VENI, VINDy, VICI: a generative reduced-order modeling framework with uncertainty quantification

TL;DR

This paper introduces VENI, VINDy, VICI, a data-driven reduced-order modeling framework that combines variational autoencoders and a novel sparse identification method to create interpretable, reliable models with uncertainty quantification for complex PDE systems.

Contribution

The work presents a new probabilistic, non-intrusive ROM framework that integrates variational autoencoders with a variational SINDy for interpretable dynamics and uncertainty quantification.

Findings

Effective identification of dynamical systems with noisy data

Accurate reduced-order models for PDE benchmarks

Uncertainty quantification with Certainty Intervals

Abstract

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 governing equations are unknown or partially known, typically ROMs lack interpretability and reliability of the predicted solutions. In this work we present a data-driven, non-intrusive framework for building ROMs where the latent variables and dynamics are identified in an interpretable manner and uncertainty is quantified. 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 newly introduced, variational version of sparse identification of nonlinear dynamics (SINDy), which we refer to as Variational Identification of Nonlinear Dynamics (VINDy). 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 VINDy. Once trained offline, the identified model can be queried for new parameter instances and new initial conditions to compute the corresponding full-time solutions. The probabilistic setup enables uncertainty quantification as the online testing consists of Variational Inference naturally providing Certainty Intervals (VICI). In this work we showcase the effectiveness of the newly proposed VINDy method in identifying interpretable and accurate dynamical system for the Roessler system with different noise intensities and sources. Then the performance of the overall method - named VENI, VINDy, VICI - is tested on PDE benchmarks including structural mechanics and fluid dynamics.