Homepage Research Research topics Machine Learning & Uncertainty Quantification (MLQ)
Machine Learning & Uncertainty Quantification (MLQ)
Research in the MLQ theme follows three main axes:
- developing a Bayesian approach to uncertainty quantification for industrial turbulence models;
- CFD-driven learning of accurate RANS turbulence models for non-equilibrium flows;
- sparse, symbolic or rank-constrained regression methods for nonlinear system identification and reduced-order modelling.
Axis 1. Uncertainty quantification and industrial turbulence modelling
Despite the exponential growth of computing resources (processor power, parallel computing, architectures), Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES) remain impractical in industry, where flows are mostly at high Reynolds numbers. Industry therefore relies mainly on the Reynolds-Averaged Navier-Stokes (RANS) equations. Yet despite decades of research, there is still no consensus on modelling the Reynolds stress tensor, and the assumption that it is aligned with the strain-rate tensor regularly fails. Building on this, DynFluid applies and develops approaches to:
- Quantify the uncertainty of RANS predictions: to account for uncertainties, all research directions have been developed in a Bayesian framework, around Bayesian model mixtures. This led to a space-varying model aggregation method that relaxes the fixed-weight constraint of more classical approaches such as BMA, while drawing on Mixture of Experts and model aggregation. The method is currently being extended to include the calibration of the uncertain parameters of the models in the mixture.
- Correct the anisotropic part of the Reynolds stress tensor: corrections are derived from a database of canonical flows using a Bayesian symbolic identification algorithm (Sparse Bayesian Learning). They are then analysed in a space of flow features, and a learning algorithm (Random Forests, Gaussian Mixture Models) selects the best correction to apply in each region of the flow.
Axis 2. CFD-driven machine learning of RANS turbulence models
Within the PERTINENT project funded by the Carnot ARTS Institute, machine-learning methods were used to build accurate RANS turbulence models for non-equilibrium turbulence. A deterministic, CFD-driven symbolic identification algorithm (online approach) for learning explicit algebraic Reynolds stress models (EARSM) from high-fidelity data was developed, based on the offline learning algorithm SpaRTA [Schmelzer et al. 2020]. Corrections to the Reynolds stress tensor and to the production of transported turbulent quantities of a baseline linear eddy-viscosity model (LEVM) are expressed as functions of tensor polynomials selected from a library of candidate functions. CFD-driven training amounts to solving a black-box optimisation problem in which candidate EARSM models are assessed by running RANS simulations (online approach). Models can thus be trained against any target quantity of interest that the CFD model can output. Unlike offline training, the method is not restricted to datasets for which full high-fidelity fields, including second-order flow statistics, are available. It does, however, require solving an expensive, high-dimensional black-box optimisation problem, and several steps reduce the computational cost. First, a sensitivity analysis identifies the most influential terms and reduces the dimension of the search space. Then the CORS algorithm (Constrained Optimization using Response Surface), which approximates the black-box cost function with a response surface built from a limited number of CFD runs, finds the optimal model parameters. Model discovery and cross-validation are performed on three 2D turbulent separated flows in channels of varying cross-section, with different training datasets, to show the flexibility of the method. The discovered models are then applied to predict an unseen 2D separated flow at a higher Reynolds number and with a different geometry. Their predictions for this new case are more accurate not only than the baseline LEVM, but also than a general-purpose EARSM derived from purely physical arguments. The proposed deterministic symbolic identification approach is a promising way to build accurate, robust RANS models for a given class of flows with non-equilibrium turbulence at moderate computational cost. Such flows are found in many industrial sectors.
Axis 3. Statistical learning and reduced-order modelling
Over the past decade, statistical learning has attracted renewed interest, notably with the development and spread of methods based on deep neural networks. In fluid dynamics, this has led to many applications in turbulence modelling (see above) and to new methodologies for reduced-order modelling, particularly with flow control in mind.
Among statistical learning techniques for system identification and reduced-order modelling, particular emphasis is placed on sparse, symbolic and rank-constrained regression. The first two require the intrinsic dimension (e.g. the number of degrees of freedom) of the system to be fairly low, so they must be combined with dimensionality-reduction techniques (e.g. POD, DMD, autoencoders) to be applicable. Over the past five years, this approach has successfully produced low-dimensional, interpretable models of canonical flows such as the two-dimensional von Kármán vortex street, the flow over an open cavity, and the chaotic reversals of a convection cell in an annular thermosyphon. Compared with more classical approaches such as POD-Galerkin, these models proved more stable, more accurate and interpretable by construction. The importance of accounting for nonlinear correlations between the coherent structures used as the projection basis (e.g. the different POD modes) was also highlighted for the first time. This research was accompanied by the development of PySINDy, an open-source Python library that makes these approaches widely available. Rank-constrained multivariate regression is at the core of the research currently led by Jean-Christophe Loiseau on linear system identification, parametric dependencies and optimal sensor and actuator placement.
Among statistical learning techniques for system identification and reduced-order modelling, particular emphasis is placed on sparse, symbolic and rank-constrained regression. The first two require the intrinsic dimension (e.g. the number of degrees of freedom) of the system to be fairly low, so they must be combined with dimensionality-reduction techniques (e.g. POD, DMD, autoencoders) to be applicable. Over the past five years, this approach has successfully produced low-dimensional, interpretable models of canonical flows such as the two-dimensional von Kármán vortex street, the flow over an open cavity, and the chaotic reversals of a convection cell in an annular thermosyphon. Compared with more classical approaches such as POD-Galerkin, these models proved more stable, more accurate and interpretable by construction. The importance of accounting for nonlinear correlations between the coherent structures used as the projection basis (e.g. the different POD modes) was also highlighted for the first time. This research was accompanied by the development of PySINDy, an open-source Python library that makes these approaches widely available. Rank-constrained multivariate regression is at the core of the research currently led by Jean-Christophe Loiseau on linear system identification, parametric dependencies and optimal sensor and actuator placement.
