*Article* **A Stochastic FE<sup>2</sup> Data-Driven Method for Nonlinear Multiscale Modeling**

**Xiaoxin Lu <sup>1</sup> , Julien Yvonnet 2,\*, Leonidas Papadopoulos <sup>3</sup> , Ioannis Kalogeris <sup>3</sup> and Vissarion Papadopoulos <sup>3</sup>**

	- lew.papado@hotmail.com (L.P.); yianniskalogeris@gmail.com (I.K.); vissarion.papadopoulos@gmail.com (V.P.)

**Abstract:** A stochastic data-driven multilevel finite-element (FE<sup>2</sup> ) method is introduced for random nonlinear multiscale calculations. A hybrid neural-network–interpolation (NN–I) scheme is proposed to construct a surrogate model of the macroscopic nonlinear constitutive law from representativevolume-element calculations, whose results are used as input data. Then, a FE<sup>2</sup> method replacing the nonlinear multiscale calculations by the NN–I is developed. The NN–I scheme improved the accuracy of the neural-network surrogate model when insufficient data were available. Due to the achieved reduction in computational time, which was several orders of magnitude less than that to direct FE<sup>2</sup> , the use of such a machine-learning method is demonstrated for performing Monte Carlo simulations in nonlinear heterogeneous structures and propagating uncertainties in this context, and the identification of probabilistic models at the macroscale on some quantities of interest. Applications to nonlinear electric conduction in graphene–polymer composites are presented.

**Keywords:** data-driven; multiscale; nonlinear; stochastics; neural networks
