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 ...

We demonstrate the synthesis of sparse sampling and machine learning to characterize and model complex, nonlinear dynamical systems over a range of bifurcation parameters. First, we construct modal libraries using the classical proper orthogonal decomposition to uncover dominant low-rank coherent structures. Here, nonlinear libraries are also constructed in order to take advantage of the discrete empirical interpolation method and projection that allows for the approximation of nonlinear terms in a low-dimensional way. The selected sampling points are shown to be nearly optimal sensing locations for characterizing the underlying dynamics, stability, and bifurcations of complex systems. The use of empirical interpolation points and sparse representation facilitate a family of local reduced-order models for each physical regime, rather than a higher-order global model, which has the benefit of physical interpretability of energy transfer between coherent structures. In particular, the discrete interpolation points and nonlinear modal libraries are used for sparse representation to classify the dynamic bifurcation regime in the complex Ginzburg-Landau equation. It is shown that nonlinear point measurements are more effective than linear measurements when sensor noise is present.

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