Accelerating Bayesian inverse design in computational fluid dynamics using neural operators

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.

Neural surrogate models for computational fluid dynamics (CFD) are typically trained as forward operators that map explicit problem specifications, such as geometry and boundary conditions, to solution fields. This ties the model to the conditioning variables seen during training and limits reuse under boundary-condition shifts or local geometry changes. We propose to reformulate steady CFD inference as an inpainting problem: instead of training on explicit boundary conditions, we learn a self-supervised prior over velocity fields and impose boundary constraints only during inference by fixing known regions such as inlet, outlet or unchanged regions from previous simulations. To scale this idea to large 3D meshes, we introduce a local neighbourhood tokeniser that represents high-resolution velocity fields as compact spatial latent tokens and train latent flow-matching and masked-autoencoder models on these tokens. On intracranial aneurysm hemodynamics, our method reconstructs full velocity fields from sparse boundary context, outperforms supervised neural surrogates under boundary-condition and dataset shift and enables local geometry editing by reusing unchanged simulation context. These results suggest that viewing CFD inference as context-conditioned inpainting can turn neural surrogates from task-specific predictors into reusable flow priors.

Computational Fluid Dynamics (CFD) is central to race-car aerodynamic development, yet its cost -- tens of thousands of core-hours per high-fidelity evaluation -- severely limits the design space exploration feasible within realistic budgets. AI-based surrogate models promise to alleviate this bottleneck, but progress has been constrained by the limited complexity of public datasets, which are dominated by smoothed passenger-car shapes that fail to exercise surrogates on the thin, complex, highly loaded components governing motorsport performance. This work presents three primary contributions. First, we introduce a high-fidelity RANS dataset built on a parametric LMP2-class CAD model and spanning six operating conditions (map points) covering straight-line and cornering regimes, generated and validated by aerodynamics experts at Dallara to preserve features relevant to industrial motorsport. Second, we present the Gauge-Invariant Spectral Transformer (GIST), a graph-based neural operator whose spectral embeddings encode mesh connectivity to enhance predictions on tightly packed, complex geometries. GIST guarantees discretization invariance and scales linearly with mesh size, achieving state-of-the-art accuracy on both public benchmarks and the proposed race-car dataset. Third, we demonstrate that GIST achieves a level of predictive accuracy suitable for early-stage aerodynamic design, providing a first validation of the concept of interactive design-space exploration -- where engineers query a surrogate in place of the CFD solver -- within industrial motorsport workflows.