Inferring unknown initial states in shock-dominated compressible flows from sparse and noisy measurements is a challenging ill-posed inverse problem due to nonlinear wave interactions and limited sensing. In this work, we develop a non-intrusive reduced-order modeling framework for efficient Bayesian initial-state inversion with uncertainty quantification. The framework combines a convolutional autoencoder with a learned latent-space forward operator. The autoencoder compresses high-dimensional flow fields into a compact nonlinear latent representation, while the forward operator predicts final-time latent states from encoded initial conditions. This AE-ROM surrogate enables rapid forward evaluations and is embedded within a No-U-Turn Sampler (NUTS) for posterior exploration. The framework is demonstrated using 500 high-fidelity Sod shock tube simulations generated through Latin hypercube sampling and solved using a fifth-order WENO scheme. The inverse problem seeks to recover unknown left and right density and pressure states from sparse noisy observations of final-time density and pressure fields. Results show that the AE-ROM accurately reconstructs key shock-tube structures, including the rarefaction wave, contact discontinuity, and shock front. A latent dimension of 32 provides an effective balance between reconstruction accuracy and reduced-space compactness, while 250 training simulations are sufficient for accurate reconstruction. Increasing observation density significantly contracts posterior uncertainty, reducing the mean posterior standard deviation by approximately 78% for density and 76% for pressure. Overall, the proposed framework provides a computationally efficient and uncertainty-aware approach for inverse analysis of shock-dominated flows, with potential extensions to multidimensional compressible-flow and digital-twin applications.
Bayesian inverse design provides a principled framework for inferring aerodynamic geometries from sparse flow observations while quantifying uncertainty. However, its practical use in computational fluid dynamics (CFD) is severely limited by the cost of repeated high-fidelity simulations required for gradient-based Markov chain Monte Carlo (MCMC) sampling. While surrogate models are commonly proposed to reduce this cost, their effect on posterior geometry and uncertainty, especially for shock-dominated flows, remains poorly understood. In this work, we demonstrate that neural operator surrogates can be embedded directly within the MCMC inference loop while preserving posterior structure. Using a fully Bayesian inverse formulation of quasi-one-dimensional nozzle flow, we demonstrate that geometry parameterization plays a decisive role in identifiability and posterior conditioning, with cubic B-splines yielding stable and physically meaningful uncertainty estimates. Building on this formulation, a Deep Operator Network trained on CFD-generated data is substituted for the CFD solver within a No-U-Turn Sampler, while keeping the likelihood model, priors, and sampling configuration unchanged. Across sparse to fully observed regimes, surrogate-based inference reproduces the posterior geometry and uncertainty trends of the CFD reference. As a result of surrogate integration, total inference time is reduced to under one second, corresponding to a speedup exceeding three orders of magnitude. In addition, a direct inverse neural operator is examined as a deterministic alternative for inverse design, enabling single-shot geometry reconstruction without posterior sampling. These results demonstrate that neural operator-accelerated Bayesian inference enables practical, uncertainty-aware inverse design workflows for aerodynamic applications.
We propose latent-space diffusion posterior sampling (L-DPS), an approximate Bayesian framework for high-dimensional inverse problems governed by partial differential equations (PDEs). The method addresses three challenges in PDE-constrained inversion: implicit sample-based priors without tractable densities, high-dimensional spatially distributed parameters, and the high cost of repeated forward-model evaluations during posterior sampling. L-DPS combines a variational autoencoder, an unconditional latent diffusion model, diffusion posterior sampling, and a differentiable neural surrogate. The VAE maps the parameter field to a lower-dimensional latent space, the diffusion model learns an implicit prior score in this latent space, and DPS combines this learned prior with likelihood-based guidance. The likelihood gradient is evaluated through the decoder-surrogate composition, avoiding repeated calls to the full numerical PDE solver. We evaluate the method on an inverse Darcy flow problem with an unknown spatially distributed permeability field inferred from sparse and noisy pressure observations. L-DPS produces accurate and robust inverse solutions, reduces inference cost relative to full-space DPS, and outperforms amortized inverse baselines such as conditional latent diffusion and inverse FNO in sparse and noisy regimes. We further compare L-DPS with a KLE-MAP baseline and study mixed-prior generalization and the sensitivity of inversion accuracy to surrogate forward-model error.
The ensemble Kalman filter (EnKF) is widely adopted for sequential data assimilation, but fails for solutions with discontinuities, such as shocks in compressible flows. Uncertainty in shock location induces multimodal ensemble statistics that violate the Gaussian assumptions underlying the EnKF, producing large-scale spurious oscillations in the analysis state. We introduce a feature-preserving latent-EnKF that performs the ensemble update in a learned low-dimensional latent space, where shock and flow features admit a smooth manifold representation, thereby preserving sharp features during EnKF analysis. The updated latent state is mapped back to physical state through a shared decoder for all ensemble members. The algorithm eliminates the member-specific ordered training and positivity flooring used in prior approaches. Numerical experiments on a Sod shock tube and Mach 2 shock interaction with a 2D cylinder, using sparse and noisy observations, show accurate feature recovery of shocks and contact discontinuities without spurious oscillations.