Speaker
Description
A wide variety of complex phenomena in engineering and science require the solution of expensive, high-dimensional systems of partial differential equations (PDEs). In order to overcome the computational burden and accelerate the calculations, reduced order models (ROMs) have been developed. Non-intrusive methods have been shown to be effective in scenarios involving experimental measurements or under constrained access to full-order solvers. However, these approaches often lack interpretability, physicality, and uncertainty quantification (UQ) of the predicted solutions. To resolve this compromise, we present a data-driven, non-intrusive, reduced-order modelling scheme that follows a generative process to compute new solutions while maintaining physical consistency. In particular, a variational autoencoder is used for dimensionality reduction and a variational adaptation of sparse identification of nonlinear dynamics (SINDy) is developed to proficiently construct reduced-order models (ROMs) from limited amount of high-dimensional noisy data. This identifies latent dynamics in an interpretable manner while inherently incorporating UQ. The proposed method is composed of three building blocks: Variational Encoding of Noisy Inputs (VENI): A generative model utilizing variational autoencoders (VAEs) is applied to transform high-dimensional, noisy data into a low-dimensional latent space representation that is suitable to describe the dynamics of the system. Variational Identification of Nonlinear Dynamics (VINDy): On the time series data expressed in the new set of latent coordinates, a probabilistic dynamical model is learned by a variational version of SINDy. In this process, the distributions of the coefficients that determine the contribution of terms from a predefined set of candidate functions are identified. Variational Inference with Certainty Intervals (VICI): Variational inference is used to construct physically consistent ROMs for new parameter instances and initial conditions in a generative manner. The probabilistic framework inherently facilitates the quantification of uncertainty in the model coefficients, while providing certainty intervals for temporal solutions. The performance of the proposed method is validated on a diverse set of partial differential equation benchmarks including structural mechanics and fluid dynamics examples.
Reference
P. Conti, J. Kneifl, A. Manzoni, A. Frangi, J. Fehr, S. L. Brunton, and J. N. Kutz. VENI, VINDy, VICI: a variational reduced-order modeling framework with uncertainty quantification, 6 2024.
