Tucker-based model speeds up repeated simulations for design and analysis
A Product-Grassmann Tucker Surrogate for Parametric Field Prediction
Computational Engineering, Finance, and Science
Summary
Simulating complex physical phenomena often requires time-consuming computer calculations repeated many times, such as for design or uncertainty testing. The authors developed a mathematical shortcut that breaks down these simulations into parts and smoothly combines them, preserving important structures in space and time. This approach reduces the amount of data to store and speeds up making predictions while keeping errors very low. It works well across various example problems including fluid flow, making repeated full predictions much faster and smaller in size.
What this means in practice
- •For simulation engineers: Accelerate repeated full-field predictions in PDE simulations during design optimization workflows with reduced data size and faster query times.
- •For digital twin developers: Use the proposed Tucker surrogate to quickly update and query physical state predictions for digital twins relying on complex PDE models.
Authors
Melika Baghi, Xiao Liu, Kamran Paynabar
Abstract
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.