This work proposes a method for model reduction of finite-volume models that guarantees the resulting reduced-order model is conservative, thereby preserving the structure intrinsic to finite-volume discretizations. The proposed reduced-order models associate with optimization problems characterized by a minimum-residual objective function and nonlinear equality constraints that explicitly enforce conservation over subdomains. Conservative Galerkin projection arises from formulating this optimization problem at the time-continuous level, while conservative least-squares Petrov-Galerkin (LSPG) projection associates with a time-discrete formulation. We equip these approaches with hyper-reduction techniques in the case of nonlinear flux and source terms, and also provide approaches for handling infeasibility. In addition, we perform analyses that include deriving conditions under which conservative Galerkin and conservative LSPG are equivalent, as well as deriving a posteriori error bounds. Numerical experiments performed on a parameterized quasi-1D Euler equation demonstrate the ability of the proposed method to ensure not only global conservation, but also significantly lower state-space errors than nonconservative reduced-order models such as standard Galerkin and LSPG projection. (d) a posteriori bounds (Section 5.3) for the error in the quantities conserved over subdomains (Theorem 5.3), in the null space (Lemma 5.1) and row space (Lemma 5.2) of the constraints, in the full state (Theorem 5.2), and in the conserved quantities (Lemma 5.3 and Theorem 5.3).3. Numerical experiments on a parameterized quasi-1D Euler equation associated with modeling inviscid compressible flow in a converging-diverging nozzle (Section 6). These experiments demonstrate the merits of the proposed method and illustrate the importance of ensuring reduced-order models are globally conservative.We remark that this work was first presented publically at the "Recent Developments in Numerical Methods for Model Reduction" workshop at the Institut Henri Poincaré on November 10, 2016.Other works have also explored formulating reduced-order models that associate with constrained optimization problems. Zimmermann et al.[59] equip equality 'aerodynamic constraints' to ROMs applied to steady-state external flows, where the constraints associate with matching experimental data or target performance metrics in a design setting. Recently, Reddy et al. [45] propose equipping the time-discrete Galerkin ROM with inequality constraints that enforce solution positivity or a bound on the gas-void fraction. Relatedly, Fick et al. [26] proposed a modified Galerkin optimization problem applicable to the incompressible Navier-Stokes equations, where the inequality constraints associate with bounds on the generalized coordinates; these bounds correspond to the extreme values of the generalized coordinates arising during the training simulations.The remainder of this paper is organized as follows. Section 2 describes finite-volume discretizations of conservation ...

This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space-time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space-time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted 2 -norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time Gauss-Newton with Approximated Tensors (GNAT) variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov-Galerkin projection include a reduction of both the spatial and temporal dimensions of the dynamical system, and a priori error bounds that bound the solution error by the best space-time approximation error and whose stability constants exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy for a fixed spatio-temporal discretization.

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.

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