Optimization and Generation in Aerodynamics Inverse Design

TL;DR

This work reframes aerodynamic inverse design as a dual optimization-guided generation problem, bridging point-design and distributional design through a divergence-based framework and a shape-prior that preserves plausible geometries. It introduces a symmetric KL-based loss for cost predictors, a density-gradient optimization that leverages a learned data prior, and a unified guided-generation approach. To address high-dimensional covariance estimation, it proposes SA-MC, a time- and memory-efficient low-rank covariance estimator enabling scalable guidance in diffusion-like generation and flow matching. Experiments on controlled 2D and high-fidelity 3D CFD tasks (cars and aircraft), validated with OpenFOAM simulations and wind-tunnel tests, show consistent improvements in both optimization and generation, with offline RL results supporting generality and robustness.

Abstract

Inverse design with physics-based objectives is challenging because it couples high-dimensional geometry with expensive simulations, as exemplified by aerodynamic shape optimization for drag reduction. We revisit inverse design through two canonical solutions, the optimal design point and the optimal design distribution, and relate them to optimization and guided generation. Building on this view, we propose a new training loss for cost predictors and a density-gradient optimization method that improves objectives while preserving plausible shapes. We further unify existing training-free guided generation methods. To address their inability to approximate conditional covariance in high dimensions, we develop a time- and memory-efficient algorithm for approximate covariance estimation. Experiments on a controlled 2D study and high-fidelity 3D aerodynamic benchmarks (car and aircraft), validated by OpenFOAM simulations and miniature wind-tunnel tests with 3D-printed prototypes, demonstrate consistent gains in both optimization and guided generation. Additional offline RL results further support the generality of our approach.

Figures (11)

• Figure 1: Inverse-design optimality and covariance-aware guidance. The left panel illustrates two classical paradigms in inverse design, point design and distributional design, and highlights their connection. The blue boxes indicate the organization of our theoretical sections. The right panel compares four representative guidance mechanisms, showing their computational graphs and differences. The schematic plots on the far right visualize the distinct trajectory patterns induced by different methods. Circles and ellipses denote variance.
• Figure 2: Method overview.a, Latent shape-preserving optimization in the Shape-VAE space. b, Guided flow-matching generation with a cost predictor. c, Cost predictor conditioned on geometry, operating conditions and $\lambda$. d, Detailed modules in Transformer predictor.
• Figure 3: 2D validation experiment.a, Base distribution $p$, cost field, and target distributions ($q^\star$ (Eq. \ref{['best-dis']})) via ground-truth cost ($q_{\text{real}}$), cost predictors trained with MSE ($q_\text{MSE}$) and SKL ($q_\text{SKL}$). b, Flow matching from noise to $p$ (blue: noise; red: samples) and guided sampling with DPS, LGD-MC, SIM-MC, and SA-MC using Real/MSE/SKL costs. c, Cost-gradient vs. density-gradient optimization using MSE- or SKL-trained cost predictors.
• Figure 4: Vehicle aerodynamic optimization and generation.a, Drag reduction after optimization under different gradient and predictor training losses. b, Shape fidelity during optimization (Chamfer distance); dashed line marks an empirical plausibility boundary. c, Generation performance across guidance scales $\lambda$ for different methods and training losses. d, OpenFOAM velocity magnitude before/after optimization, showing a reduced low-speed wake. e, Smoke visualization of 3D-printed shapes in a miniature wind tunnel, corroborating weaker wake turbulence after optimization. f, Example meshes generated without/with guidance. g, OpenFOAM velocity magnitude, SA-MC reduces low-speed wake. h, Wind-tunnel tests of guided generations, showing a smaller turbulent region in the wake.
• Figure 5: Aircraft aerodynamic optimization and generation.a, Optimized drag-to-lift ratio ($C_d/C_l$); cost-gradient guidance can yield negative lift ($<0$). b, Shape fidelity during optimization (Chamfer distance); dashed line marks an empirical plausibility boundary. c, Best generated $C_d/C_l$ over $\lambda\!\in\!\{1,10,100\}$ for each guidance method. d, Runtime and GPU memory for a single sample. e, Definition of ${C_p}\text{-}\mathrm{up}$ and ${C_p}\text{-}\mathrm{down}$ . f, Velocity magnitude and Pressure before/after optimization, showing smoother $C_p$ and reduced low-velocity region in the wake. g, Comparison of unguided and different guidance schemes; SA-MC yields lower ${C_p}\text{-}\mathrm{up}$ and higher ${C_p}\text{-}\mathrm{down}$.

Theorems & Definitions (14)

• Theorem 2.1
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• Theorem 2.3
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• Theorem 2.5
• proof
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