Fig 1.
The proposed multifidelity concepts toward hybrid ROM/FOM coupling.
Dashed blocks refer to the interface learning approaches introduced in the present work: (a) Direct Prolongation Interface (DPI), (b) Prolongation followed by a machine learning Correction Interface (PCI), (c) machine learning Correction followed by a Prolongation Interface (CPI), and (d) Uplifted Prolongation Interface (UPI) where the latent space is enhanced through machine learning before we apply the prolongation operator.
Fig 2.
Schematic illustration of the methodologies introduced to utilize ROM to economically provide sound interface conditions in a multifidelity domain decomposition problems.
Galerkin ROM yields inaccurate predictions (represented by rough curve), and direct prolongation of these results might be not efficient. PCI adds a correction effect to the prolonged solution in FOM space. Instead, CPI and UPI introduce the correction at ROM level before prolongation. UPI adds an extra superresolution effect to augment solution quality.
Fig 3.
Velocity at the interface obtained by considering ROM in the left part of the domain, with r = 2 (top) and r = 4 (bottom).
FOM solution corresponds to solving the governing equation over the entire domain, while TP denotes the projection of the FOM solution in the left zone onto the corresponding POD subspace.
Table 1.
ℓ2 norm for the deviation of the velocity at the interface with respect to its FOM value for t ∈ [0, 2].
Fig 4.
Spatio-temporal velocity profile obtained from applying high fidelity (FOM) solver onto the right subdomain (0.75 ≤ x ≤ 1), fed with interface boundary from a low-fidelity (ROM) solution with r = 2.
Fig 5.
Temperature field at different time instances for 2D Boussinesq problem using 4096 × 512 grid and Δt = 0.0005.
Fig 6.
Final temperature fields as obtained from different ROM-FOM coupling approached, compared to the FOM solution.
We note that the PCI becomes infeasible for higher dimensional systems.
Fig 7.
Projection of the predicted temperature fields at different times from FOM, DPI, CPI and UPI onto the POD basis function.