**7. Conclusions**

A stochastic data-driven multilevel finite-element (FE<sup>2</sup> ) method was proposed to solve nonlinear heterogeneous structures with uncertainties at both the micro- and the macrolevel. A hybrid neural-network–interpolation (NN–I) scheme was developed to improve the accuracy of NN surrogate models, allowing for the use of a lower number of representative volume element (RVE) nonlinear calculations, which serve as a database to train the neural networks. This NN–I surrogate model was used to develop a data-driven method for nonlinear heterogeneous conduction in a stochastic framework: uncertainties can be included on both the micro- and the macrolevel. More specifically, the drastic reduction in computational costs brought by the NN-I surrogate model allows Monte Carlo simulations of nonlinear heterogeneous structures. This framework was applied to propagate uncertainties in such nonlinear multiscale models, and demonstrated that it can be used to identify probabilistic models related to some quantities of interest at the macroscale in a fully nonlinear, anisotropic context.

**Author Contributions:** Conceptualization, J.Y and V.P.; software, X.L. and L.P.; formal analysis, J.Y. and V.P.; investigation, X.L., J.Y., V.P., L.P. and I.K; writing—original draft preparation, J.Y., X.L. and V.P.; writing—review and editing, X.L., J.Y., V.P., L.P. and I.K.; supervision, J.Y.; project administration, J.Y.; funding acquisition, X.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** Xiaoxin LU thanks the support from SIAT Innovation Program for Excellent Young Researchers 346 (E1G045).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author upon reasonable request.

**Acknowledgments:** Xiaoxin Lu thanks the SIAT Innovation Program for Excellent Young Researchers (E1G045) for the support.

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

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