In the present paper we consider a 2-D shallow-water equations (SWE) model on a β-plane solved using an alternating direction fully implicit (ADI) finite-difference scheme on a rectangular domain. ...The scheme was shown to be unconditionally stable for the linearized equations.
The discretization yields a number of nonlinear systems of algebraic equations. We then use a proper orthogonal decomposition (POD) to reduce the dimension of the SWE model. Due to the model nonlinearities, the computational complexity of the reduced model still depends on the number of variables of the full shallow – water equations model. By employing the discrete empirical interpolation method (DEIM) we reduce the computational complexity of the reduced order model due to its depending on the nonlinear full dimension model and regain the full model reduction expected from the POD model.
To emphasize the CPU gain in performance due to use of POD/DEIM, we also propose testing an explicit Euler finite difference scheme (EE) as an alternative to the ADI implicit scheme for solving the swallow water equations model.
We then proceed to assess the efficiency of POD/DEIM as a function of number of spatial discretization points, time steps, and POD basis functions. As was expected, our numerical experiments showed that the CPU time performances of POD/DEIM schemes are proportional to the number of mesh points. Once the number of spatial discretization points exceeded 10000 and for 90 DEIM interpolation points, the CPU time decreased by a factor of 10 in case of POD/DEIM implicit SWE scheme and by a factor of 15 for the POD/DEIM explicit SWE scheme in comparison with the corresponding POD SWE schemes. Moreover, our numerical tests revealed that if the number of points selected by DEIM algorithm reached 50, the approximation errors due to POD/DEIM and POD reduced systems have the same orders of magnitude, thus supporting the theoretical results existing in the literature.
This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key ...requirement for a successful reduced order solution is that reduced order Karush–Kuhn–Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for proper orthogonal decomposition (POD) ROM data assimilation using both Galerkin and Petrov–Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov–Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.
To help create a comfortable and healthy indoor and outdoor environment in which to live, there is a need to understand turbulent air flows within the urban environment. To this end, building on a ...previously reported method 1, we develop a fast-running Non-Intrusive Reduced Order Model (NIROM) for predicting the turbulent air flows found within an urban environment. To resolve larger scale turbulent fluctuations, we employ a Large Eddy Simulation (LES) model and solve the resulting computational model on unstructured meshes. The objective is to construct a rapid-running NIROM from these results that will have ‘similar’ dynamics to the original LES model. Based on Proper Orthogonal Decomposition (POD) and machine learning techniques, this Reduced Order Model (ROM) is six orders of magnitude faster than the high-fidelity LES model and we demonstrate how ‘similar’ it can be to the high-fidelity model by comparing statistical quantities such as the mean flows, Reynolds stresses and probability densities of the velocities. We also include validation of the high-fidelity model against data from wind tunnel experiments.
This paper represents a key step towards the use of reduced order modelling for operational purposes with the tantalising possibility of it being used in place of Gaussian plume models, and the potential for greatly improved model fidelity and confidence.
•A Non-Intrusive Reduced Order Model (NIROM) is constructed using a Gaussian Regression Process machine learning method.•NIROM is used to predict the turbulent flows found within an urban neighbourhood of the Elephant and Castle area of London.•First implementation of such a model into an advanced 3D unstructured finite element mesh fluid model (Fluidity).•The model is several orders of magnitude faster than the high-fidelity Large Eddy Simulation (LES) model.
This paper presents a new nonlinear non‐intrusive reduced‐order model (NL‐NIROM) that outperforms traditional proper orthogonal decomposition (POD)‐based reduced order model (ROM). This improvement ...is achieved through the use of auto‐encoder (AE) and self‐attention based deep learning methods. The novelty of this work is that it uses stacked auto‐encoder (SAE) network to project the original high‐dimensional dynamical systems onto a low dimensional nonlinear subspace and predict fluid dynamics using an self‐attention based deep learning method. This paper introduces a new model reduction neural network architecture for fluid flow problem, as well as, a linear non‐intrusive reduced order model (L‐NIROM) based on POD and self‐attention mechanism. In the NL‐NIROM, the SAE network compresses high‐dimensional physical information into several much smaller sized representations in a reduced latent space. These representations are expressed by a number of codes in the middle layer of SAE neural network. Then, those codes at different time levels are trained to construct a set of hyper‐surfaces using self‐attention based deep learning methods. The inputs of the self‐attention based network are previous time levels' codes and the outputs of the network are current time levels' codes. The codes at current time level are then projected back to the original full space by the decoder layers in the SAE network. The capability of the new model, NL‐NIROM, is demonstrated through two test cases: flow past a cylinder, and a lock exchange. The results show that the NL‐NIROM is more accurate than the popular model reduction method namely POD based L‐NIROM.
