Research advances in Uncertainty Quantification and Design Optimization for Flight Vehicles
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摘要: 飞行器不确定性设计分析方法作为一种先进的飞行器设计范式, 为解决不确定性精确表征、量化传播与优化设计提供了系统的方法体系. 本文系统梳理了飞行器不确定性设计相关的核心概念与关键技术, 归纳了飞行器关键系统设计与重要环境条件预示涉及的不确定性问题. 在此基础上, 总结提炼了该领域的五个重要发展方向: (1) 高维不确定性量化与高效传播: 通过融合输入空间降维、函数表示稀疏化与低秩张量分解等技术, 构建自适应高维不确定性量化框架, 有效应对“维度灾难”带来的挑战; (2) 混合不确定性量化与高效传播: 构建兼容概率、区间、模糊、证据理论等多种不确定性类型的统一量化分析框架, 并结合代理模型与主动学习策略提升复杂多源不确定性问题的计算效率; (3) 多层级多保真度不确定性量化框架: 融合广义近似控制变量、自适应多指标随机配点等技术, 实现计算资源在不同保真度模型上的动态优化配置; (4) 基于不确定性的设计优化算法与框架: 将概率约束与鲁棒性度量统一到不确定性条件下的多目标优化与决策框架, 通过单层循环与解耦式优化策略, 实现性能、可靠性与鲁棒性的综合权衡优化; (5) 基于人工智能技术的不确定性设计分析: 以物理信息神经网络为核心, 融合物理知识与多源数据, 构建智能不确定性量化与优化框架.
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关键词:
- 不确定性设计分析 /
- 高维不确定性量化 /
- 混合不确定性量化 /
- 基于可靠性的设计分析 /
- 鲁棒性设计优化
Abstract: Uncertainty Quantification (UQ) and Uncertainty-Based Design Optimization (UBDO), as an emerging design paradigm for flight vehicles, provides a systematic methodological framework for addressing the precise characterization, propagation, and design optimization of uncertainties. This paper reviews the core concepts and key technologies in this field. It summarizes the uncertainty challenges associated with critical systems and significant environmental conditions of flight vehicles. Based on the latest research progress, five key research directions are identified: (1) High-dimensional uncertainty quantification and efficient propagation: Constructing an adaptive high-dimensional UQ framework by integrating techniques such as dimensionality reduction, compressed sensing, and low-rank tensor decomposition to effectively address the "curse of dimensionality." (2) Hybrid uncertainty quantification and efficient propagation: A unified framework is established to accommodate various types of uncertainties—including probabilistic, interval, fuzzy, and evidence theory. The computational efficiency for complex, multi-source uncertainty problems is further enhanced by incorporating surrogate modeling and active learning strategies. (3) Multi-level and multi-fidelity UQ framework: Achieving dynamic and optimal allocation of computational resources across models of varying fidelities by integrating techniques like generalized approximate control variates and adaptive multi-index stochastic collocation. (4) Uncertainty-based design optimization algorithms and frameworks: Unifying probabilistic constraints and robustness metrics within a multi-objective