10:00 Opening (Sashi and Volker)
10:10 Ulrich Wilbrandt
Optimization of stabilization parameters for convection-dominated convection-diffusion equations
10:40 Derk Frerichs
On reducing spurious oscillations in discontinuous Galerkin methods for convection-diffusion equations
11:10 Sangeeta Yadav
SPDE-Net: Predict a robust stabilization parameter for a singularly perturbed PDE using deep learning
11:40 break
12:00 Subodh Joshi
Global sensitivity analysis of multilayer perceptron hyperparameters for application to stabilization of
high-order numerical schemes for singularly perturbed PDEs
12:30 Zahra Lakdawala
A starter on how CFD meets machine learning using a Bayesian framework
13:00 Closing
*** Ulrich Wilbrandt
Optimization of stabilization parameters for convection-dominated convection-diffusion equations
Solutions to convection-dominated convection-diffusion equations typically have spurious oscillations and other non-physical properties. To overcome these difficulties in the Finite Element context one often uses stabilizations, which are additional terms in the weak formulation. We will consider the streamline upwind Petrov-Galerkin (SUPG) and a spurious oscillations at layers diminishing (SOLD) method. These involve a user-chosen parameter on each cell and it is unknown how to choose these parameters exactly. The approach presented in this talk is to let an optimization algorithm decide on these parameters which leaves the choice of an appropriate objective to the user. We will show some objectives and their motivation and discuss a reduction of the rather large control space.
*** Derk Frerichs
On reducing spurious oscillations in discontinuous Galerkin methods for convection-diffusion equations
Standard discontinuous Galerkin methods for discretizing steady-state convection-diffusion-reaction equations produce often very sharp layers in the convection-dominated regime, but also show large spurious oscillations.
These nonphysical oscillations can be reduced in a computationally cheap way using post-processing techniques that replace the solution in the vicinity of layers by linear or constant approximations. This talk presents some known post-processing methods from the literature, and proposes several generalizations as well as novel modifications. Numerical results are presented showing advantages and disadvantages of the methods.
*** Sangeeta Yadav
SPDE-Net: Predict a robust stabilization parameter for a singularly perturbed PDE using deep learning
Numerical techniques for solving Singularly Perturbed Differential Equations
(SPDE) suffer low accuracy and high numerical instability in presence of interior
and boundary layers. Stabilization techniques are often employed to reduce
the spurious oscillations in the numerical solution. Such techniques are highly
dependent on user chosen stabilization parameter. Streamline Upwind Petrov
Galerkin technique is one such residual based stabilization technique. Here we
propose SPDE-Net, a novel neural network based technique to predict the value
of optimal stabilization parameter for SUPG technique. The prediction task is
modeled as a regression problem and is solved using semi supervised learning.
The training is based on error between the Finite Element Method and the
analytical solution of SPDE. Global and local variants of stabilization param-
eter τ are demonstrated. Experiments on a benchmark case of 1-dimensional
convection diffusion equation show a reasonable performance of semi supervised
training as compared to the conventional supervised training. This makes the
proposed technique eligible for extension to higher dimensions and other cases
where the analytical formula for stabilization parameter is unknown, therefore
making supervised learning impossible.
*** Subodh Joshi
Global sensitivity analysis of multilayer perceptron hyperparameters for application to stabilization of
high-order numerical schemes for singularly perturbed PDEs
Advection dominated flows are often modeled by singularly perturbed partial differential
equations. High-order finite element schemes are typically used for numerical solution of
such equations. In order to obtain a stable and accurate solution, the right amount of
artificial dissipation needs to be added to the flux residual of the numerical scheme. In this
work, we explore the possibility of using a supervised learning approach to automatically
decide the right amount of artificial dissipation to be added to the numerical flux residual
for optimal speed, accuracy and stability. We deploy a network of multilayer perceptrons
(also known as artificial neural networks or ANNs) for ‘learning’ the correct value of the
dissipation coefficient.
In literature, we often find the methodology adopted by researchers for selection of hy-
perparameters of ANNs somewhat heuristic. Although there are a few studies which try
to address this issue for the general classification or regression problems, when it comes
to data-driven scientific computing, a void remains. In this work, we try to address this
issue by analyzing a large number of ANN architectures and carrying out a global sensi-
tivity analysis of the hyperparameters. In order to do this, we use the Gaussian process
regression (Kriging) to obtain a metamodel of the multilayer perceptrons. This enables
us to generate a large number of input data points and corresponding network predic-
tions. Thereafter, we perform the Analysis of variance (ANOVA) based on Sobol’ indices.
The total order Sobol’ indices highlight the relative sensitivity of each hyperparameter
towards the network accuracy as quantified by several metrics. The ANOVA technique
also helps reduce the input space dimension by eliminating hyperparameters which have
a marginal or no effect on the network accuracy for further optimizations.
In this work, we analyse effect of several hyperparameters on the overall accuracy such
as the number of hidden layers, number of neurons per hidden layer, activation functions
used etc. The statistical invarience of the results is verified for ensuring the objectivity of
the analysis. Based on this analysis, we identify the most effective network architectures
and hyperparameters for our application. Finally, we demonstrate the efficacy of this
approach for 1D advection dominated flows.
*** Zahra Lakdawala
A starter on how CFD meets machine learning using a Bayesian framework
Recent models applying deep learning to flow and transport equations will
be discussed in the context of learning partial differential equations using
sparse data. Such models use the physical laws to recognize patterns from
data from experiments. An overview of a Bayesian framework, introducing
the concept of numerical Gaussian processes with covariance functions
(resulting from spatio-temporal nonlinear PDEs), is presented to balance
the complexity and data fitting. I will discuss the modeling framework
in ([1], [2]) and demonstrate the build-up of learning machines with the
ability to extract patterns from complex domains without requiring large
amounts of data, and yet are informed by the physics that generated
the data. All of this will be done using examples of nonlinear partial
differential equations, such as Burger's and Navier-Stokes equations.
References
[1] Raissi, Maziar & Perdikaris, Paris & Karniadakis, George. (2017). Numerical
Gaussian Processes for Time-Dependent and Nonlinear Partial Differential
Equations. SIAM Journal on Scientific Computing 40 (1), A172-A198
[2] Raissi, Maziar & Karniadakis, George. (2018). Hidden physics models: Machine
learning of nonlinear partial differential equations. Journal of Computational
Physics 357, 125-141