A machine learning approach for efficient uncertainty quantification using multiscale methods

Several multiscale methods account for sub-grid scale features using coarse scale basis functions. For example, in the Multiscale Finite Volume method the coarse scale basis functions are obtained by solving a set of local problems over dual-grid cells. We introduce a data-driven approach for the estimation of these coarse scale basis functions. Specifically, we employ a neural network predictor fitted using a set of solution samples from which it learns to generate subsequent basis functions at a lower computational cost than solving the local problems. The computational advantage of this approach is realized for uncertainty quantification tasks where a large number of realizations has to be evaluated. We attribute the ability to learn these basis functions to the modularity of the local problems and the redundancy of the permeability patches between samples. The proposed method is evaluated on elliptic problems yielding very promising results.Comment: Journal of Computational Physics (2017

Kernel Analog Forecasting: Multiscale Test Problems

Data-driven prediction is becoming increasingly widespread as the volume of data available grows and as algorithmic development matches this growth. The nature of the predictions made, and the manner in which they should be interpreted, depends crucially on the extent to which the variables chosen for prediction are Markovian, or approximately Markovian. Multiscale systems provide a framework in which this issue can be analyzed. In this work kernel analog forecasting methods are studied from the perspective of data generated by multiscale dynamical systems. The problems chosen exhibit a variety of different Markovian closures, using both averaging and homogenization; furthermore, settings where scale-separation is not present and the predicted variables are non-Markovian, are also considered. The studies provide guidance for the interpretation of data-driven prediction methods when used in practice.Comment: 30 pages, 14 figures; clarified several ambiguous parts, added references, and a comparison with Lorenz' original method (Sec. 4.5

Forecasting the geomagnetic activity of the Dst Index using radial basis function networks

The Dst index is a key parameter which characterises the disturbance of the geomagnetic field in magnetic storms. Modelling of the Dst index is thus very important for the analysis of the geomagnetic field. A data-based modelling approach, aimed at obtaining efficient models based on limited input-output observational data, provides a powerful tool for analysing and forecasting geomagnetic activities including the prediction of the Dst index. Radial basis function (RBF) networks are an important and popular network model for nonlinear system identification and dynamical modelling. A novel generalised multiscale RBF (MSRBF) network is introduced for Dst index modelling. The proposed MSRBF network can easily be converted into a linear-in-the-parameters form and the training of the linear network model can easily be implemented using an orthogonal least squares (OLS) type algorithm. One advantage of the new MSRBF network, compared with traditional single scale RBF networks, is that the new network is more flexible for describing complex nonlinear dynamical systems

Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics