European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   MS44: Uncertainty quantification with physics-informed surrogate models (Part 2)
Chair: Benjamin Sanderse
25 mins
Consistency and generalization in physics-informed data-driven surrogate models
Karthik Duraisamy
Abstract: This talk will present recent developments in adopting statistical inference and machine learning to extract quantify and reduce model form errors in physics-based models. The emphasis is on enforcing consistency between the data and prediction environments by treating field inference and machine learning in an integrated fashion, rather than as distinct steps. To promote tractability of the procedure, a novel optimization algorithm is introduced. Results will be demonstrated in turbulence modeling and fuel cell modeling . problems.
25 mins
Bayesian field-inversion for turbulence anisotropy with informative priors
Richard Dwight, Arent van Korlaar
Abstract: Reynolds-Averaged Navier-Stokes (RANS) and its associated turbulence closure models will continue to be necessary in order to perform computationally affordable flow simulations for the foreseeable future. The errors inherent in the semi-empirical turbulence closures, means that significant uncertainty is associated with the results. In this work we use data from expensive LES and DNS simulations to train new closure models for RANS, similar to previous deterministic work. Our procedure is a two-step process: (a) First we find a local turbulence anisotropy correction to the RANS model, which causes the model to reproduce the given LES/DNS mean field. (b) Secondly we regress the local correction discovered in part (a) as a function of the local mean-flow, using machine-learning techniques. Step (a) is a field-inversion problem with a large space of valid solutions. Rather than using Tikhonov regularization, we introduce an informative prior on the correction derived using random matrix theory. This prior guarantees the positive-definiteness of the Reynolds stress tensor, and encourages physically reasonable corrections in the posterior. After doing this for several flows, in Step (b) we fit a stochastic regression model to the resulting data set. The posterior covariance, approximated using the Hessian of the simulation code, is exploited to give the regression model realistic uncertainty. Unseen flows are then predicted with this model, which include now informed estimates of model uncertainty.
25 mins
PDE-constrained neural network for turbulent Rayleigh-Bénard convection
Atul Agrawal, Didier Lucor, Yann Fraigneau, Bérengère Podvin, Anne Sergent
Abstract: The motivation of this work stems from the limited understanding of the physical mechanisms responsible for the heat transfer enhancement in rough turbulent Rayleigh-Bénard convection. Indeed, strong dynamic interactions between a large spatial/time scale range from mean wind to small-scale plumes remain difficult to comprehend with a single numerical or experimental emulator \cite{Rusaouen2018}. The development of data-driven surrogate models for the prediction of complex physical phenomena, in place of more standard numerical simulations, is an ongoing challenge in various fields and may help for this particular application \cite{Podvin2017,Kutz2017}. Here, we decide to rely on deep neural networks (DNN) which are known to be performant in capturing transient and intermittent phenomenon with the possibility of handling translations, rotations and other invariances. More specifically, the approach retained in this project is the one of training a DNN based on a cost function that involves a set of partial differential equations (PDEs). The idea is to incorporate prior scientific knowledge to be used as a guideline for designing efficient deep learning models \cite{Raissi2019,Zhu2019}. In particular, a reasonable approach is to incorporate (some of) the governing equations of the physical model (e.g. mass/momentum/energy conservation) at the core of the DNN, i.e. in the loss/likelihood functions. We propose to investigate how this additional information effectively regularizes the minimization procedure in the training of DNN, and enables them to generalize well with fewer training samples. More specifically, we will report on the influence of the choice of the domain of interest for data acquisition as well as subsequent training and predictions, in relation to the problem geometry and initial/boundary conditions. Finally, we will report on the DNNs training attempt on large direct numerical simulations database acquired for turbulent convective flow in rectangular cavity with rough bottom plate.
25 mins
Reduced model-error source terms for fluid flows
Wouter Edeling, Daan Crommelin
Abstract: It is well known that the wide range of spatial and temporal scales present in geophysical flow problems represents a (currently) insurmountable computational bottleneck, which must be circumvented by a coarse-graining procedure. The effect of the unresolved fluid motions enters the coarse-grained equations as an unclosed forcing term, denoted as the 'eddy forcing'. Traditionally, the system is closed by approximate deterministic closure models, i.e. so-called parameterizations. Instead of creating a deterministic parameterization, some recent efforts have focused on creating a stochastic, data-driven surrogate model for the eddy forcing from a (limited) set of reference data, with the goal of accurately capturing the long-term flow statistics. Since the eddy forcing is a dynamically evolving field, a surrogate should be able to mimic the complex spatial patterns displayed by the eddy forcing. Rather than creating such a (fully data-driven) surrogate, we propose to precede the surrogate construction step by a procedure that replaces the eddy forcing with a new model-error source term which: i) is tailor-made to capture spatially-integrated statistics of interest, ii) strikes a balance between physical insight and data-driven modelling , and iii) significantly reduces the amount of training data that is needed. Instead of creating a surrogate for an evolving field, we now only require a surrogate model for one scalar time series per statistical quantity-of-interest. We derive the model-error source terms, and construct the reduced surrogate using an ocean model of two-dimensional turbulence in a doubly periodic square domain.