Machine Learning (ML) is increasingly used to accelerate modelling and design in computational mechanics. This contribution in particular delves into improved accuracy of these aspects through the development of equivariant and flow-matching machine learning tools with incorporated physical constraints within the context of mechanical metamaterials operating in highly non-linear regimes including bifurcations and instabilities. A similarity-equivariant graph neural network for homogenization of metamaterials' mechanical response will be presented first. The architecture embeds translational, rotational, reflectional, and scaling equivariance directly into the network architecture, ensuring physically consistent predictions of scalars, as well as second- and fourth-order tensorial quantities. Compared to conventional ML-based surrogates, our architecture achieves improved data efficiency, accuracy, and robustness [1]. The equivariant formulation is next extended to account for bifurcations and multistability. Using flow-matching learning, the proposed method predicts distributions over multiple solution trajectories while respecting the underlying equivariances. Such a probabilistic formulation avoids averaging across trajectories during training, improves accuracy, and has the potential to faithfully predict behaviour of bifurcating metamaterials [2]. Integrating the above aspects, elements of a generative inverse design towards new metamaterial geometries will finally be demonstrated next. A large-scale openly available dataset for training will be presented, of two-dimensional architected materials organized by wallpaper symmetry groups [3]. The dataset systematically captures microstructural geometry, non-linear mechanical responses, and pattern-transforming behaviour, providing a benchmark for data-driven modelling and a foundation for learning structure-property relationships. By explicitly encoding symmetry classes, the dataset improves consistent training and evaluation of machine learning models. Relying on flow-matching learning, discussion of generative design against target effective properties trained on the bespoke dataset will conclude this talk. References [1] Hendriks F., et al., 10.1016/j.cma.2025.117867. [2] Hendriks F., et al., arXiv:2509.03340v2. [3] Hendriks F., et al., 10.1038/s41597-025-06150-x.