发表机构
Université Paris-Saclay; CNRS; ENS Paris-Saclay; Centre Borelli（巴黎-萨克雷大学; 法国国家科学研究中心; 巴黎-萨克雷高等师范学校; 博雷利中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出结合Stiefel流形扰动与共形风险控制的框架，量化NIROMs的模型形式不确定性，在参数化PDE基准和工业压延工艺上验证其可提供可靠的局部不确定性量化
AI 中文摘要
非侵入式降阶模型（NIROMs）已成为从计算机实验设计中近似参数化偏微分方程的标准工具，同时能显著降低计算成本。然而，评估其预测的可靠性仍然是一个重大挑战，尤其是在 extrapolation（外推）机制或训练数据有限的情况下。在本研究中，我们提出了一种框架，通过将降阶基的扰动随机表示与无分布的共形类方法相结合，来量化NIROMs中的模型形式不确定性。从快照矩阵构建的确定性降阶基出发，我们在Stiefel流形上定义沿被丢弃模态的随机扰动来建模不确定性，得到的随机降阶近似所诱导的方差反映了基截断误差。传输近似给出了闭式后验方差，将基诱导的不确定性与回归诱导的不确定性分离开来，无需重新训练底层高斯过程。我们将该后验方差纳入共形风险控制校准框架，该框架提供具有坐标未覆盖保证的预测集。该框架生成的校准因子本身是不确定性估计质量的可解释标量诊断指标。我们在参数化PDE基准和工业轮胎制造压延工艺上对该方法进行了评估。数值实验表明，该方法能提供可靠、局部信息丰富的不确定性量化，其性能超出了高斯预测方差。
英文摘要
Non-intrusive reduced-order models (NIROMs) have become a standard tool for approximating parametric partial differential equations from computer design of experiments while significantly reducing computational costs. However, assessing the reliability of their predictions remains a major challenge, particularly in extrapolation regimes or under limited training data. In this work, we introduce a framework for quantifying model-form uncertainty in NIROMs by combining a perturbative stochastic representation of reduced bases with distribution-free conformal-type methods. Starting from a deterministic reduced basis constructed from snapshot matrices, we model uncertainty through random perturbations defined on the Stiefel manifold, directed along the discarded modes, yielding stochastic reduced-order approximations whose induced variance reflects the basis-truncation error. A transport approximation gives a closed-form posterior variance that separates basis-induced from regression-induced uncertainty, without re-training the underlying Gaussian processes. We include this posterior variance within a conformal risk control calibration framework, that provides prediction sets with coordinate miscoverage guarantees. The calibration factor produced by this framework is itself an interpretable, scalar diagnostic of the quality of the uncertainty estimate. The methodology is evaluated on parametric PDE benchmarks and an industrial tire-manufacturing calendering process. Numerical experiments demonstrate reliable, locally informative uncertainty quantification that goes beyond the Gaussian predictive variance.
