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

TL;DR

The paper introduces VENI, VINDy, and VICI, a probabilistic, non-intrusive reduced-order modeling framework that learns latent coordinates and dynamics from noisy data while providing uncertainty quantification. It integrates a variational encoder (VENI) for latent state distributions, a variational sparse-dynamics learner (VINDy) for latent dynamics, and online credibility intervals (VICI) to propagate uncertainty to full-field predictions. The authors demonstrate accurate state reconstruction, interpretable latent dynamics, and robust UQ on the Rössler system, a reaction–diffusion PDE, and a MEMS resonator, including extrapolation to unseen parameters. The framework offers scalable, interpretable ROMs with uncertainty-aware predictions and potential for online updating and validation on real-world data, highlighting its practical relevance for PDE-guided engineering and science.

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.

Figures (5)

• Figure 1: Overview of the VENI, VINDy, VICI procedure. High-dimensional, noisy data are mapped through a variational encoder to low-dimensional, latent random variables (VENI). Simultaneously, in a joint offline-training, the dynamics of the latent variables are learned by VINDy. Once this offline phase is concluded, noise-free, full-field solutions are computed through the online generative process together with the predictions uncertainty bounds (VICI).
• Figure 2: Schematic representation of the online VICI procedure to generate new solutions for a given initial condition $\bm{x}_0$ and set of parameters $\bm{\beta}$. We first sample multiple instances of the corresponding latent initial conditions and coefficients of the dynamical model, each defining an ODE system. Then each dynamical system is integrated in time through standard time-stepping schemes, resulting in multiple latent trajectories. These latter are finally processed by the decoder mean to obtain full state trajectories. Predictions and the corresponding UQ are computed directly from the statistical properties of the approximated solution trajectories.
• Figure 3: Approximation results for the Rössler system. Different types of noisy data (each represented by a different color) a) are generated by the Rössler system, b) serve as basis for identifying the distributions of coefficients, and c) are compared against the true values (indicated by triangles at the bottom of each axis). Coefficients appearing in the original equations are emphasized with bold red axes, while the PDF threshold is marked by a dashed line. The resulting temporal evolutions of the identified systems, including UQ, are shown in d) for an initial condition (IC) within the training regime and in e) for an IC outside the training domain (out-of-distribution inference).
• Figure 4: Results of the offline training a), consisting of the VENI and VINDy steps, for the reaction-diffusion problem. In the VINDy box, the identified posterior distributions of the coefficients are displayed. We note that the mixed linear terms, which are the dominant terms for the observed dynamics, are accurately identified as significant. In the VENI box, we highlight how the method is capable of learning an encoding function from noisy data, resulting in two latent variables that exhibit the expected oscillatory behavior. A comparison of solution fields between the approximation and the numerical reference solution is given in b) for a training $t=15$ and extrapolated time instances $t \in \{20,35\}$, while forward uncertainty quantification at the different reduction levels, performed with the VICI procedure.
• Figure 5: Schematic representation of the beam MEMS resonator a) with the mesh used in the FOM simulations. The physical reconstruction of latent mean trajectories and corresponding uncertainties for a node in the middle of the beam where the most dynamics occur are given in b). For training, only the first $T_\text{train}=1091\,\mu$s are used; the remaining time for which the model extrapolates is indicated with a red background. The identified coefficient distribution c) show that the terms appearing in the normal form of the observed high-dimensional system (framed in bold red, correct values that result from the natural frequency and the damping of the system are marked with a black triangle at the bottom of the according axis, the PDF threshold value is indicated as dashed line) are correctly identified by our method.