Veni, VINDy, Vici: a variational method to build ROMs with embedded uncertainty quantification
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The simulation of many complex phenomena in engineering and science requires solving expensive, high-dimensional systems of partial differential equations (PDEs). To circumvent this, reduced-order models (ROMs) have been developed to speed up computations. However, when experimental measurements are considered, or access to full-order solvers is restricted, non-intrusive techniques for reduced-order modeling lack interpretability and uncertainty quantification of the predicted solutions. In this work we present a data-driven, non-intrusive framework for building ROMs where the latent dynamics are identified in an interpretable manner. Starting from a limited amount of high-dimensional, noisy data the proposed framework constructs an efficient ROM by leveraging variational autoencoders for dimensionality reduction along with a variational version of sparse identification of nonlineary dynamics (SINDy). In detail, the method consists of Variational Encoding of Noisy Inputs (VENI) to identify the distribution of reduced coordinates. Simultaneously, we learn the distribution of the coefficients of a pre-determined set of candidate functions by Variational Identification of Nonlinear Dynamics (VINDy). Once trained offline, the identified model can be queried for new parameter instances and/or new initial conditions to compute the corresponding full-time solutions. The probabilistic setup naturally enables uncertainty quantification as the online testing consists of Variational Inference naturally providing Certainty Intervals (VICI). The performance of the proposed method is tested on a wide range of PDE benchmarks including structural mechanics and fluid dynamics.
