[Paper Review] Physics-agnostic and Physics-infused machine learning for thin films flows: modeling, and predictions from small data

Cristina P. Martin-Linares, Yorgos M. Psarellis+3 more|arXiv (Cornell University)|Jan 29, 2023

Fluid Dynamics and Turbulent Flows40 references4 citations

TL;DR

This paper proposes a physics-agnostic and physics-infused machine learning framework for modeling thin film flows using small datasets. It leverages non-harmonic diffusion maps and geometric harmonics to extract low-dimensional manifolds from data, enabling accurate predictions even with limited training examples, with key results showing robust performance on thin film dynamics without explicit physical priors.

ABSTRACT

Numerical simulations of multiphase flows are crucial in numerous engineering applications, but are often limited by the computationally demanding solution of the Navier-Stokes (NS) equations. Here, we present a data-driven workflow where a handful of detailed NS simulation data are leveraged into a reduced-order model for a prototypical vertically falling liquid film. We develop a physics-agnostic model for the film thickness, achieving a far better agreement with the NS solutions than the asymptotic Kuramoto-Sivashinsky (KS) equation. We also develop two variants of physics-infused models providing a form of calibration of a low-fidelity model (i.e. the KS) against a few high-fidelity NS data. Finally, predictive models for missing data are developed, for either the amplitude, or the full-field velocity and even the flow parameter from partial information. This is achieved with the so-called "Gappy Diffusion Maps", which we compare favorably to its linear counterpart, Gappy POD.

Motivation & Objective

• To develop a machine learning framework that models thin film flows with minimal data.
• To integrate physical principles into data-driven models without requiring large-scale simulations or labeled data.
• To enable accurate predictions of thin film dynamics using small, sparse datasets.
• To demonstrate the effectiveness of geometric harmonics and diffusion maps in capturing the intrinsic dynamics of thin film systems.

Proposed method

• The method constructs a non-harmonic diffusion map (NDM) to extract low-dimensional representations from thin film flow data.
• It uses a modified affinity matrix based solely on the non-harmonic DMAP coordinates to preserve geometric structure.
• Geometric harmonics are applied to project functions onto a truncated set of eigenvectors from the NDM, enabling function approximation.
• The framework supports both physics-agnostic and physics-infused learning by incorporating physical constraints into the learned manifold.
• The approach uses a truncated expansion $ f \rightarrow P_{\delta}f = \sum_{\beta \in \overline{S}_{\delta}} \langle f, \overline{\psi}_{\beta} \rangle \overline{\psi}_{\beta} $ to approximate functions on the manifold.

Experimental results

Research questions

• RQ1 Can a physics-agnostic machine learning model accurately predict thin film flow dynamics from small datasets?
• RQ2 How do non-harmonic diffusion maps improve the representation of thin film flow manifolds compared to standard methods?
• RQ3 To what extent can physical constraints be embedded into data-driven models without requiring large-scale training data?
• RQ4 What is the predictive accuracy of geometric harmonics on low-dimensional manifolds derived from thin film simulations?

Key findings

• The non-harmonic diffusion map effectively captures the intrinsic geometry of thin film flow dynamics from small datasets.
• Geometric harmonics enable accurate function approximation on the learned manifold, supporting stable predictions.
• The framework achieves high predictive accuracy even with limited training data, demonstrating robustness to data scarcity.
• The integration of physical priors into the manifold learning process improves generalization and fidelity of predictions.