Sharp interface approaches and deep learning techniques for multiphase flows

Many machine/deep learning artificial neural networks are trained to simply be interpolation functions that map input variables to output values interpolated from the training data in a linear/nonlinear fashion. Even when the input/output pairs of the training data are physically accurate (e.g. the results of an experiment or numerical simulation), interpolated quantities can deviate quite far from being physically accurate. Although one could project the output of a network into a physically feasible region, such a postprocess is not captured by the energy function minimized when training the network; thus, the final projected result could incorrectly deviate quite far from the training data. We propose folding any such projection or postprocess directly into the network so that the final result is correctly compared to the training data by the energy function. Although we propose a general approach, we illustrate its efficacy on a specific convolutional neural network that takes in human pose parameters (joint rotations) and outputs a prediction of vertex positions representing a triangulated cloth mesh. While the original network outputs vertex positions with erroneously high stretching and compression energies, the new network trained with our physics prior remedies these issues producing highly improved results. there will be large errors inf (w, x T ) when compared to y T . On the other hand, although one could create a network with many degrees of freedom in order to capture y T =f (w, x T ) as accurately as desired, even exactly,f (w, x) could oscillate wildly and inaccurately when x is not equal to x T , i.e. overfitting. See, e.g. [14,15,16,17]. One needs to take great care when designing the network architecture in order to avoid underfitting while still allowing for enough regularization to also avoid overfitting. Likewise, the form of the energy function and nature of the numerical optimization techniques also need careful consideration. Some of the most popular methods include variants of BFGS [18,19,20] and a number of methods based on gradient descent [21,22] (see also [23] and the references therein) or interpreting gradient descent as a numerical approximation to an ordinary differential equation to be solved via various approaches motivated by order of accuracy [24,25] and adaptive time-stepping [26,27,28,29,30].Devising a network architecture with enough representative capability to alleviate underfitting while still being amenable to the regularization required to avoid overfitting, and subsequently applying numerical optimization techniques to an adequately designed energy in order to find reasonable parameters w is a quite difficult and mostly experimental endeavour. Thus, much of the progress made by the community emanates from the laborious creation of data sets that many researchers can consider in order to design network architectures and find suitable parameters w, see e.g. [31]. This is typically driven by a community (rather than an individual or group) effort, and state-of-the-art r...

Section: Buckling and Inextensibility Priormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Coercing machine learning to output physically accurate results

Geng

Johnson

Fedkiw

2020

“…Remark 2.4. In case the boundary ∂Ω is split into subsets ∂Ω D and ∂Ω M where on ∂Ω D we have Dirichlet boundary conditions on u and on ∂Ω M we have the mixed boundary conditions in equation (7), then the forms a(t , u, v) and l (t , v, q) change to…”

Section: A Weak Formulationmentioning

confidence: 99%

A cut finite element method for incompressible two-phase Navier–Stokes flows

Frachon

Zahedi

2019

We present a space-time Cut Finite Element Method (CutFEM) for the time-dependent Navier-Stokes equations involving two immiscible incompressible fluids with different viscosities, densities, and with surface tension. The numerical method is able to accurately capture the strong discontinuity in the pressure and the weak discontinuity in the velocity field across evolving interfaces without re-meshing processes or regularization of the problem. We combine the strategy proposed in [P. Hansbo, M. G. Larson, S. Zahedi, Appl. Numer. Math. 85 (2014), 90-114] for the Stokes equations with a stationary interface and the space-time strategy presented in [P. Hansbo, M. G. Larson, S. Zahedi, Comput. Methods Appl. Mech. Engrg. 307 (2016), . We also propose a strategy for computing high order approximations of the surface tension force by computing a stabilized mean curvature vector. The presented space-time CutFEM uses a fixed mesh but includes stabilization terms that control the condition number of the resulting system matrix independently of the position of the interface, ensure stability and a convenient implementation of the space-time method based on quadrature in time. Numerical experiments in two and three space dimensions show that the numerical method is able to accurately capture the discontinuities in the pressure and the velocity field across evolving interfaces without requiring the mesh to be conformed to the interface and with good stability properties.

“…For viscous incompressible flows, like bubble flows where the curvature of the interface controls the dynamics, it seems that ML techniques are established techniques now [37]: we refer to Zaleski et al [28,2] for the reconstruction of the curvature of interfaces and to [16] for an extension to compressible effects. The exact curvature is function of the second derivative of the function that defines the interface, and a comprehensive review centered on incompressible flows with surface tension is [17]. More general references can be found in [29].…”

Section: Introductionmentioning

confidence: 99%

Machine Learning design of Volume of Fluid schemes for compressible flows

Després

Jourdren

2020

Our aim is to establish the feasibility of Machine-Learning-designed Volume of Fluid algorithms for compressible flows. We detail the incremental steps of the construction of a new family of Volume of Fluid-Machine Learning (VOF-ML) schemes adapted to bi-material compressible Euler calculations on Cartesian grids. An additivity principle is formulated for the Machine Learning datasets. We explain a key feature of this approach which is how to adapt the compressible solver to the preservation of natural symmetries. The VOF-ML schemes show good accuracy for advection of a variety of interfaces, including regular interfaces (straight lines and arcs of circle), Lipschitz interfaces (corners) and non Lipschitz triple point (the Trifolium test problem). Basic comparisons with a SLIC/Downwind scheme are presented together with elementary bi-material calculations with shocks.