European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   MS44: Uncertainty quantification with physics-informed surrogate models (Part 1)
Chair: Benjamin Sanderse
25 mins
Multi-fidelity neural networks for efficient surrogate modeling
Yous van Halder, Benjamin Sanderse, Barry Koren
Abstract: A novel approach for constructing parametric solutions for PDEs is proposed. Multifidelity uncertainty quantification has gained much attention in recent years. Using high-fidelity solvers to perform UQ for high- dimensional problems is often infeasable. A multi-fidelity approach uses a combination of low and high- fidelity solvers to significantly decrease the computational cost of performing UQ. In our approach a neural network is used to generate a mapping between a low and high-fidelity solver, which can be used as a post- processing tool for enhancing low-fidelity solutions in the future. The neural network structure and training procedure in our approach results in a neural network that is capable of enhancing low-fidelity solutions outside the training set, which removes the need for a high-fidelity solver after training, as the enhanced low- fidelity solutions are perfectly suited for constructing surrogate models or extracting proper statistical moments. Our approach is demonstrated on a range of test-cases, ranging from the linear advection equation to the non-linear Navier-Stokes equations.
25 mins
Enhancing RANS with LES for Variable-Fidelity Optimization
Yu Zhang, Rizhard Dwight, Martin Schmelzer, Javier Gómez, Stefan Hickel
Abstract: Reynolds-Averaged Navier-Stokes (RANS) based aerodynamic optimization is widely used in aerospace engineering due to its acceptable computational cost and turn-around time. However, in many important situations, such as friction-drag reducing surfaces (e.g. dimples), and designs where separation is critical, the physics demands scale-resolving simulations of turbulence such as large-eddy simulation (LES). LES is often useful for a final analysis, but its high computation cost precludes its use within a design loop. Variable-fidelity methods with high-fidelity (hi-fi) from LES and low-fidelity (low-fi) from RANS are possible, but this class of methods rely for their efficiency on a high-correlation between the low-fi and hi-fi models, and still require a large number of hi-fi evaluations – whereas our target is ∼ 3 LES simulations per optimization. Therefore, to obtain hi-fi optimum within a limited computational budget, we propose an efficient variable-fidelity optimization strategy based on LES and an enhanced RANS model. The enhanced RANS model is obtained by using stochastic machine-learning methodologies to provide a prediction of the turbulence anisotropy tensor (b i j ), and the turbulence kinetic-energy (k), in terms of the mean-flow quantities resolved by RANS, in a manner analogous to algebraic Reynolds-stress models. The optimization is then performed on the surrogate using an EGO-like sampling procedure [1] [2]. The framework is tested on periodic hill case with parameters controlling the hills’ shape [3]. We investigate stochastic versions of several machine-learning methods, in particular the Tensor Basis Random Forest (TBRF) [4], the Tensor Basis Neural Network (TBNN) [5], and a symbolic regression method [6]. All the methods are trained for a specific periodic hill [7], and then used for predicting the flow field of periodic hills with different geometries to evaluate the statistic error. With the best-performed method, the variable-fidelity optimization using LES as the hi-fi analysis and enhanced RANS as low-fi analysis is tested. Comparison with single-fidelity optimization based on RANS and variable-fidelity optimization based on LES and standard RANS will be given. [1] D. R. Jones, M. Schonlau, and W. J. Welch, "Efficient global optimization of expensive black-box functions," Journal of Global Optimization 13(4), 455-492 (1998). [2] Y. Zhang, Z. H. Han, and K. S. Zhang, "Variable-fidelity expected improvement method for efficient global optimization of expensive functions," Structural and Multidisciplinary Optimization 58(4), 1431-1451 (2018). [3] J. F. Gómez, Multi-fidelity Co-kriging optimization using hybrid injected RANS and LES (Master thesis, Delft University of Technology, 2018). [4] M. Kaandorp, Machine learning for data-driven RANS turbulence modelling (Master thesis, Delft University of Technology, 2018). [5] J. Ling, A. Kurzawski, and J. Templeton, "Reynolds averaged turbulence modelling using deep neural networks with embedded invariance," Journal of Fluid Mechanics 807, 155-166 (2016). [6] M. Schmelzer, R. Dwight, and P. Cinnella, "Data-Driven Deterministic Symbolic Regression of Nonlinear Stress-Strain Relation for RANS Turbulence Modelling," In Fluid Dynamic Conference, 1-13 (AIAA, Atlanta, Georgia, 2018). [7] M. Breuer, N. Peller, Ch. Rapp, and M. Manhart, "Flow over periodic hills - Numerical and experimental study in a wide range of Reynolds numbers," Computers & Fluids 38, 433-457 (2009).
25 mins
Data-driven stochastic modeling for multiscale dynamical systems
Daan Crommelin
Abstract: Modeling and simulation of multiscale dynamical systems (e.g. atmosphere and ocean) is challenging due to the wide range of spatio-temporal scales that need to be taken into account. To tackle this issue, one can employ stochastic models to represent the feedback from dynamical processes at the small/fast scales onto processes at larger scales. I will discuss ongoing work on extracting stochastic subgrid-scale models from data, including results with resampling methods and discrete models. A key aspect is the two-way coupling of the data-driven subgrid-scale model to a given (e.g. physics-based) large-scale model. Furthermore, the involved systems often display spatial and temporal correlations (or even memory) that should be accounted for.
25 mins
Hidden Physics Models: Machine Learning of Non-Linear Partial Differential Equations
Maziar Raissi
Abstract: A grand challenge with great opportunities is to develop a coherent framework that enables blending conservation laws, physical principles, and/or phenomenological behaviors expressed by differential equations with the vast data sets available in many fields of engineering, science, and technology. At the intersection of probabilistic machine learning, deep learning, and scientific computations, this work is pursuing the overall vision to establish promising new directions for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data. To materialize this vision, this work is exploring two complementary directions: (1) designing data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and non-linear differential equations, to extract patterns from high-dimensional data generated from experiments, and (2) designing novel numerical algorithms that can seamlessly blend equations and noisy multi-fidelity data, infer latent quantities of interest (e.g., the solution to a differential equation), and naturally quantify uncertainty in computations. The latter is aligned in spirit with the emerging field of probabilistic numerics.