This article presents a new reduced order model based upon proper orthogonal decomposition (POD) for solving the Navier–Stokes equations. The novelty of the method lies in its treatment of the ...equation's non-linear operator, for which a new method is proposed that provides accurate simulations within an efficient framework. The method itself is a hybrid of two existing approaches, namely the quadratic expansion method and the Discrete Empirical Interpolation Method (DEIM), that have already been developed to treat non-linear operators within reduced order models. The method proposed applies the quadratic expansion to provide a first approximation of the non-linear operator, and DEIM is then used as a corrector to improve its representation. In addition to the treatment of the non-linear operator the POD model is stabilized using a Petrov–Galerkin method. This adds artificial dissipation to the solution of the reduced order model which is necessary to avoid spurious oscillations and unstable solutions.
A demonstration of the capabilities of this new approach is provided by solving the incompressible Navier–Stokes equations for simulating a flow past a cylinder and gyre problems. Comparisons are made with other treatments of non-linear operators, and these show the new method to provide significant improvements in the solution's accuracy.
Deep learning techniques for fluid flow modelling have gained significant attention in recent years. Advanced deep learning techniques achieve great progress in rapidly predicting fluid flows without ...prior knowledge of the underlying physical relationships. However, most of existing researches focused mainly on either sequence learning or spatial learning, rarely on both spatial and temporal dynamics of fluid flows (Reichstein et al., 2019). In this work, an Artificial Intelligence (AI) fluid model based on a general deep convolutional generative adversarial network (DCGAN) has been developed for predicting spatio-temporal flow distributions. In deep convolutional networks, the high-dimensional flows can be converted into the low-dimensional “latent” representations. The complex features of flow dynamics can be captured by the adversarial networks. The above DCGAN fluid model enables us to provide reasonable predictive accuracy of flow fields while maintaining a high computational efficiency. The performance of the DCGAN is illustrated for two test cases of Hokkaido tsunami with different incoming waves along the coastal line. It is demonstrated that the results from the DCGAN are comparable with those from the original high fidelity model (Fluidity). The spatio-temporal flow features have been represented as the flow evolves, especially, the wave phases and flow peaks can be captured accurately. In addition, the results illustrate that the online CPU cost is reduced by five orders of magnitude compared to the original high fidelity model simulations. The promising results show that the DCGAN can provide rapid and reliable spatio-temporal prediction for nonlinear fluid flows.
•A deep convolutional GAN (DCGAN) is developed for large data-driven fluid modelling.•First use of DCGANs for predicting spatio-temporal nonlinear fluid flows.•Predictive results from DCGAN and high fidelity model are in a good agreement.•Using DCGAN the computational cost is reduced by five orders of magnitude.•The DCGAN is a robust and efficient tool for predictive modelling of fluid flows.
This article presents the first Reduced Order Model (ROM) that efficiently resolves the angular dimension of the time independent, mono-energetic Boltzmann Transport Equation (BTE). It is based on ...Proper Orthogonal Decomposition (POD) and uses the method of snapshots to form optimal basis functions for resolving the direction of particle travel in neutron/photon transport problems. A unique element of this work is that the snapshots are formed from the vector of angular coefficients relating to a high resolution expansion of the BTE's angular dimension. In addition, the individual snapshots are not recorded through time, as in standard POD, but instead they are recorded through space. In essence this work swaps the roles of the dimensions space and time in standard POD methods, with angle and space respectively.
It is shown here how the POD model can be formed from the POD basis functions in a highly efficient manner. The model is then applied to two radiation problems; one involving the transport of radiation through a shield and the other through an infinite array of pins. Both problems are selected for their complex angular flux solutions in order to provide an appropriate demonstration of the model's capabilities. It is shown that the POD model can resolve these fluxes efficiently and accurately. In comparison to high resolution models this POD model can reduce the size of a problem by up to two orders of magnitude without compromising accuracy. Solving times are also reduced by similar factors.
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