对比四种模型在湍流模拟外推时的不确定性量化能力，发现集成神经网络表现最稳。
Quantifying Out-of-Training Uncertainty of Neural-Network based Turbulence Closures
- 用三种神经网络与高斯过程对比外推不确定性，基于已有代数模型验证
- 集成神经网络在外部输入下误差最小，负对数似然最优，且计算成本远低于高斯过程
- 尽管结构简单，集成神经网络仍提供直观可靠的不确定性估计，适合工程应用
基于神经网络的湍流闭合方法被开发为传统湍流模型的预训练替代品，旨在提升计算流体动力学（CFD）仿真的计算效率和预测精度。其广泛应用的瓶颈在于缺乏对模型不确定性的量化，尤其是针对训练数据范围外输入的不确定性。本文采用已发表的代数湍流闭合模型，比较三种神经网络方法（深度集成、蒙特卡洛丢弃、随机变分推断）与高斯过程（GP）在表征认知不确定性方面的表现。在训练范围内，精确高斯过程表现最佳，均方根误差（RMSE）为 $2.14 \cdot 10^{-5}$，其次为深度集成（RMSE $4.59 \cdot 10^{-4}$）。在外部输入区域，高斯过程依然最优，但深度集成表现相近。在不确定性校准方面，随机变分推断与深度集成在某一案例中误校准误差最低；而深度集成在两种外部情况下的负对数似然均最优。总体准确度排序为：高斯过程 > 深度集成 > 随机变分推断 > 蒙特卡洛丢弃。深度集成虽为朴素集成，但结果稳健，且计算复杂度仅为 $O(n^3)$，远低于高斯过程的开销。
原文摘要 · Abstract (English)
Neural-Network (NN) based turbulence closures have been developed for being used as pre-trained surrogates for traditional turbulence closures, with the aim to increase computational efficiency and prediction accuracy of CFD simulations. The bottleneck to the widespread adaptation of these ML-based closures is the relative lack of uncertainty quantification (UQ) for these models. Especially, quantifying uncertainties associated with out-of-training inputs, that is when the ML-based turbulence closures are queried on inputs outside their training data regime. In the current paper, a published algebraic turbulence closure1 has been utilized to compare the quality of epistemic UQ between three NN-based methods and Gaussian Process (GP). The three NN-based methods explored are Deep Ensembles (DE), Monte-Carlo Dropout (MCD), and Stochastic Variational Inference (SVI). In the in-training results, we find the exact GP performs the best in accuracy with a Root Mean Squared Error (RMSE) of $2.14 \cdot 10^{-5}$ followed by the DE with an RMSE of $4.59 \cdot 10^{-4}$. Next, the paper discusses the performance of the four methods for quantifying out-of-training uncertainties. For performance, the Exact GP yet again is the best in performance, but has similar performance to the DE in the out-of-training regions. In UQ accuracy for the out-of-training case, SVI and DE hold the best miscalibration error for one of the cases. However, the DE performs the best in Negative Log-Likelihood for both out-of-training cases. We observe that for the current problem, in terms of accuracy GP > DE > SV I > MCD. The DE results are relatively robust and provide intuitive UQ estimates, despite performing naive ensembling. In terms of computational cost, the GP is significantly higher than the NN-based methods with a $O(n^3)$ computational complexity for each training step
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