做 CFD 逆设计或不确定性量化的研究者终于有了一个计算可行的方案——神经算子加速后，贝叶斯推断从小时级降到秒级，建议直接复现论文中的喷管案例。
贝叶斯逆设计为从稀疏流场观测中推断气动几何形状并量化不确定性提供了理论框架，但传统方法依赖高保真 CFD 模拟进行 MCMC 采样，计算成本极高。本研究将神经算子代理模型直接嵌入 MCMC 推断循环，在准一维喷管流动的贝叶斯逆问题中，证明其能保持后验几何与不确定性分布。使用 Deep Operator Network 替代 CFD 求解器后，总推断时间降至 1 秒以内，加速超过三个数量级。研究还发现几何参数化（如三次 B 样条）对可辨识性和后验条件化起决定性作用。该方法为航空工程中不确定性感知的逆设计提供了实用工作流。
Accelerating Bayesian inverse design in computational fluid dynamics using neural operators
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.