(4hb) Data Driven Development of Approximate Inertial Forms and Closures for Coarse-Scale Modeling of Multiphase Flows

Learning closures for coarse-graining fine scale models in multiscale systems is relevant when the macroscopic physics must be accurately described without having to explicitly resolve the fine scales [1]. Traditional constitutive modeling to close the equations and solve for the variables that describe the relevant fields is now evolving due to machine learning techniques for data-driven modeling [2][3]. This modeling approach can also take advantage of further reduced latent spaces, based on the idea of approximate inertial manifolds [4], learning reduced order nonlinear representations that are not easily physically interpretable. The aim of this work is to learn coarse grained PDEs as well as reduced order models of coarse-scale PDEs for multiphase flows using a data-driven approach. In the first portion of this work, we train a neural network to learn an approximate inertial form: a few ODEs for the system behavior projected onto a few proper orthogonal decomposition (POD) modes obtained from the fine scale simulations [5]: From 2D Direct Numerical Simulations of a multiphase bubbly flow in a vertical channel, we average in the direction parallel to the overall flow to create a dataset of one-spatial-dimension, time-dependent profiles, corresponding to the vertical velocity of the liquid and the void fraction fields [6]. We perform POD to reduce the high-dimensional averaged snapshot data to a truncated set of 10 leading-mode amplitude coefficients, and further reduce these to two latent coordinates (through an autoencoder) which are one to one with the first two POD modes, as illustrated in Fig.1(a).

We train a neural network to approximate the continuous-time dynamics of the system in terms of the amplitudes of the first two, “determining“ POD modes. Since the higher POD modes are functions (on our data) of the first two ”determining“ ones, that parametrize our approximate inertial manifold, we reconstruct the full solution from these two time series through a network that has learned the manifold, i.e. the higher 8 POD coefficients as a function of the 2 ”determining“ ones, as depicted in Fig.1(b).

In the second portion of this work, we attempt to learn the right-hand-side operator of the averaged PDE through (a) a black box model and (b) a learned ”grey-box“ model that exploits the known parts/structure of the operator: everything else but the closure. To evolve the relevant fields, a pair of unknown closure terms must be modelled, Fl (x, t), Gl (x, t) = C(system state(x, t)) for the wall-normal liquid flux and summed dissipative terms, respectively. These terms are learned from coarse evolution data [7]: we train a network to approximate these closure terms C away from the wall, using only x-local information. Close to the wall, we learn a smooth non-decreasing function that corrects the effect of the drag. The wall function is zero in a region close to the wall, and asymptotes to one far from it. Before applying this closure-learning method to the data derived from the full 2D multiphase flow, we validate the approach on a simplified version of the problem with an explicitly known closure function (which we show we can accurately recover).

[1] C. Meneveau and J. Katz. Scale-invariance and turbulence models for large-eddy simulation. Annual Review of Fluid Mechanics, 32(1):1–32, 2000.

[2] F. J. Alexander, G. Johnson, G. L. Eyink, and I. G. Kevrekidis. Equation-free implementation of statistical moment closures. Physical Review E, 77:026701, 2008.

[3] S. Ansumali, C. E. Frouzakis, I. V. Karlin, and I. G. Kevrekidis. Exploring Hydrodynamic Closures for the Lid-driven Micro-cavity. arXiv:0502018 [cond-mat], 2005. [4] C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, and E. S. Titi. On the computation of inertial manifolds. Physics Letters A, 131(7):433–436, 1988.

[5] R. Rico-Martinez, K. Krischer, I. G. Kevrekidis, M. C. Kube, and J. L. Hudson. Discrete vs. continuous-time nonlinear signal processing of cu electrodissolution data. Chemical Engineering Communications, 118(1):25–48, 1992.

[6] M. Ma, J. Lu, and G. Tryggvason. Using statistical learning to close two-fluid multiphase flow equations for bubbly flows in vertical channels. International Journal of Multiphase Flow, 85:336–347, 2016.

[7] S. Lee, M. Kooshkbaghi, K. Spiliotis, C. I. Siettos, and I. G. Kevrekidis. Coarse- scale PDEs from fine-scale observations via machine learning. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(1):013141, 2020.

Research Interests

I am interested in the modeling of multiscale phenomena in space and time, using continuum mechanics to understand the physics of the problem, and computational science to develop innovative tools. I am also interested in applying these modeling techniques in biology. My Ph.D. is in Mechanical engineering at Johns Hopkins University. I have a strong background in mechanistic modeling. More specifically, I have worked on the development of a nonlinear viscoelastic constitutive model for liquid crystal elastomers as well as the numerical implementation. My current work involves using machine learning tools in fluid mechanics problems for which the continuum model is expensive and a reduced-order representation accurately represents the physics. Applying this new math has been exciting for me, especially, learning more about the connection between nonlinear dynamics and machine learning.

Teaching Interests

I am interested in educating and transmitting knowledge about my research to the new generations. Particularly, I would like to teach core classes in continuum mechanics and nonlinear dynamics but connecting the concepts to research applications so that students can put the theory into practice and develop critical thinking. I am also particularly interested in promoting and encouraging diversity and have already taken steps in that direction by mentoring minority students. Regarding my teaching experience, I have been the primary instructor of an undergraduate level class in mechanics of materials for three semesters, teaching assistant, and guest lecturer in a graduate-level class in continuum mechanics and will be teaching and developing a new class in nonlinear dynamics next semester. I volunteered to teach a lecture about continuum mechanics to high school students at Johns Hopkins University.

Figure 1: Autoencoder that learns the identity with input and output the first 10 POD coefficients: (Cˆ1 , Cˆ2 , ..., Cˆ10 ) = dec(enc(C1 , C2 , ..., C10 )), and a bottleneck with 2 neurons: (N1 , N2 ) = enc(C1 , C2 , ..., C10 ). The first 2 POD modes are one-to-one with the autoencoder bottleneck neurons (a). As a result, we could learn the dynamics of the first 2 POD modes and predict the remaining 8 modes using the slaving Multilayer Perceptron (MLP) ρ (b).