Reduced order modeling of fluid flows: Machine learning, Kolmogorov barrier, closure modeling, and partitioning

In this paper, we put forth a long short-term memory (LSTM) nudging framework for the enhancement of reduced order models (ROMs) of fluid flows utilizing noisy measurements. We build on the fact that in a realistic application, there are uncertainties in initial conditions, boundary conditions, model parameters, and/or field measurements. Moreover, conventional nonlinear ROMs based on Galerkin projection (GROMs) suffer from imperfection and solution instabilities due to the modal truncation, especially for advection-dominated flows with slow decay in the Kolmogorov width. In the presented LSTM-Nudge approach, we fuse forecasts from a combination of imperfect GROM and uncertain state estimates, with sparse Eulerian sensor measurements to provide more reliable predictions in a dynamical data assimilation framework. We illustrate the idea with the viscous Burgers problem, as a benchmark test bed with quadratic nonlinearity and Laplacian dissipation. We investigate the effects of measurements noise and state estimate uncertainty on the performance of the LSTM-Nudge behavior. We also demonstrate that it can sufficiently handle different levels of temporal and spatial measurement sparsity. This first step in our assessment of the proposed model shows that the LSTM nudging could represent a viable realtime predictive tool in emerging digital twin systems

Symbiotic Ocean Modeling Using Physics-Controlled Echo State Networks

We introduce a “symbiotic” ocean modeling strategy that leverages data-driven and machine learning methods to allow high- and low-resolution dynamical models to mutually benefit from each other. In this work we mainly focus on how a low-resolution model can be enhanced within a symbiotic model configuration. The broader aim is to enhance the representation of unresolved processes in low-resolution models, while simultaneously improving the efficiency of high-resolution models. To achieve this, we use a grid-switching approach together with hybrid modeling techniques that combine linear regression-based methods with nonlinear echo state networks. The approach is applied to both the Kuramoto–Sivashinsky equation and a single-layer quasi-geostrophic ocean model, and shown to simulate short-term and long-term behavior better than either purely data-based methods or low-resolution models. By maintaining key flow characteristics, the hybrid modeling techniques are also able to provide higher quality initial conditions for high-resolution models, thereby improving their efficiency.</p

Symbiotic Ocean Modeling Using Physics-Controlled Echo State Networks

We introduce a “symbiotic” ocean modeling strategy that leverages data-driven and machine learning methods to allow high- and low-resolution dynamical models to mutually benefit from each other. In this work we mainly focus on how a low-resolution model can be enhanced within a symbiotic model configuration. The broader aim is to enhance the representation of unresolved processes in low-resolution models, while simultaneously improving the efficiency of high-resolution models. To achieve this, we use a grid-switching approach together with hybrid modeling techniques that combine linear regression-based methods with nonlinear echo state networks. The approach is applied to both the Kuramoto–Sivashinsky equation and a single-layer quasi-geostrophic ocean model, and shown to simulate short-term and long-term behavior better than either purely data-based methods or low-resolution models. By maintaining key flow characteristics, the hybrid modeling techniques are also able to provide higher quality initial conditions for high-resolution models, thereby improving their efficiency.</p

Endogenous Persistence in an Estimated DSGE Model under Imperfect Information

We provide a tool for estimating DSGE models by BayesianMaximum-likelihood methods under very general information assumptions. This framework is applied to a New Keynesian model where we compare the standard approach, that assumes an informational asymmetry between private agents and the econometrician, with an assumption of informational symmetry. For the former, private agents observe all state variables including shocks, whereas the econometrician uses only data for output, inflation and interest rates. For the latter both agents have the same imperfect information set and this corresponds to what we term the 'informational consistency principle'. We first assume rational expectations and then generalize the model to allow some households and firms to form expectations adaptively. We find that in terms of model posterior probabilities, impulse responses, second moments and autocorrelations, the assumption of informational symmetry by rational agents significantly improves the model fit. We also find qualified empirical support for the heterogenous expectations model. JEL Classification: C11, C52, E12, E32.Imperfect Information, DSGE Model, Rational versus Adaptive Expectations, Bayesian Estimation

Reliability-based design optimization of shells with uncertain geometry using adaptive Kriging metamodels

Optimal design under uncertainty has gained much attention in the past ten years due to the ever increasing need for manufacturers to build robust systems at the lowest cost. Reliability-based design optimization (RBDO) allows the analyst to minimize some cost function while ensuring some minimal performances cast as admissible failure probabilities for a set of performance functions. In order to address real-world engineering problems in which the performance is assessed through computational models (e.g., finite element models in structural mechanics) metamodeling techniques have been developed in the past decade. This paper introduces adaptive Kriging surrogate models to solve the RBDO problem. The latter is cast in an augmented space that "sums up" the range of the design space and the aleatory uncertainty in the design parameters and the environmental conditions. The surrogate model is used (i) for evaluating robust estimates of the failure probabilities (and for enhancing the computational experimental design by adaptive sampling) in order to achieve the requested accuracy and (ii) for applying a gradient-based optimization algorithm to get optimal values of the design parameters. The approach is applied to the optimal design of ring-stiffened cylindrical shells used in submarine engineering under uncertain geometric imperfections. For this application the performance of the structure is related to buckling which is addressed here by means of a finite element solution based on the asymptotic numerical method