  Numerical modeling of localizations is a challenging task due to the evolving
rough solution in which the localization paths are not predefined. Despite
decades of efforts, there is a need for innovative discretization-independent
computational methods to predict the evolution of localizations. In this work,
an improved version of the neural network-enhanced Reproducing Kernel Particle
Method (NN-RKPM) is proposed for modeling brittle fracture. In the proposed
method, a background reproducing kernel (RK) approximation defined on a coarse
and uniform discretization is enriched by a neural network (NN) approximation
under a Partition of Unity framework. In the NN approximation, the deep neural
network automatically locates and inserts regularized discontinuities in the
function space. The NN-based enrichment functions are then patched together
with RK approximation functions using RK as a Partition of Unity patching
function. The optimum NN parameters defining the location, orientation, and
displacement distribution across location together with RK approximation
coefficients are obtained via the energy-based loss function minimization. To
regularize the NN-RK approximation, a constraint on the spatial gradient of the
parametric coordinates is imposed in the loss function. Analysis of the
convergence properties shows that the solution convergence of the proposed
method is guaranteed. The effectiveness of the proposed method is demonstrated
by a series of numerical examples involving damage propagation and branching.

Computational Engineering, Finance, and Science;Machine Learning;Numerical Analysis;Numerical Analysis