The impact of observation density on Bayesian inversion of latent dynamics in shock-dominated flows

Abstract: 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.

• The paper develops an autoencoder-based reduced-order surrogate that replaces repeated CFD solves during NUTS sampling, using a 32-dimensional latent space and 250 high-fidelity Sod shock-tube simulations.
• The paper finds that increasing observations from 5 to 100 reduces posterior mean RMSE by approximately 11–12% but contracts mean posterior standard deviations by roughly 76–78%, making uncertainty reduction the primary benefit of denser sensing.
• The paper shows that the surrogate recovers rarefaction waves, contact discontinuities, and shocks with errors concentrated near discontinuities, while highlighting the need for discrepancy-aware likelihoods and validation in multidimensional, time-dependent flows.

Problem setting and motivation

The paper addresses the inverse problem of recovering unknown initial states of a shock-dominated compressible flow from sparse, noisy measurements of the final-time flow field. This task is ill-posed under limited sensing and nonlinear wave interactions, and it is computationally prohibitive when the forward model is a high-fidelity CFD solver, since Bayesian posterior exploration requires thousands to millions of forward evaluations. The authors' response is a non-intrusive parametric reduced-order modeling (ROM) framework: a convolutional autoencoder compresses high-dimensional flow fields into a compact latent space, and a learned latent-space forward operator maps encoded initial conditions to final-time latent states. The resulting surrogate (AE-ROM) is embedded directly within a No-U-Turn Sampler (NUTS) for posterior exploration with uncertainty quantification (2605.19076).

The choice of the one-dimensional Sod shock tube as the benchmark is deliberate: it contains the canonical wave structures of interest — a rarefaction wave, contact discontinuity, and shock front — while remaining tractable enough for controlled parametric studies. The broader motivation is digital-twin applications in high-speed aerodynamics, where near-real-time inversion from operational data requires fast, uncertainty-aware surrogates.

Data generation and surrogate architecture

A dataset of 500 high-fidelity Sod shock tube simulations was generated via Latin hypercube sampling over four parameters: left-state density and pressure ($0.5 \le \rho_L, p_L \le 1.5$) and right-state density and pressure ($0.05 \le \rho_R, p_R \le 0.15$), with zero initial velocity throughout. Each case was solved on 1000 uniform grid points using a fifth-order WENO reconstruction, an HLLC Riemann solver, and third-order Runge–Kutta time integration to $t = 0.2$ s.

Each flow state is a tensor of shape $(3, 1000)$ comprising density, velocity, and pressure. A one-dimensional convolutional autoencoder with strided convolutions maps this state to an $N_z$-dimensional latent vector and reconstructs it through a mirrored decoder, trained by minimizing mean squared reconstruction error. After freezing the encoder, a multilayer-perceptron forward operator $\mathcal{F}$ is trained on paired latent states to approximate $z_T \approx \mathcal{F}(z_0)$, yielding the full surrogate mapping

$$\hat{X}_f(\boldsymbol{\theta}) = \Psi\left[\mathcal{F}\left(\Phi(X_0(\boldsymbol{\theta}))\right)\right],$$

where $\boldsymbol{\theta} = [\rho_L, p_L, \rho_R, p_R]^T$. Because this pipeline consists solely of neural-network forward evaluations, it removes the CFD bottleneck inside MCMC sampling.

Latent dimensionality and data efficiency

Two design studies establish the surrogate configuration. A sweep over latent dimensions $N_z \in \{4, 8, 16, 32, 64\}$ shows that the largest single improvement occurs between $0.05 \le \rho_R, p_R \le 0.15$0 and $0.05 \le \rho_R, p_R \le 0.15$1 (an 86.01% error reduction), with continued gains at $0.05 \le \rho_R, p_R \le 0.15$2 (52.70%) and $0.05 \le \rho_R, p_R \le 0.15$3 (51.87%), but diminishing returns beyond ($0.05 \le \rho_R, p_R \le 0.15$4 yields only 31.62%). Qualitatively, $0.05 \le \rho_R, p_R \le 0.15$5 reconstructions exhibit oscillations and smeared discontinuities, whereas $0.05 \le \rho_R, p_R \le 0.15$6 recovers the shock location, contact discontinuity, and rarefaction profile accurately across all three variables. The authors select $0.05 \le \rho_R, p_R \le 0.15$7 as the operating point.