optimization and decision-making framework under uncertainty, enabling trade-off optimization among performance, reliability, and robustness through single-loop and decoupled optimization strategies. (5) Uncertainty design and analysis based on artificial intelligence techniques: Centered on physics-informed neural networks, this direction incorporates physical knowledge and multi-source data to establish intelligent frameworks for uncertainty quantification and optimization. -
图 1 不同类别湍流模型以及其中的不确定性来源 (Duraisamy et al. 2019, 陈鑫 等 2021)
图 2 不确定性影响下的载荷辨识示意图 (Wang et al. 2021)
图 3 美国DARPA高超声速国际飞行研究与实验项目 (Hypersonic International Flight Research and Experimentation, HIFiRE) 飞行试验载荷与地面直连试验装置 (HDCR) 示意图 (Geraci et al. 2019)
图 6 类型Ⅱ概率−区间混合不确定性传播示意图 (Jiang 等 2017)
图 8 多保真度方法的三种模型管理策略: 自适应、融合和抽样 (Peherstorfer 等 2018)
图 10 基于贝叶斯优化的高斯过程模型融合飞试数据和CFD仿真数据的物理信息数据融合框架 (Romeo 等 2024)
表 1 高维不确定性量化技术方法对比分析
方法名称 输入空间降维 函数表示稀疏化 低秩张量分解 计算过程优化 核心思想 高维模型输出的绝大部分变化由少数几个输入变量的线性/非线性组合所主导. 高维模型响应函数可由一个庞大的函数基项中的少数几个关键基函数精确表示. 高维模型内在结构存在变量间的弱耦合性, 可分解为多个低维核心张量的乘积. 复杂模型的计算流程中存在大量因变量依赖稀疏性而产生的冗余计算. 优化对象 输入参数空间: 从高维输入空间识别并投影到一个低维的活跃子空间. 函数表达式 (代理模型) : 在固定的高维输入空间上, 寻找一个系数稀疏、形式简洁的函数表达式. 函数的数据结构: 将表示高维函数的全尺寸张量, 分解为一系列低维 (低秩) 核心张量的组合. 计算执行流程: 重构模型的计算图, 消除在张量网格求值时的重复操作. 关键假设 低维结构假设: 存在一个低维子空间, 模型函数在该空间方向上变化最剧烈, 而在其正交方向上近乎恒定. 稀疏性假设: 在某个合适的正交多项式基下, 模型响应的展开系数向量是稀疏的 (大部分为零或接近零). 低秩/可分离性假设: 高维函数可以近似为一个可分离变量的函数之和/积, 即变量间的耦合程度不高. 计算稀疏性假设: 模型中的许多中间计算步骤仅依赖于小部分输入变量, 计算图存在操作级的稀疏依赖. 技术途径 将高维空间UQ问题转化到降维后的低维空间中进行, 从根本上规避了“维度灾难”. 通过正则化回归等手段, 从组合爆炸的候选基函数中筛选出有效基项, 使所需样本数不依赖于输入维度, 而依赖于解的稀疏度. 通过将高维函数近似为一系列低维核心张量的结构化乘积, 将指数级增长的计算复杂度分解为随维度线性或多项式增长. 通过消除张量化输入下的重复计算, 避免了模型评估成本随维度指数级增长. 优势长处 物理概念直观. 降维效果显著, 识别出子空间之后, 其后续分析成本低. 可与其他代理模型技术 (如Kriging, PCE) 无缝结合. 不改变原始输入空间, 保留了参数的物理意义. 能够处理重要变量非线性组合的情况. 在样本数远少于基函数数量时依然可获得精确解. 对一大类具有可分离结构的函数非常有效. 能够自适应的从少量函数响应样本中构建代理模型. 在求解PDE等问题上效率极高. 不依赖输入空间、函数本身或数据结构的任何特定假设. 不引入模型近似误差, 结果与原始模型完全一致. 局限挑战 识别子空间可能需要梯度信息, 计算成本较高. 降维后的变量通常是原始变量的线性组合, 失去了直接的物理意义. 若模型的低维结构不明显, 则方法失效. 依赖于基函数的选择, 若函数在所选基下不稀疏, 则效果不佳. 正则化超参数的选择对结果影响很大, 需要精细调优. 要求函数具有低秩结构, 对于变量强耦合的函数不适用. 高阶张量分解的理论和算法相对复杂. 主要针对张量网格类输入, 对蒙特卡洛等随机采样输入的加速效果有限. 对于包含高成本隐式操作 (如迭代求解器) 的模型, 加速效果受限. 实现复杂, 需要对模型代码进行深入分析和转换. 代表方法 活跃子空间、主成分
分析压缩感知、正则化回归 张量列车分解 (TT-Cross算法) 计算图转换 (AMTC) 
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表 2 概率−模糊混合不确定性下的可靠性分析三类问题
序号 分类 解释说明 1 PROFUST问题 失效判据具有模糊性, 而参数只具有随机性, 基于模糊状态假设与概率假设进行混合不确定性下的可靠性分析. 2 POSBIST问题 失效判据不具有模糊性, 而参数具有模糊性, 基于双态假设与可能性假设进行混合不确定性下的可靠性分析. 3 POSFUST问题 失效判据与参数均具有模糊性时, 基于模糊状态假设与可能性假设进行混合不确定性下的可靠性分析.
