Bayesian operator learning for nonlocal damage mechanics and internal length scale identification

Abstract

We develop a Bayesian operator-learning framework for nonlocal damage mechanics aimed at calibrating uncertain internal length scales and quantifying predictive uncertainty from limited observations. The deterministic forward model is based on an AT2-type gradient damage regularization, which introduces a finite localization width and provides a mesh-objective continuum description of strain-softening behavior. The resulting boundary-value problem is interpreted as a parametric solution map from loading and material parameters to displacement and damage fields in Sobolev spaces. To accelerate the repeated forward evaluations required in Bayesian inversion, we introduce a learned surrogate approximation of this nonlinear operator. The present numerical implementation employs a computationally efficient convolutional surrogate with Monte Carlo-dropout uncertainty quantification, while the overall framework is formulated in an architecture-agnostic manner and can be extended to richer neural-operator constructions in future work. A principal contribution of the paper concerns parameter identifiability in the pre-critical regime. We show that when only global traction–displacement observations are available, the Fisher information associated with the internal length scale may become small. Consequently, the inverse problem can be practically nonidentifiable: materially distinct parameter values may generate nearly indistinguishable global responses, yielding diffuse or biased posterior distributions despite apparently accurate curve fits. Numerical experiments on a 1D benchmark validate this mechanism. The posterior distribution of the internal length scale is shown to remain weakly informative under monotone pre-critical loading, even though the global traction–displacement response is predicted accurately. In addition, field-level predictive uncertainty is observed to concentrate near mechanically active damage zones, highlighting spatial regions where additional measurements would be most informative. The proposed framework provides a mathematically consistent and computationally efficient route toward probabilistic calibration of nonlocal damage models. The most strongly validated conclusion of the present study is that accurate global response prediction does not necessarily imply reliable identification of internal length scales. Extensions to post-peak snap-back regimes, instability-enhanced identifiability, and multidimensional operator-learning architectures are left for future work.