发表机构
School of Mathematics and Physics, North China Electric Power University; School of Energy Power and Mechanical Engineering, North China Electric Power University（华北电力大学数理学院; 华北电力大学能源动力与机械工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对概率DeepONet仅能建模对角条件协方差的局限，提出两步MV-DeepONet，通过两步训练与高斯概率建模转移，在保留单次推理的同时准确恢复非对角相关模式，提升了不确定性传播建模性能。
AI 中文摘要
复杂物理系统中的前向不确定性传播可在场值输出间诱导结构化协方差。对于概率代理模型，总预测协方差包含输入实现下的条件均值协方差与平均条件预测协方差。概率DeepONet（Prob-DeepONet）通过单次前向传播预测逐点高斯均值与方差，实现轻量型不确定性量化，但其条件预测协方差仅为对角形式。为在不显式参数化完整高维协方差矩阵的情况下表示跨位置条件依赖，我们通过两项主要改进开发了两步均值-方差DeepONet（两步MV-DeepONet）：其一，采用两步训练将输出基学习与输入到系数的解耦，结合基正交化与子空间旋转；其二，将高斯概率建模从高维物理输出空间转移至低维旋转系数空间。通过共享基映射这些概率系数，可在保留单次推理的同时，在物理输出空间中诱导一般非对角的条件预测协方差。Frobenius范数误差分解及对应上界表明，低秩协方差可压缩性、主干子空间近似、有限样本统计误差与系数空间协方差估计是决定协方差恢复的主要因素。针对三个偏微分方程（PDE）控制的代表性问题及高超声速钝体气动热问题的数值实验显示，与Prob-DeepONet相比，所提方法的泛化能力提升、不确定性带更结构化，且能准确恢复非对角相关模式。
英文摘要
Forward uncertainty propagation in complex physical systems can induce structured covariance across field-valued outputs. For a probabilistic surrogate, the total predictive covariance comprises the covariance of conditional means across input realizations and the average conditional predictive covariance. Probabilistic DeepONet (Prob-DeepONet) provides lightweight uncertainty quantification by predicting pointwise Gaussian means and variances in a single forward pass, but its conditional predictive covariance is restricted to a diagonal form. To represent cross-location conditional dependence without explicitly parameterizing a full high-dimensional covariance matrix, we develop a two-step mean-variance DeepONet (two-step MV-DeepONet) through two principal modifications. First, two-step training is used to decouple output-basis learning from the input-to-coefficient mapping, together with basis orthogonalization and subspace rotation. Second, Gaussian probabilistic modeling is transferred from the high-dimensional physical output space to the low-dimensional rotated coefficient space. Mapping these probabilistic coefficients through the shared basis induces a generally non-diagonal conditional predictive covariance in the physical output space while retaining single-pass inference. A Frobenius-norm error decomposition and corresponding upper bound identify low-rank covariance compressibility, trunk-subspace approximation, finite-sample statistical error, and coefficient-space covariance estimation as the principal factors governing covariance recovery. Numerical experiments on three representative problems governed by partial differential equations (PDEs) and a hypersonic blunt-body aerothermal problem show improved generalization, more structured uncertainty bands, and accurate recovery of off-diagonal correlation patterns compared with Prob-DeepONet.