A scaling study varying training-set size from 20 to 500 simulations shows validation MSE decaying rapidly up to $0.05 \le \rho_R, p_R \le 0.15$8, reaching order $0.05 \le \rho_R, p_R \le 0.15$9, then saturating while training time grows nearly linearly. Accordingly, 250 high-fidelity simulations are adopted as the data budget — a notable result indicating that accurate nonlinear reduced manifolds for hyperbolic problems can be learned from a modest number of full-order solves.

On a held-out test case, the complete AE-ROM prediction closely tracks the high-fidelity solution in smooth regions, with errors concentrated near the shock and contact discontinuity — the expected localization for reduced-order approximations of hyperbolic systems, where small errors in discontinuity placement produce sharp residual peaks.

Bayesian formulation and observation-density study

The inverse problem infers the four-parameter vector $t = 0.2$0 from observations of final-time density and pressure at $t = 0.2$1 uniformly distributed locations on $t = 0.2$2, corrupted by independent Gaussian noise with standard deviation $t = 0.2$3. Uniform priors are assigned over ranges slightly wider than the training domain, and NUTS explores the resulting posterior.

The central empirical finding concerns how observation density affects inference. As $t = 0.2$4 increases from 5 to 100:

Metric	Density	Pressure
Posterior mean RMSE	$t = 0.2$5 (~12% reduction)	$t = 0.2$6 (~11% reduction)
Mean posterior std. dev.	~77.6% reduction	~76.1% reduction

The asymmetry between these two rows constitutes the paper's principal claim: denser observations primarily improve parameter identifiability and confidence rather than shifting the posterior mean. Posterior probability densities contract sharply with increasing $t = 0.2$7, and spatially resolved 95% credible intervals narrow systematically around a posterior mean that remains close to the ground-truth piecewise structure even in the sparsest configuration. For practitioners, this implies that sensor-placement decisions in shock-dominated flows should be evaluated primarily by their effect on credible-interval width rather than point-estimate accuracy.

Limitations and open questions

The paper is explicit about several constraints on its results. First, the study is restricted to a one-dimensional Riemann problem with piecewise-constant initial conditions and zero initial velocity, which simplifies both the parameterization and the latent manifold. Second, the forward operator predicts only a single fixed-horizon final state ($t = 0.2$8 s); time-continuous latent dynamics over arbitrary horizons are not learned. Third, and most consequential for interpretation: the likelihood assumes independent Gaussian noise and omits any surrogate-model discrepancy term, so the reported credible intervals quantify uncertainty conditioned on the AE-ROM and the prescribed noise model. Given that surrogate errors concentrate precisely near discontinuities — the regions most informative about the inferred states — posteriors may be overconfident in exactly the regimes of greatest interest; the magnitude of this effect is not quantified in the paper. Finally, whether the autoencoder's latent representation remains learnable and stable in multidimensional flows with complex shock interactions and evolving topology remains an open question.

Conclusion

This work demonstrates a complete, non-intrusive workflow — LHS-sampled WENO data generation, convolutional autoencoder compression, latent-space forward modeling, and NUTS-based Bayesian inversion — validated on the Sod shock tube problem. With a latent dimension of 32 and 250 training simulations, the AE-ROM reproduces dominant wave structures with errors localized at discontinuities, and the embedded sampler recovers unknown initial density and pressure states with quantified uncertainty. The quantitative outcome that a twentyfold increase in observation density contracts posterior uncertainty by roughly 76–78% while improving mean accuracy by only 11–12% identifies confidence calibration, not bias reduction, as the dominant benefit of denser sensing in this setting. Extensions to multidimensional flows, discrepancy-aware likelihoods, and time-aware latent dynamics remain open.