VENI, VINDy, VICI – a variational reduced-order modeling framework with uncertainty quantification

The paper proposes VENI–VINDy–VICI: a VAE-based encoder/decoder for snapshots (VENI), a variational SINDy latent-dynamics identification (VINDy) with Laplace priors for sparsity and a Gaussian likelihood on ż, plus an online sampling procedure (VICI) to integrate latent ODEs and decode trajectories with uncertainty bounds. These are clearly described, including Gaussian q(z|x), Laplace priors for Ξ, a closed-form Laplace–Laplace KL, joint offline training, zero-pdf thresholding, and the online Monte Carlo sampling/integration/decoding pipeline (e.g., VENI/VINDy/VICI overview and problem setup; Gaussian encoder; VINDy model ż|Ξ,z ∼ N(Θ(z,β)Ξ, σ^2I) with Laplacian priors and KL in closed form; joint training sketch; zero-pdf thresholding; VICI sampling and integration) . The candidate solution matches the architecture and key ingredients: VAE ELBO on snapshots, a variational latent-dynamics term with Laplace prior on Ξ and closed-form Laplace–Laplace KL, zero-pdf thresholding, and online Monte Carlo prediction/uncertainty. Its main difference is modeling a discrete-time latent transition p(z_{t+1}|z_t,Ξ) via an Euler step, whereas the paper models a continuous-time derivative likelihood p(ż|z,Ξ); these are consistent discretization variants. The candidate also omits the extra KL(q(z)||p(z)) term that appears when VENI and VINDy are combined (giving a factor of 2 on the KL in the paper’s sketch), and it does not include the optional full-dynamics regularizer the paper uses to stabilize scaling; however, those are weighting/regularization choices rather than conceptual contradictions. Overall, both present a consistent variational ROM with sparsity and UQ; the candidate’s is a discrete-time variant of the paper’s continuous-time formulation.