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Approximation frameworks for phase-field models of brittle fracture are presented and compared in this work. Such methods aim to address the computational cost associated with conducting full-scale simulations of brittle fracture in heterogeneous materials where material parameters, such as fracture toughness, can vary spatially. They proceed by combining a dimension reduction with learning between function spaces. Two classes of approximations are considered. In the first class, deep learning models are used to perform regression in ad hoc latent spaces. PCA-Net and Fourier neural operators are specifically presented for the sake of comparison. In the second class of techniques, statistical sampling is used to approximate the forward map in latent space, using conditioning. To ensure proper measure concentration, a reduced-order Hamiltonian Monte Carlo technique (namely, probabilistic learning on manifold) is employed. The accuracy of these methods is then investigated on a proxy application where the fracture toughness is modeled as a non-Gaussian random field. It is shown that the probabilistic framework achieves comparable performance in the 𝐿2 sense while enabling the end-user to bypass the art of defining and training deep learning models.