**4. Conclusions**

The present study proves that effective properties can be associated with microstructures with complex morphological and topological features. For this purpose, those features are extracted by using TDA, post-compressed by using linear dimensionality reduction (PCA) which output represents the parameters employed by the nonlinear *Code2Vect* regression that finally assign a effective property (here the effective thermal conductivity) to a given microstructure.

The procedure demonstrated its robustness and performance in the low-data limit, as well as its capacity to provide better predictions when considering larger training sets. It successfully combines physics-based data for learning purposes, with almost real-time inference based on the topological analysis of images.

**Author Contributions:** Conceptualization, software and validation, M.Y. and C.A.; methodology, J.L.D., E.C. and F.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors acknowledge the ESI Chairs at Arts et Métiers Institute of Technology and the University of Zaragoza, as well as the French ANR through the DataBEST project.

**Conflicts of Interest:** The authors declare no conflict of interest.
