Stochastic Reduced Order Modeling

The selection of interatomic potentials governing atom interactions is a crucial ingredient in Molecular Dynamics (MD) simulations, often poisoned by uncertainties in the calibrated parameters and functional forms selected for the aforementioned potentials. In this context, performing uncertainty quantification with model-type uncertainties remains extremely challenging, due to the form of the equations to be solved.

In this project, we have proposed an approach where uncertainties are accounted for through a Stochastic Reduced Order Model (SROM). First, a classical Reduced Order Model (ROM) is constructed by the method of snapshots in the space of admissible configurations. Second, a SROM is introduced by defining a stochastic reduced order basis that properly fluctuates around the deterministic ROB. Hyperparameters can subsequently be identified by solving statistical inverse problems, based on some quantities of interest (which can be microscopic or macroscopic, and time-dependent).

To demonstrate the relevance of the framework, three applications on Graphene sheets were proposed. Specifically, it was shown that the SROM allows us to model uncertainties raised by the truncation of the ROB, and that it enables the treatment of uncertainties induced by interatomic potential selection. A few examples are shown below that qualitatively illustrate the performance of the SROM.

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Graphene sheet subjected to periodic loading: Vertical displacement of all atoms after 60,000 fs, predicted by different high-fidelity models, and associated 98% confidence interval estimated with the SROM (from 1,000 independent samples).

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Graphene sheet subjected to tension: Confidence region and evolution of the potential energy function in the armchair and zigzag directions (continuum-mechanics model, molecular dynamics results for all potentials and SROM predictions).