Projection-Based ROMs & Hyperreduction

• Projection-based ROMs are techniques that construct low-dimensional approximations of high-dimensional dynamical systems using tailored Galerkin or Petrov–Galerkin projections, preserving stability and structure.
• Hyperreduction methods reduce online computational cost by approximating nonlinear operators and residuals through techniques like DEIM, ECSW, and machine learning surrogates, making the cost independent of the full-order model.
• Combined, ROMs and hyperreduction enable significant speedups in simulating nonlinear structural and fluid mechanics, achieving accelerations from 10× up to 10^5× in practical large-scale applications.

Projection-based reduced order models (ROMs) and hyperreduction techniques are integral to enabling efficient simulation of large-scale nonlinear dynamical systems, particularly for parametric PDEs in structural mechanics, fluids, and multiphysics settings. Projection-based ROMs construct low-dimensional models via Petrov–Galerkin or Galerkin projection onto carefully selected trial and test subspaces (often using Proper Orthogonal Decomposition, POD), but their computational speedup can be limited when nonlinear and/or parameter-dependent operators are present—because projected residuals, forces, or Jacobians still require evaluation at the full-order (high-dimensional) scale. Hyperreduction overcomes this bottleneck via auxiliary approximations (e.g., empirical interpolation, energy-conserving quadrature, machine learning surrogates) that compress online evaluation of these quantities, making the cost independent of the full state dimension. This article reviews the mathematical structure, key algorithms, stability theory, state-of-the-art software, and research advances in projection-based ROMs and hyperreduction, with specific reference to large-scale nonlinear structural and fluid mechanics applications.

1. Mathematical Structure of Projection-Based ROMs

At the core of projection-based model reduction is the approximation of high-dimensional states $u(t;\mu)\in\mathbb{R}^N$ as $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$, where $V\in\mathbb{R}^{N\times r}$ is the reduced basis, and $a(t;\mu)\in\mathbb{R}^r$ are the reduced coordinates. For dynamical and nonlinear systems, the full model typically takes the semidiscrete form

$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$

with $M$ (mass), $C$ (damping), and $K$ (stiffness) symmetric positive-definite matrices, or, in general nonlinear form, as $\dot{u} = f(u, t; \mu)$. The projection step enforces that the residual of the reduced approximation is orthogonal (Galerkin) or optimal (Petrov–Galerkin/LSPG) with respect to a test basis $W$ (\cite{(Rizzi et al., 2020)}, \cite{(Bach et al., 2018)}). The reduced system thus has the form

$u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$0

with similar forms for nonlinear $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$1, where the reduction of nonlinear terms may require additional care for efficiency.

For time-discrete or steady residuals $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$2, the reduced residuals $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$3 are solved at each time step or Newton-iteration step, with the Jacobian appropriately projected as $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$4. In least-squares Petrov–Galerkin (LSPG), the test basis depends on the time-discrete Jacobian (\cite{(Rizzi et al., 2020)}).

Projection-based ROMs rigorously preserve essential structure, such as symmetry, positive-definiteness, and the Hamiltonian/Lagrangian formulation, provided the bases are constructed appropriately (\cite{(Bach et al., 2018)}, \cite{(Barnett et al., 2022)}). These inherited properties are crucial for ensuring stability and accuracy when deploying explicit or implicit time integration.

2. Hyperreduction: Concept and Main Techniques

While ROMs lower the number of dynamical variables, for general nonlinear systems the cost of evaluating $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$5 or $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$6, and thus the projected residual or Jacobian, often remains $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$7—the size of the full-order model (FOM). Hyperreduction eliminates this $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$8-dependence, enabling true online speedup by approximating these projected quantities within the reduced basis (\cite{(Rizzi et al., 2020)}, \cite{(Grimberg et al., 2020)}).

