Data-driven low-dimensional model of a sedimenting flexible fiber

TL;DR

This work addresses the computational cost of simulating sedimenting flexible filaments by introducing a data-driven, low-dimensional framework. It combines an undercomplete autoencoder to compress filament shapes into a four-dimensional latent space with a neural ODE that evolves the latent state (and center position) under the elasto-gravitational parameter $\mathcal{B}$, enabling accurate forecasting in Stokes flow. The approach reproduces both filament shapes and center-bead trajectories across unseen $\mathcal{B}$ and initial angles, with ensemble errors below $0.1$ and terminal shapes matching the physics-driven terminal states. This method offers fast, generalizable predictions for deformable fibers and lays groundwork for extending to more complex flows and multi-particle systems, potentially via graph-based architectures.

Abstract

The dynamics of flexible filaments entrained in flow, important for understanding many biological and industrial processes, are computationally expensive to model with full-physics simulations. This work describes a data-driven technique to create high-fidelity low-dimensional models of flexible fiber dynamics using machine learning; the technique is applied to sedimentation in a quiescent, viscous Newtonian fluid, using results from detailed simulations as the data set. The approach combines an autoencoder neural network architecture to learn a low-dimensional latent representation of the filament shape, with a neural ODE that learns the evolution of the particle in the latent state. The model was designed to model filaments of varying flexibility, characterized by an elasto-gravitational number $\mathcal{B}$, and was trained on a data set containing the evolution of fibers beginning at set angles of inclination. For the range of $\mathcal{B}$ considered here (100-10000), the filament shape dynamics can be represented with high accuracy with only four degrees of freedom, in contrast to the 93 present in the original bead-spring model used to generate the dynamic trajectories. We predict the evolution of fibers set at arbitrary angles and demonstrate that our data-driven model can accurately forecast the evolution of a fiber at both trained and untrained elasto-gravitational numbers.

Figures (10)

• Figure 1: A flexible filament of length $L$ settling under an external force $\mathbf{F}_g$ in a quiescent Newtonian fluid. The filament begins at an arbitrary initial inclination, $\theta_0$, relative to the $x$-axis, and evolves until it reaches a terminal shape. The fiber is modeled as a series of $N$ beads of radius $a$ connected by springs, with the center bead, which has position $\mathbf{c}(t)$, shown in red.
• Figure 2: The evolution of the shape of a filament settling in a quiescent Newtonian fluid from an initial orientation to a common terminal shape at $\mathcal{B} = 1000$ for all initial angles of orientation within the training data set.
• Figure 3: The trajectories $\mathbf{c}(t)$ of the center bead of a filament settling in a quiescent Newtonian fluid at $\mathcal{B} = 1000$ for all initial angles of inclination within the training data set. The initial positions are denoted by symbol “$\blacktriangledown$” and the terminal positions are denoted by the symbol “$\times$”.
• Figure 4: The evolution of the shape of a filament settling in a quiescent Newtonian fluid from a common initial angle of orientation of $\pi/4$ to a terminal shape for all $\mathcal{B}$ within the training data set.
• Figure 5: Block diagram for data-driven model combining the autoencoder and temporal-evolution scheme. The temporal-evolution neural network, expanded in red, can be separated into two distinct neural networks forecasting the evolution of latent representation of the shape, $\mathbf{h}(t+\tau)$, and the shape-dependent change in position, $\mathbf{c}$; in practice, these can be forecasted by a single neural network.