Physics-based and physics-agnostic reduced order modeling

In this work we describe dimensionality reduction methods to solve parametric problems governed by partial differential equations (PDE). The goal of these methodologies is to elucidate and reduce the dimensionality of the manifold containing the family of solutions of a parametric problem. This leads to a reduced system that can often be solved in real time. These techniques are being successfully applied in many fields in science and engineering, e.g. [1,2] as well as in geodynamics [3,4].

The application of these techniques involves two phases: i) creation of a reduced space, often done via a sampling of the parametric space and a singular value decomposition, and ii) use of the reduced space to find a new solution within the family.

Here we want to compare methodologies that, in their second step, include or neglect the physics described by the PDEs. We call these, physics-based and physics-agnostic approaches. Reduced Basis methods being an example of the first, and surrogate modelling one of the second.

Applications of flow in porous media are used as examples to test the strengths and weaknesses of the different approaches. These methodologies are a potential tool to be used in situations where it is unaffordable to obtain a very large training set (big data).

REFERENCES

[1] Rocas M., A. García-González, S. Zlotnik, X. Larráyoz and P. Díez. Nonintrusive Uncertainty Quantification for automotive crash problems with VPS/Pamcrash. Finite Elements in Analysis & Design, Vol. 193, doi:10.1016/j.finel.2021.103556, 2021.

[2] Muixí A., S. Zlotnik, P. Calvet, M. Espanol, I. Lodoso-Torrecilla, M.P. Ginebra, P. Díez and A. García-González. A multiparametric advection-diffusion reduced-order model for molecular transport in scaffolds for osteoinduction. Biomechanics and Modeling in Mechanobiology, doi:10.1007/s10237-022-01577-2, 2022.

[3] Ortega O., S. Zlotnik, J.C. Afonso and P. Díez. Fast Stokes flow simulations for geophysical-geodynamic inverse problems and sensitivity analyses based on reduced order modeling. Journal of Geophysical Research: Solid Earth, Vol. 125, 1–25, doi:10.1029/2019JB018314, 2020.

[4] Manassero M., J.C. Afonso, F. Zyserman, A. Jones, S. Zlotnik and I. Fomin. A Reduced Order Approach for Probabilistic Inversions of 3D Magnetotelluric Data II: Joint inver- sion of MT and Surface-Wave Data. Journal of Geophysical Research - Solid Earth, doi:10.1029/2021JB021962, 2021.