Core hyperreduction techniques include:

• Empirical Interpolation Method (EIM)/Discrete Empirical Interpolation Method (DEIM): Nonlinear vectors (forces, residuals) are interpolated via a collateral basis and a select set of "magic points," enabling their recovery from fewer degrees of freedom (\cite{(Cicci et al., 2022)}, \cite{(Rizzi et al., 2020)}).
• Energy-Conserving Sampling and Weighting (ECSW): Projects the reduced quantities via a weighted quadrature over a small set of mesh entities, preserving SPD structure and energy balance (\cite{(Grimberg et al., 2020)}, \cite{(Barnett et al., 2022)}, \cite{(Saccani et al., 2024)}, \cite{(Biondic et al., 7 Apr 2025)}).
• Gappy POD, GNAT: Residuals (or other nonlinear terms) are approximated from fewer sampled entries (gappy) and a collateral basis, often using a least-squares fit for the selection matrix (\cite{(Rodriguez et al., 2021)}, \cite{(Rizzi et al., 2020)}, \cite{(Zucatti et al., 2021)}).
• DNN/ML Surrogates for Reduced Operators: Instead of sampling from the FOM, train neural networks to directly map reduced states and parameters to reduced residuals/Jacobians; enables nonintrusive ROMs with runtime cost independent of $u(t;\mu)\approx u_0(\mu)+V a(t;\mu)$9 (\cite{(Cicci et al., 2022)}, \cite{(Bai et al., 2021)}).

These approaches aim to balance offline cost, online accuracy, structural preservation, and code intrusiveness.

3. Inherited Stability and Structure of ROMs and Hyperreduced Models

For explicit time-integration (e.g., central difference in structural mechanics), the stability of the ROM inherits critical properties from the FOM, provided the subspaces and hyperreduction preserve symmetry and positive-definiteness. The Galerkin-reduced matrices $V\in\mathbb{R}^{N\times r}$0, $V\in\mathbb{R}^{N\times r}$1, $V\in\mathbb{R}^{N\times r}$2 retain symmetric positive-definite character, while the largest eigenvalue of the reduced stiffness matrix $V\in\mathbb{R}^{N\times r}$3 is always less than or equal to that of the FOM by the Cauchy interlacing theorem (\cite{(Bach et al., 2018)}). The critical time step $V\in\mathbb{R}^{N\times r}$4 for explicit integration thus satisfies: $V\in\mathbb{R}^{N\times r}$5 where $V\in\mathbb{R}^{N\times r}$6 is the largest reduced eigenvalue and $V\in\mathbb{R}^{N\times r}$7 the Rayleigh damping parameter.

Hyperreduction via ECSW or symmetrized DEIM inherits this monotonic increase in time step and stability, provided the hyperreduced mass, damping, and stiffness matrices remain SPD. Conversely, nonsymmetric or purely collocative hyperreduction (e.g., Jacobian collocation without weights) can break this property and lead to unconditional instability (\cite{(Bach et al., 2018)}).

For iterative implicit solves (e.g., Newton in nonlinear structural mechanics or CFD), hyperreduction schemes such as ECSW ensure stability and monotonic convergence in LSPG by guaranteeing that the reduced Jacobian remains SPD (for ECSW) and consistent with the discrete optimality conditions (\cite{(Grimberg et al., 2020)}, \cite{(Biondic et al., 7 Apr 2025)}).

4. Hyperreduction Algorithms and Computational Workflow

Hyperreduction methods generally follow a two-phase workflow:

Offline phase:

• Selection of training data: snapshot states, residuals, or operator samples.
• Collateral basis construction (e.g., SVD or POD of residuals).
• Sampling set selection: greedy DEIM, QR for interpolation points, NNLS for ECSW weights.
• Precomputation of reduced operators or tensors for polynomial systems.

Online phase:

• Evaluation of large-scale quantities (forces, residuals, Jacobians) only at selected sample points.
• Reconstruction via sparse quadrature (ECSW), interpolation (DEIM), or regression surrogate (DNN).
• Reduced residual and Jacobian assembly, followed by reduced solve (explicit or Newton iterations).

