VENI, VINDy, VICI: a variational reduced-order modeling framework with uncertainty quantification
🤖 AI Summary
Traditional reduced-order models (ROMs) suffer from limited interpretability and inadequate uncertainty quantification when applied to high-dimensional partial differential equations (PDEs), especially under expensive simulations, unknown or partially known governing equations. Method: This paper proposes a non-intrusive Bayesian ROM framework that jointly couples variational autoencoders (VAEs) with variational sparse identification of nonlinear dynamics (VINDy) to infer both the latent-space dynamics distribution and the sparse coefficient distribution—naturally yielding prediction-based uncertainty quantification via Variational Inference Confidence Intervals (VICI). The method operates on limited, noisy high-dimensional data without requiring prior knowledge of the original PDEs, integrating a pre-specified candidate function library with probabilistic modeling. Results: Evaluated on the Rössler system and benchmark PDEs from structural mechanics and fluid dynamics, the framework demonstrates significantly improved noise robustness, enhanced accuracy in dynamical system identification, and superior predictive reliability.
Application Category
📝 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 R""ossler 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.
Problem
Research questions and friction points this paper is trying to address.
Developing interpretable reduced-order models for complex PDE systems
Quantifying uncertainty in predictions from limited noisy data
Identifying unknown dynamics when governing equations are incomplete
Innovation
Methods, ideas, or system contributions that make the work stand out.
Variational autoencoders reduce dimensionality from noisy data
Variational sparse identification learns interpretable dynamical systems
Probabilistic framework enables uncertainty quantification through variational inference
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