Uncertainty Quantification at the Microscale: A Data-Driven Multi-Scale Approach

Data-driven formulations are currently developed to deal with the complexity of the multiphysics governing the response of microelectromechanical systems (MEMS) to external stimuli and can be extremely helpful. Such devices are in fact characterized by a hierarchy of length and timescales, which are difficult to fully account for in a purely model-based approach. In this work, we specifically refer to a (single-axis) Lorentz force micro-magnetometer designed for navigation purposes. Due to an alternating current flowing in a slender mechanical part (beam) and featuring an ad hoc set frequency, the microsystem is driven into resonance so that its sensitivity to the magnetic field is improved. A reduced-order physical model was formerly developed for the aforementioned movable part of the device; this model was then used to feed and speed up a multi-physics and multi-objective topology optimization procedure, aiming to design a robust and performing magnetometer. The stochastic effects, which are responsible for the scattering in the experimental data at the microscale, were not accounted for in such a model-based approach. A recently proposed formulation is here discussed and further extended to allow for such stochastic effects. The proposed multi-scale deep learning approach features: at the material scale, a convolutional neural network adopted to learn the scattering in the mechanical properties of polysilicon, induced by its morphology; and, at the device scale, two feedforward neural networks, one adopted to upscale the mechanical properties, while the other learns a microstructure-informed mapping between the geometric imperfections induced by the microfabrication process and the effective response of the movable part of the magnetometer. The data-driven models are linked through the physical model to provide a kind of hybrid solution to the problem. Results relevant to different neural network architectures are here discussed, along with a proposal to frame the approach as a multi-fidelity, uncertainty quantification procedure.