Many-query applications such as design optimization, uncertainty quantification and digital twins call for repeated full-field predictions from expensive partial differential equation (PDE) simulations. Grassmannian reduced-order models interpolate parameter-dependent subspaces but commonly vectorize the spatial field, and the ambient dimension then grows with the product of the grid sizes. Tensor-based models preserve the multiway structure but often use shared mode-wise bases or a fixed core across parameters. We propose a Tucker-based surrogate that keeps the spatial directions and time as separate modes. The factor subspaces are interpolated on a product Grassmann manifold, Procrustes alignment places the Tucker cores in one coordinate system, and a truncated singular value decomposition models their variation. Across five PDE benchmarks in two and three spatial dimensions, including public computational fluid dynamics data, the interpolation size is reduced by factors of 13 to 482 and the stored model by factors of 13 to 464 relative to a single-subspace Grassmannian surrogate with the same local interpolation. Cached queries run 3.3 to 8.5 times faster, the mean errors over ten resampled splits differ by at most 0.22 percentage points (0.36 when every model choice is re-selected inside each split), and the prediction-error terms of the two methods differ by less than 0.1 percentage points. The remaining accuracy differences are consistent with the larger representation error of the Tucker format. In a three-dimensional grid study the interpolation-size ratio rises to 2,260 and the query speedup to 11.9 at 64^3.
High-fidelity simulations such as computational fluid dynamics and finite element analysis are essential for modelling complex engineering systems but remain prohibitively expensive for tasks including parametric studies, optimisation and real-time control. Projection-based reduced-order models (ROMs) alleviate this cost by projecting full-order dynamics onto low-dimensional subspaces. On the other hand, projection-based ROMs can become less robust under parameter variation, and one approach is to adapt the projection basis (that is, the subspace) as the parameters change. We propose constrained extreme gradient boosting (cXGBoost) for predicting the proper orthogonal decomposition (POD) basis at given parameter settings. In particular, given the observed parameter-subspace pairs, we map the subspaces from the Grassmann manifold to a Euclidean space. Then, cXGBoost is trained in this Euclidean space with a constraint that the norm of the predicted vectors is bounded to ensure the injectivity of the mapping between the Euclidean space and the Grassmann manifold. On four parametric problems, a cylinder wake, Kolmogorov flow, the Kuramoto-Sivashinsky equation and a forced beam, cXGBoost has lower mean reconstruction errors than existing approaches, namely the projected Gaussian process and Grassmann interpolation.
A scalar measurement can determine a robust decision without identifying the latent covariance or every decision risk. We study finite Gaussian projection experiments with a fixed mean-plus-standard-deviation target. The observation laws define a set of indistinguishable moment models. We exhibit decisions that agree throughout that set despite unidentified risk values, and give a confidence-set certificate that returns a decision or an unresolved outcome. We prove soundness, necessary abstention for indistinguishable models with incompatible optima, and a fixed-checkpoint certification guarantee. A scalar profile reduces nonlinear risk comparisons to convex inner problems. Across seven synthetic instances, jointly calibrated moment regions give sufficient budgets 21 to 52 times explicit expected-sample information lower bounds. Separately measured stopping times are lower, with zero decision errors in 1,900 simulated streams. These comparisons do not constitute matching bounds, and the guarantees require Gaussian replication.