In this work, we follow the data-driven (DD) approach [1] as an alternative to material modeling, and develop a DD finite element framework for nonlinear diffusion problems, such as heat transfer in porous media. In particular, we consider nuclear graphite which has nonlinear dependence of the heat flux on temperature and its gradient resulting from the irradiated graphite microstructure. DD approach allows us to avoid empirical material models by using the experimental material data directly in the numerical simulations [2]. Furthermore, for a dataset with imperfections (e.g. with noise or missing data) we can estimate the uncertainty of the obtained solution at every integration point. We use the finite element method applied to a weaker mixed formulation [3] to satisfy the conservation laws and boundary conditions. In particular, the temperature is approximated in the L2 space while the heat flux is in the H(div) (Raviart-Thomas) space. Such a formulation enforces normal flux continuity across inner boundaries and provides a posteriori error estimates that enable adaptive refinement. As a result, the number of unknowns required to achieve the desired accuracy is reduced, and, accordingly, the number of searches through the material dataset is minimised. The framework has been implemented in the open-source finite element software MoFEM [4]. REFERENCES [1] T. Kirchdoerfer, M. Ortiz (2016): Data-driven computational mechanics. Computer Methods in Applied Mechanics and Engineering 304, 81-101. [2] A. Kuliková, A. G. Shvarts, L. Kaczmarczyk, and C. J. Pearce (2021): Data-driven finite element method. Proceedings of UKACM 2021 conference, doi: 10.17028/rd.lboro.14588577.v1. [3] D. Boffi, F. Brezzi, M. Fortin et al. (2013): Mixed finite element methods and applications. Springer, vol. 44. [4] L. Kaczmarczyk et al. (2020): MoFEM: An open source, parallel finite element library. Journal of Open Source Software, 5(45), 1441.