12,263 research outputs found
Machine learning based data-driven discovery of nonlinear phase-field dynamics
One of the main questions regarding complex systems at large scales concerns
the effective interactions and driving forces that emerge from the detailed
microscopic properties. Coarse-grained models aim to describe complex systems
in terms of coarse-scale equations with a reduced number of degrees of freedom.
Recent developments in machine learning (ML) algorithms have significantly
empowered the discovery process of the governing equations directly from data.
However, it remains difficult to discover partial differential equations (PDEs)
with high-order derivatives. In this paper, we present new data-driven
architectures based on multi-layer perceptron (MLP), convolutional neural
network (CNN), and a combination of CNN and long short-term memory (CNN-LSTM)
structures for discovering the non-linear equations of motion for phase-field
models with non-conserved and conserved order parameters. The well-known
Allen--Cahn, Cahn--Hilliard, and the phase-field crystal (PFC) models were used
as the test cases. Two conceptually different types of implementations were
used: (a) guided by physical intuition (such as local dependence of the
derivatives) and (b) in the absence of any physical assumptions (black-box
model). We show that not only can we effectively learn the time derivatives of
the field in both scenarios, but we can also use the data-driven PDEs to
propagate the field in time and achieve results in good agreement with the
original PDEs
Data-driven discovery of stochastic dynamical equations of collective motion
Coarse-grained descriptions of collective motion of flocking systems are
often derived for the macroscopic or the thermodynamic limit. However, many
real flocks are small sized (10 to 100 individuals), called the mesoscopic
scales, where stochasticity arising from the finite flock sizes is important.
Developing mesoscopic scale equations, typically in the form of stochastic
differential equations, can be challenging even for the simplest of the
collective motion models. Here, we take a novel data-driven equation learning
approach to construct the stochastic mesoscopic descriptions of a simple
self-propelled particle (SPP) model of collective motion. In our SPP model, a
focal individual can interact with k randomly chosen neighbours within an
interaction radius. We consider k = 1 (called stochastic pairwise
interactions), k = 2 (stochastic ternary interactions), and k equalling all
available neighbours within the interaction radius (equivalent to Vicsek-like
local averaging). The data-driven mesoscopic equations reveal that the
stochastic pairwise interaction model produces a novel form of collective
motion driven by a multiplicative noise term (hence termed, noise-induced
flocking). In contrast, for higher order interactions (k > 1), including
Vicsek-like averaging interactions, yield collective motion driven primarily by
the deterministic forces. We find that the relation between the parameters of
the mesoscopic equations describing the dynamics and the population size are
sensitive to the density and to the interaction radius, exhibiting deviations
from mean-field theoretical expectations. We provide semi-analytic arguments
potentially explaining these observed deviations. In summary, our study
emphasizes the importance of mesoscopic descriptions of flocking systems and
demonstrates the potential of the data-driven equation discovery methods for
complex systems studies
Sub-grid modelling for two-dimensional turbulence using neural networks
In this investigation, a data-driven turbulence closure framework is
introduced and deployed for the sub-grid modelling of Kraichnan turbulence. The
novelty of the proposed method lies in the fact that snapshots from
high-fidelity numerical data are used to inform artificial neural networks for
predicting the turbulence source term through localized grid-resolved
information. In particular, our proposed methodology successfully establishes a
map between inputs given by stencils of the vorticity and the streamfunction
along with information from two well-known eddy-viscosity kernels. Through this
we predict the sub-grid vorticity forcing in a temporally and spatially dynamic
fashion. Our study is both a-priori and a-posteriori in nature. In the former,
we present an extensive hyper-parameter optimization analysis in addition to
learning quantification through probability density function based validation
of sub-grid predictions. In the latter, we analyse the performance of our
framework for flow evolution in a classical decaying two-dimensional turbulence
test case in the presence of errors related to temporal and spatial
discretization. Statistical assessments in the form of angle-averaged kinetic
energy spectra demonstrate the promise of the proposed methodology for sub-grid
quantity inference. In addition, it is also observed that some measure of
a-posteriori error must be considered during optimal model selection for
greater accuracy. The results in this article thus represent a promising
development in the formalization of a framework for generation of
heuristic-free turbulence closures from data
Stratification relieves constraints from steric hindrance in the generation of compact acto-myosin asters at the membrane cortex
Recent in-vivo studies have revealed that several membrane proteins are driven to form nanoclusters by active contractile flows arising from F-actin and myosin at the cortex. The mechanism of clustering was shown to be arising from the dynamic patterning of transient contractile platforms (asters) generated by actin and myosin. Myosin-II, which assemble as minifilaments consisting of tens of myosin heads, are rather bulky structures and hence a concern could be that steric considerations might obstruct the emergence of nanoclustering. Here, using coarse-grained, agent-based simulations that respect the size of constituents, we find that in the presence of steric hindrance, the patterns exhibited by actomyosin in two dimensions, do not resemble the steady state patterns observed in our in-vitro reconstitution of actomyosin on a supported bilayer. We then perform simulations in a thin rectangular slab, allowing the separation of a layer of actin filaments from those of myosin-II minifilaments. This recapitulates the observed features of in-vitro patterning. Using super resolution microscopy, we find direct evidence for stratification in our in-vitro system. Our study suggests the possibility that molecular stratification may be an important organising feature of the cortical cytoskeleton in-vivo
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