Scattered Data Approximation

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena, such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physics-informed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 1D and 2D Burgers' equations. A speedup of up to 2.6 for 1D Burgers' and a speedup of 11.7 for 2D Burgers' equations are achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique. Finally, a posteriori error bounds for the NM-ROMs are derived that take account of the hyper-reduced operators.

A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, such as advection-dominated flow phenomena, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed an efficient nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models (FOMs). The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 2D Burgers' equations with a high Reynolds number. A speed-up of up to 11.7 for 2D Burgers' equations is achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique.

Efficient nonlinear manifold reduced order model.

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Operator inference for non-intrusive model reduction with quadratic manifolds

In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations

We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov–Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.

/pdf/space-time-registration-based-model-reduction-of-1nquiuq89j.pdf

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov-Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.

We present a general approach for the treatment of parameterized geometries in projection-based model order reduction. During the offline stage, given (i) a family of parameterized domains $\{ \Omega_{\mu}: \mu \in \mathcal{P} \} \subset \mathbb{R}^D$ where $\mu \in \mathcal{P} \subset \mathbb{R}^P$ denotes a vector of parameters, (ii) a parameterized mapping ${\Phi}_{\mu}$ between a reference domain $\Omega$ and the parameter-dependent domain $\Omega_{\mu}$, and (iii) a finite element triangulation of $\Omega$, we resort to an empirical quadrature procedure to select a subset of the elements of the grid. During the online stage, we first use the mapping to "move" the nodes of the selected elements and then we use standard element-wise residual evaluation routines to evaluate the residual and possibly its Jacobian. We discuss how to devise an online-efficient reduced-order model and we discuss the differences with the more standard "map-then-discretize" approach (e.g., Rozza, Huynh, Patera, ACME, 2007); in particular, we show how the discretize-then-map framework greatly simplifies the implementation of the reduced-order model. We apply our approach to a two-dimensional potential flow problem past a parameterized airfoil, and to the two-dimensional RANS simulations of the flow past the Ahmed body.

A discretize-then-map approach for the treatment of parameterized geometries in model order reduction.

A discretize-then-map approach for the treatment of parameterized geometries in model order reduction

We present a general -- i.e., independent of the underlying equation -- registration procedure for parameterized model order reduction. Given the spatial domain $\Omega \subset \mathbb{R}^2$ and the manifold $\mathcal{M}= \{ u_{\mu} : \mu \in \mathcal{P} \}$ associated with the parameter domain $\mathcal{P} \subset \mathbb{R}^P$ and the parametric field $\mu \mapsto u_{\mu} \in L^2(\Omega)$, our approach takes as input a set of snapshots $\{ u^k \}_{k=1}^{n_{\rm train}} \subset \mathcal{M}$ and returns a parameter-dependent bijective mapping ${\Phi}: \Omega \times \mathcal{P} \to \mathbb{R}^2$: the mapping is designed to make the mapped manifold $\{ u_{\mu} \circ {\Phi}_{\mu}: \, \mu \in \mathcal{P} \}$ more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition $\{ \Omega_{q} \}_{q=1} ^{N_{\rm dd}}$ of the domain $\Omega$ such that $\Omega_1,\ldots,\Omega_{N_{\rm dd}}$ are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain, a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.

Lei Zhang

Papers

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

A discretize-then-map approach for the treatment of parameterized geometries in model order reduction.

A discretize-then-map approach for the treatment of parameterized geometries in model order reduction

Registration-based model reduction in complex two-dimensional geometries