For example, ECSW selects a reduced mesh and positive weights by solving a convex NNLS problem constrained to match reduced operator values (with early stopping for sparsity), and hyperreduced residuals/jacobians are then computed by summing only over these reduced mesh entities (\cite{(Grimberg et al., 2020)}, \cite{(Biondic et al., 7 Apr 2025)}, \cite{(Saccani et al., 2024)}).

For polynomial-nonlinearity systems where all full-order operators are available, all reduced tensor contractions can be precomputed, yielding a "hyper-reduction-free" (HRF) ROM where online cost depends solely on the reduced dimension and polynomial order (\cite{(Magargal et al., 3 Mar 2026)}). For machine learning surrogates, two neural networks are trained to approximate the reduced residual and Jacobian directly from reduced coordinates and parameters, bypassing physical sampling entirely (\cite{(Cicci et al., 2022)}, \cite{(Bai et al., 2021)}).

5. Performance, Scalability, and Practical Applications

Projection-based ROMs coupled with hyperreduction achieve wall-clock and CPU speedups that scale from $V\in\mathbb{R}^{N\times r}$8 (for basic DEIM/gappy-POD and small basis sizes) up to $V\in\mathbb{R}^{N\times r}$9–$a(t;\mu)\in\mathbb{R}^r$0 for ECSW in large-scale turbulent flow, structural, and design optimization applications (\cite{(Rizzi et al., 2020)}, \cite{(Grimberg et al., 2020)}, \cite{(Biondic et al., 7 Apr 2025)}, \cite{(Saccani et al., 2024)}).

Representative performance from recent studies:

Application	FOM size	Method	Speedup	Relative error	Reference
Blottner-sphere CFD	$a(t;\mu)\in\mathbb{R}^r$1	LSPG+ECSW, $a(t;\mu)\in\mathbb{R}^r$2	$a(t;\mu)\in\mathbb{R}^r$3	$a(t;\mu)\in\mathbb{R}^r$4–$a(t;\mu)\in\mathbb{R}^r$5	(Rizzi et al., 2020)
Ventricle inflation (elastodynamics)	$a(t;\mu)\in\mathbb{R}^r$6	Deep-HyROMnet	$a(t;\mu)\in\mathbb{R}^r$7	$a(t;\mu)\in\mathbb{R}^r$8	(Cicci et al., 2022)
Ahmed body wake (DES)	$a(t;\mu)\in\mathbb{R}^r$9	ECSW (local LSPG)	$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$0	$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$1	(Grimberg et al., 2020)
Aeronautical panel (structural)	$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$2	EED-ECSW (offline)	$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$3	indistinguishable PSD	(Saccani et al., 2024)
NACA0012 airfoil (DG)	$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$4–$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$5	LSPG+ECSW	$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$6–$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$7	$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$8	(Biondic et al., 7 Apr 2025)
N-body Biot–Savart vortex	$M\,\ddot{u}(t) + C\,\dot{u}(t) + K\,u(t) = f(t)\,,$9	PTROM (LSPG+GNAT)	$M$0	$M$1	(Rodriguez et al., 2021)

Hyperreduction enables ROM-based optimization with error certification and global convergence (via adaptive trust-region and on-the-fly empirical quadrature), yielding $M$2–$M$3 end-to-end speedups for complex PDE-constrained shape optimization (\cite{(Wen et al., 2022)}). Non-intrusive ROMs via operator learning (ML surrogates) achieve up to $M$4–$M$5 speedup for canonical problems, fully detaching online cost from the FOM dimension (\cite{(Bai et al., 2021)}).

