Learning fast, accurate, and stable closures of a kinetic theory of an active fluid

Important classes of active matter systems can be modeled using kinetic theories. However, kinetic theories can be high dimensional and challenging to simulate. Reduced-order representations based on tracking only low-order moments of the kinetic model serve as an efficient alternative, but typically require closure assumptions to model unrepresented higher-order moments. In this study, we present a learning framework based on neural networks that exploit rotational symmetries in the closure terms to learn accurate closure models directly from kinetic simulations. The data-driven closures demonstrate excellent a-priori predictions comparable to the state-of-the-art Bingham closure. We provide a systematic comparison between different neural network architectures and demonstrate that nonlocal effects can be safely ignored to model the closure terms. We develop an active learning strategy that enables accurate prediction of the closure terms across the entire parameter space using a single neural network without the need for retraining. We also propose a data-efficient training procedure based on time-stepping constraints and a differentiable pseudo-spectral solver, which enables the learning of stable closures suitable for a-posteriori inference. The coarse-grained simulations equipped with data-driven closure models faithfully reproduce the mean velocity statistics, scalar order parameters, and velocity power spectra observed in simulations of the kinetic theory. Our differentiable framework also facilitates the estimation of parameters in coarse-grained descriptions conditioned on data

A fast Chebyshev method for the Bingham closure with application to active nematic suspensions

Continuum kinetic theories provide an important tool for the analysis and simulation of particle suspensions. When those particles are anisotropic, the addition of a particle orientation vector to the kinetic description yields a $2d-1$ dimensional theory which becomes intractable to simulate, especially in three dimensions or near states where the particles are highly aligned. Coarse-grained theories that track only moments of the particle distribution functions provide a more efficient simulation framework, but require closure assumptions. For the particular case where the particles are apolar, the Bingham closure has been found to agree well with the underlying kinetic theory; yet the closure is non-trivial to compute, requiring the solution of an often nearly-singular nonlinear equation at every spatial discretization point at every timestep. In this paper, we present a robust, accurate, and efficient numerical scheme for evaluating the Bingham closure, with a controllable error/efficiency tradeoff. To demonstrate the utility of the method, we carry out high-resolution simulations of a coarse-grained continuum model for a suspension of active particles in parameter regimes inaccessible to kinetic theories. Analysis of these simulations reveals that inaccurately computing the closure can act to effectively limit spatial resolution in the coarse-grained fields. Pushing these simulations to the high spatial resolutions enabled by our method reveals a coupling between vorticity and topological defects in the suspension director field, as well as signatures of energy transfer between scales in this active fluid model