AI summaryⓘ
The authors present a new method called two-step MV-DeepONet to better estimate uncertainty in complex systems where outputs depend on many factors. Unlike the earlier Prob-DeepONet, which assumes independent uncertainties across output locations, their approach captures dependencies between different output points by working in a simplified coefficient space. They do this by separating the learning process into two steps and rotating the data into a smaller subspace, which allows them to model correlations more accurately while keeping computations efficient. Tests on various physics problems show their method produces more realistic uncertainty estimates and captures relationships across outputs better than previous methods.
uncertainty quantificationcovariance matrixDeepONetGaussian processorthogonal basissubspace rotationpartial differential equationsprobabilistic modelinglow-rank approximationsurrogate modeling
Authors
Yupei Nie, Lei Wang, Jiasen Liu
Abstract
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.