European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   MS44: Uncertainty quantification with physics-informed surrogate models (Part 3)
Chair: Benjamin Sanderse
25 mins
Fusion plasma turbulence simulation with neural network surrogate models
Jonathan Citrin, Clarisse Bourdelle, Yann Camenen, Federico Felici, Aaron Ho, Karel van de Plassche
Abstract: Plasma energy losses due to turbulent transport in toroidal magnetic confinement devices, such as tokamaks, is one of the limiting factors for achieving viable fusion energy. Reactor design and plasma scenario optimisation demands both accurate and tractable predictive turbulence calculations. Neural network surrogate physics models provides a pathway to circumvent these conflicting constraints. High-fidelity direct numerical simulations of plasma turbulence require 10-100 MCPUh to resolve plasma evolution on discharge timescales, making routine tokamak scenario prediction infeasible. A key enabling step is the development of reduced order turbulence models. Such “quasilinear” models, such as the QuaLiKiz code [1,2] developed at DIFFER and CEA, is ×10^6 faster than nonlinear simulations and can model discharge timescales within ∼100CPUh. This capability enables routine physics analysis of heat, particle, and momentum transport, and extrapolation to future machine performance. However, the computation timescales are still insufficient for extensive scenario optimisation and control-oriented applications. Further computation acceleration is provided by neural network surrogate modelling. The computational speed of reduced turbulence models as QuaLiKiz is sufficient for the construction of extensive databases of model input-output mapping using HPC resources. These databases are then used as training sets for neural network regression, where shallow feedforward networks can provide sufficient generalisation.. A key aspect is the customisation of output regression variables and optimisation cost functions in a physics-informed manner, to capture known features of the system. The resultant neural network transport model is ×10^6 faster than QuaLiKiz itself, and ×10^12 faster than nonlinear simulations. When coupled to the RAPTOR fast tokamak simulator, near-realtime tokamak transport predictions are feasible, evolving 1s of tokamak plasma within 10 CPUs [3-4]. Validation of the surrogate model in comparison to slower simulations using QuaLiKiz itself, has been demonstrated [5]. These developments open up a plethora new possibilities in fusion science for first-principle-based scenario optimization, control-oriented applications, and uncertainty quantification, hitherto constrained by the bottleneck of turbulent transport model computations. [1] C. Bourdelle et al., 2016, Plasma Phys. Control. Fusion 58 014036 [2] J. Citrin et al. 2017, Plasma Phys. Control. Fusion 59 124005 [3] J. Citrin et al., 2015, Nucl. Fusion 55 092001 [4] F. Felici et al. 2018 Nucl. Fusion 58 096006 [5] K.L. van de Plassche, EU-US TTF Seville 2018 ; to be submitted to Nucl. Fusion [6] A. Ho et al., 2019, Nucl. Fusion 59 056007
25 mins
Structure preserving reduced order models for Hamiltonian systems on Poisson manifolds
Cecilia Pagliantini, Jan Hesthaven
Abstract: The development of stable, robust and efficient reduced basis methods for nonlinear and time-dependent problems still poses some major challenges in model order reduction. In this talk we discuss the design and analysis of structure preserving reduced basis methods for a large class of time-dependent non-dissipative problems by resorting to their semi-discrete formulation as Hamiltonian dynamical systems. The Poisson manifold structure that characterizes the phase space of Hamiltonian problems encodes the physical properties, symmetries and conservation laws of the dynamics. Failing to preserve such structure in performing a model order reduction may result in the violation of the conservation laws and in the onset of spurious artifacts and instabilities. We design structure preserving reduced basis methods for the general case of Hamiltonian system with a nonlinear degenerate Poisson structure based on a two-step approach. First, via a local approximation of the Poisson tensor we split the Hamiltonian dynamics into an "almost" symplectic component and the trivial evolution of the Casimir invariants. Second, we apply canonically symplectic reduced basis techniques only to the nontrivial component of the dynamics. With this approach the global Poisson structure and the conservation properties of the phase flow are retained by the reduced model up to errors in the Poisson tensor approximation.
25 mins
Energy-conserving reduced order models for incompressible flow
Benjamin Sanderse
Abstract: The simulation of complex fluid flows is an ongoing challenge in the scientific community. The computational cost of Direct Numerical Simulation (DNS) or Large Eddy Simulation (LES) of turbulent flows quickly becomes imperative when one is interested in control, design, optimization and uncertainty quantification. For these purposes, simplified models are typically used, such as reduced order models, surrogate models, low-fidelity models, etc. In this work we will study reduced order models (ROMs) that are obtained by projecting the fluid flow equations onto a lower-dimensional space. Classically, this is performed by using a POD-Galerkin method, where the basis for the projection is built from a proper orthogonal decomposition of the snapshot matrix of a set of high-fidelity simulations. Ongoing issues of this approach are, amongst others, the stability of the ROM, handling turbulent flows, and conservation properties [1,2]. We will address the stability of the ROM for the particular case of the incompressible Navier-Stokes equations. We propose to use an energy-conserving finite volume discretization of the Navier-Stokes equations [3] as full-order model (FOM), which has the important property that it is energy conserving in the limit of vanishing viscosity and thus possesses non-linear stability. We project this FOM on a lower-dimensional space in such a way that the resulting ROM inherits the energy conservation property of the FOM, and consequently its non-linear stability properties. The stability of this new energy-conserving ROM is demonstrated for various test cases, and its accuracy as a function of time step, Reynolds number, number of modes, and amount of snapshot data is assessed. [1] K. Carlberg, Y. Choi, and S. Sargsyan. Conservative model reduction for finite-volume models. Journal of Computational Physics, 371:280-314, 2018. [2] L. Fick, Y. Maday, A.T. Patera, and T. Taddei. A stabilized POD model for turbulent flows over a range of Reynolds numbers: Optimal parameter sampling and constrained projection, Journal of Computational Physics, 371:214-243, 2018. [3] B. Sanderse. Energy-conserving Runge-Kutta methods for the incompressible Navier-Stokes equations, Journal of Computational Physics, 233:100-131, 2013.
25 mins
Experimental Comparison of Symplectic and Non-symplectic Model Order Reduction on an Uncertainty Quantification Problem
Patrick Buchfink, Bernard Haasdonk
Abstract: Uncertainty Quantification (UQ) is an important field to quantify the propagation of uncertainties, analyze sensitivities or realize statistical inversion of a mathematical model. Sampling-based estimation techniques evaluate the model for many different parameter samples. For computationally intensive models, this might require long runtimes or even be infeasible. This so-called multi-query problem can be speeded up or even be enabled with surrogate models from model order reduction (MOR) techniques. For accurate and physically consistent MOR, structure-preserving reduction is essential. We investigate numerically how so-called symplectic model reduction techniques can improve the UQ results for Hamiltonian systems compared to conventional (non-symplectic) approaches. We conclude that the symplectic methods give better results and more robustness with respect to the size of the reduced model.