\Phi })={\int }_{{\mathcal{W}}_j}S(E,\mathrm{\Phi })dE$, evaluated over nominal $\pi$–${\pi }^{\ast }$, C–C, C–O, and M–O windows. These descriptors are relative window-integrated low-loss spectral areas, not absolute f-sum-rule effective electron numbers. For each spectral channel, a latent integrity variable $C_i(\mathrm{\Phi })$ obeys the same uncoupled power-law degradation equation in normalized dose space, $d{C}_i\mathrm{/}d\phi =-k_i{C}_i^{p_i}$, regularized by monotonicity, boundedness, and a single hierarchy prior $k_{\mathrm{C}\text{-}\mathrm{O}}\ge k_{\mathrm{C}\text{-}\mathrm{C}}$. Applied to nine dose frames spanning 152–1368~e${}^{-}$/Å${}^2$, the ensemble PINN identifies C–O and C–C as the most strongly dose-sensitive linker-associated channels, with half-integrity thresholds of approximately $1.0\times {10}^3$~e${}^{-}$/Å${}^2$. The 1–3~eV $\pi$–${\pi }^{\ast }$-labelled window increases with dose and is therefore interpreted as a mixed low-energy response, likely involving oscillator-strength redistribution rather than direct monotonic loss of a single bond population. The framework provides a dose-dependent, spectroscopy constrained description of MOF degradation while also defining the limits of what fixed-window low-loss EELS can assign without independent chemical-state validation.