Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields

Research Question and Approach

Standard probabilistic operator networks (Prob-DeepONet) quantify uncertainty by predicting pointwise Gaussian distributions at each output location. While efficient, this approach assumes conditional independence between locations, resulting in a diagonal covariance structure that fails to capture the complex, structured correlations inherent in physical systems driven by random input fields.

To address this, the authors propose the two-step MV-DeepONet. This framework decouples the learning process into two distinct stages: first, it learns an orthogonalized trunk basis to represent the output field in a low-dimensional subspace; second, it performs probabilistic Gaussian regression within this rotated coefficient space. By mapping these probabilistic coefficients back to the physical domain through the shared basis, the model induces a non-diagonal conditional predictive covariance in the output space without the need to explicitly parameterize or store a full high-dimensional covariance matrix.

The study demonstrates that the two-step MV-DeepONet framework significantly improves the representation of predictive uncertainty compared to standard Prob-DeepONet. By rotating the coefficient space to diagonalize the empirical covariance, the model effectively captures off-diagonal correlation patterns consistent with the underlying physical mechanisms of the PDEs tested (reaction-diffusion, Burgers, and Darcy equations).

Theoretical analysis via Frobenius-norm error decomposition identifies four critical factors governing the accuracy of covariance recovery: low-rank compressibility of the target covariance, the quality of the trunk-subspace approximation, finite-sample statistical errors, and the accuracy of coefficient-space covariance estimation. Numerical experiments, including a complex hypersonic aerothermal problem, confirm that the proposed method provides tighter, more structured uncertainty bands and better generalization to unseen inputs than previous lightweight probabilistic operator methods.