A Multi-Fidelity Parametric Framework for Reduced-Order Modeling using Optimal Transport-based Interpolation: Applications to Diffused-Interface Two-Phase Flows

Abstract

This work introduces a data-driven, non-intrusive reduced-order modeling (ROM) framework that leverages Optimal Transport (OT) for multi-fidelity and parametric problems. Building upon the success of displacement interpolation for data augmentation in handling nonlinear dynamics, we extend its application to more complex and practical scenarios. The framework is designed to correct a computationally inexpensive low-fidelity (LF) model to match an accurate high-fidelity (HF) one by capturing its temporal evolution via displacement interpolation while preserving the problem's physical consistency. The framework is further extended to address systems dependent on a physical parameter, for which we construct a surrogate model using a hierarchical, two-level interpolation strategy. First, it creates synthetic HF checkpoints via displacement interpolation in the parameter space. Second, the residual between these synthetic HF checkpoints and a true LF solution is interpolated in the time domain using the multi-fidelity OT-based methodology. This strategy provides a robust and efficient way to explore the parameter space and to obtain a refined description of the dynamical system. The potential of the method is discussed in the context of complex and computationally expensive diffuse interface methods for two-phase flow simulations, which are characterized by moving interfaces and nonlinear evolution, and challenging to be dealt with traditional ROM techniques.

Figures (31)

• Figure 1: Illustration of the different OT problem formulations, depicting the mass transport plan $\pi_{ij}$ proportional to the thickness of the arrows.
• Figure 2: Effect of the entropic regularization increasing the parameter $\epsilon$, from left to right the plan becomes smoother and more diffuse.
• Figure 3: Schematic of the multi-fidelity OT-ROM framework, using displacement interpolation on the residual between HF and LF models.
• Figure 4: Parametric and temporal displacement interpolation on the $(t, \mu)$ grid.
• Figure 5: Low-fidelity (top) and high-fidelity (bottom) phase-field isocontours for different initial conditions of the Rider-Kothe vortex at $t=T/2$.

Theorems & Definitions (2)

• Remark 2.1
• Remark 2.2