MoTIF: A Mode-Structured Tensor Framework for Multi-Parametric Approximation, Super-Resolution and Forecasting of Unsteady Systems

TL;DR

MoTIF introduces a tensor-based framework that combines HOSVD, Gaussian Process Regression, and neural networks for efficient multi-parametric approximation, super-resolution, and forecasting of high-dimensional unsteady systems.

Contribution

It presents a novel mode-structured tensor approach that decouples physical parameters, spatial, and temporal modes for improved surrogate modelling of complex dynamical systems.

Findings

Achieves less than 2% relative RMS error in flow reconstruction.

Effectively interpolates and extrapolates parametric and spatial modal matrices.

Successfully predicts temporal evolution of unsteady flow configurations.

Abstract

We introduce MoTIF, a mode-structured tensor framework for multi-parametric approximation, super-resolution, and temporal forecasting of high-dimensional unsteady systems. The methodology leverages High-Order Singular Value Decomposition (HOSVD) to obtain a structured multilinear representation of multi-dimensional datasets, separating physical parameters, spatial coordinates, and temporal evolution into distinct modal components. This decomposition enables the application of dedicated approximation operators to each mode. Gaussian Process Regression is employed to interpolate and extrapolate parametric and spatial modal matrices, enabling database completion and resolution enhancement, while recurrent neural networks are applied to the temporal mode to forecast system evolution. This decoupled operator-learning strategy preserves the intrinsic tensor structure while providing a flexible non-intrusive reduced-order modelling framework. The proposed methodology is validated on a database of unsteady laminar flow simulations with varying Reynolds numbers and angles of attack. Accurate reconstruction of unseen flow configurations and temporal prediction are achieved, with relative root mean square errors consistently below 2\% compared to high-fidelity simulations. The framework provides a scalable and mathematically structured alternative to conventional surrogate modelling approaches for high-dimensional parametric dynamical systems.