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表 3 混合不确定性量化与高效传播方法对比分析
文献来源 应用场景 核心思路 优势长处 聂兆伟等 (2022) 概率−区间
(Probability-Interval)主动学习Kriging + EGO
将混合不确定性下的可靠性分析转化为无约束优化问题. 利用主动学习Kriging估计多故障模式下的失效概率, 结合EGO搜索区间变量导致的失效概率边界.主动学习Kriging + EGO策略有效解决了区间变量导致的嵌套优化问题, 计算效率得到提升. Meng et al. (2022) 混合指数概率积分法 (HEPIM)
提出混合指数概率积分法. 外层用直接概率积分法 (DPIM) 求解PDF, 内层用指数凸模型处理区间/椭球边界.DPIM避免了蒙特卡洛采样, 从物理方程层面降维求解; 指数凸模型能处理变量间的相关性. You et al. (2020) 概率−模糊
(Probability-Fuzzy)自动更新极值响应面 (AUERS)
基于PSO在模糊变量的α-截集区间内寻找极限状态函数的极值点, 引入均方预测误差指导Kriging模型的局部自适应更新.只在极值点附近精细化模型, 大幅减少了计算代价高昂的功能函数的调用次数. Wang et al. (2023) 非侵入式混合PCE + 稀疏网格
利用Hermite (随机) 和Legendre (区间) 多项式构建混合PCE. 采用稀疏网格积分计算系数, 再用遗传算法在PCE上寻优.非侵入式PCE无需修改原求解器, 便于与商业软件耦合; 稀疏网格策略缓解了中等维度下的“维数灾难”, 兼顾计算精度与效率. Wang & Matthies (2020) 分布参数模糊的随机模型 + RBF + 二分法
建立分布参数模糊的随机模型, 借助λ-截集运算与模糊分解定理, 将模糊随机矩预测转化为一系列区间随机矩; 随后RBF加速从分布参数到随机矩的映射, 并采用二分点嵌套试验设计高效确定截集边界.理论框架严谨, 保留了不精确概率的数学特性; 二分法采样效率高. Elishakoff & Colombi (1993) 概率−凸集
(Probability-Convex)反优化 (Anti-optimization)
采用反优化框架, 将不确定参数约束于椭圆凸集, 并在名义值处对响应函数作一阶泰勒展开, 利用凸集极值定理推导出均方响应上下界的闭式解析解.能够给出解析解, 避免了数值仿真的高计算成本. Zhang et al. (2021) 全局敏感性 + TCR策略 + 主动学习Kriging
针对概率−凸集混合可靠性问题, 提出基于失效概率的矩独立敏感性指标. 利用截断候选域 (TCR) 缩小主动学习Kriging的搜索范围.TCR策略避免了在对结果影响较小的区域浪费采样点; 能识别多失效模式的主导变量. Du (2008) 概率−证据 (Probability-Evidence) 统一FORM框架 (FORM-UUA)
将证据理论的焦元极值分析嵌入到FORM寻找MPP的迭代过程中.解决了证据理论中焦元组合爆炸的问题; 将双层嵌套变为单层优化. Zhang et al. (2019) 针对证据理论焦元的目标失效概率分配策略
通过将目标失效概率合理分解至各焦元, 使得具有较高失效似真度的焦元在迭代过程中承担更大修正权重, 从而更有效的反映不同焦元对系统失效风险的贡献.优化了RBDO中可靠性约束向确定性问题的转换流程, 显著提升了收敛效率, 同时增强了方法的鲁棒性. Liu et al. (2024) 多种不确定性
(Multi-type)Co-Kriging多保真代理模型
面向随机−区间−模糊混合不确定性传播问题, 推导了Co-Kriging预测值的均值和方差的解析表达式. 利用增强期望改进准则平衡高/低保真采样.充分利用低成本 (低保真) 数据, 解析推导避免了后处理误差. Long et al. (2023) T-UUA/ DR/ EGO-UUA
面向概率、证据、模糊和区间四种不确定性的统一框架: 较小不确定度用T-UUA; 较大不确定度用DR/EGO-UUA.更为全面的统一框架, 涵盖4种不确定性; 分策略处理保证了效率与精度的平衡.
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表 4 多层级方法与多保真度方法对比分析
方法类别 多层级方法 多保真度方法 模型来源 同质层级: 通常由单一参数 (如网格尺寸) 系统性生成. 模型之间关系明确. 异质集合: 模型来自不同物理原理、降阶技术或数据拟合. 模型之间关系不一定明确. 理论基础 一般基于收敛率, 其分析和资源分配依赖于方差、成本和偏差的渐进收敛率. 一般基于相关性, 其分析和资源分配依赖于模型输出的统计度量. 适用性 最适用于可以系统构建模型层级的问题, 如PDE的有限元/有限体积法. 适用性更加灵活, 可以融合任何类型的低保真度模型, 即使它们的来源和特性非常不同, 甚至是黑箱模型. 模型管理策略 基于伸缩求和 (Telescoping Sum) 的模型管理, 将一个高精度计算任务分解为一个最粗糙模型的计算任务和一系列相邻模型层级之间差异的计算任务. 该策略的核心优势在于能够实现最优资源分配, 即利用已知的成本和误差收敛率, 可以精确计算每个层级需要分配多少次模拟, 从而在满足目标精度的前提下, 将总计算成本降至最低. 多样化的模型管理策略, 主要包括自适应、融合和抽样 (见图8). 例如, 通过高保真度数据在线更新低保真度模型, 或使用低保真度模型作为预筛选器来决定是否调用高保真度模型. 策略选择更加灵活.
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