6. Best Practices, Limitations, and Research Directions

Several best practices are evident:

• Structure preservation in projection (Galerkin/LSPG) and hyperreduction (e.g., ECSW with positive weights) is critical for stability, especially for explicit integration and energy-conserving systems (\cite{(Bach et al., 2018)}, \cite{(Grimberg et al., 2020)}).
• Hyperreduction error must be controlled in both reduced residual and dual-weighted error estimators to guarantee functional error tolerance, particularly for goal-oriented adaptive sampling in design applications (\cite{(Biondic et al., 7 Apr 2025)}).
• Sample selection tolerances must be carefully tuned; too loose a criterion can result in loss of convergence or degraded accuracy (\cite{(Magargal et al., 3 Mar 2026)}, \cite{(Grimberg et al., 2020)}).
• For polynomial structure, HRF ROMs provide hyperreduction-free, exact reduced models given full operator access, but offline storage can become expensive at high polynomial degree or basis size (\cite{(Magargal et al., 3 Mar 2026)}).
• Non-intrusive and data-driven techniques (operator learning/ML surrogates, Deep-HyROMnet) circumvent code modification but shift the offline cost and accuracy trade-off to ML surrogate expressivity and training (\cite{(Cicci et al., 2022)}, \cite{(Bai et al., 2021)}).

Current limitations include the mandates of full-operator access for HRF and the potential instability of poorly conditioned or non-structure-preserving hyperreduction schemes. In high-dimensional parameter spaces, offline pre-sampling remains a challenge; on-the-fly, adaptive, or globally convergent frameworks have been demonstrated to circumvent curse-of-dimensionality effects (\cite{(Wen et al., 2022)}).

Active research areas include scalable hyperreduction for high-order and multiphysics systems, hybrid model- and data-driven ROMs, robust goal-oriented error estimation, and plug-and-play multi-physics coupling using alternating Schwarz or similar domain-decomposition approaches (\cite{(Barnett et al., 2022)}).

7. Software and Frameworks

The described approaches are implemented in several open-source and production-level libraries, with Pressio exemplifying state-of-the-art capabilities for projection-based ROMs and hyperreduction (\cite{(Rizzi et al., 2020)}). Pressio provides:

• A C++11 header-only library with a uniform API, supporting direct integration with native high-performance codes as well as Python bindings (pressio4py).
• Support for Galerkin, LSPG, manifold-projected ROMs, and first-class hyperreduction (collocation, DEIM, algebraic, GNAT under development).
• Demonstrated performance on problems with $M$6 and wall-clock speedup of $M$7–$M$8.
• Comparisons with libROM, modred, pyMOR, and pyROM highlight Pressio's advantage in out-of-the-box hyperreduction, nonlinear manifolds, and HPC interoperation.

Other frameworks (libROM, pyMOR, pyROM) provide varying levels of support for POD/basis construction, Galerkin/LSPG projection, and, in some cases, hyperreduction or operator learning, but may require stricter code integration or re-compilation (\cite{(Rizzi et al., 2020)}, \cite{(Bai et al., 2021)}).

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References:

• (Bach et al., 2018): Stability conditions for the explicit integration of projection based nonlinear reduced-order and hyper reduced structural mechanics finite element models
• (Rizzi et al., 2020): Pressio: Enabling projection-based model reduction for large-scale nonlinear dynamical systems
• (Cicci et al., 2022): Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs
• (Zucatti et al., 2021): Data-Driven Closure of Projection-Based Reduced Order Models for Unsteady Compressible Flows
• (Magargal et al., 3 Mar 2026): Hyper-reduction-free reduced-order Newton solvers for projection-based model-order reduction of nonlinear dynamical systems
• (Grimberg et al., 2020): Mesh sampling and weighting for the hyperreduction of nonlinear Petrov-Galerkin reduced-order models with local reduced-order bases
• (Saccani et al., 2024): Accelerating Construction of Non-Intrusive Nonlinear Structural Dynamics Reduced Order Models through Hyperreduction
• (Biondic et al., 7 Apr 2025): A Goal-Oriented Adaptive Sampling Procedure for Projection-Based Reduced-Order Models with Hyperreduction
• (Wen et al., 2022): A globally convergent method to accelerate large-scale optimization using on-the-fly model hyperreduction: application to shape optimization
• (Rodriguez et al., 2021): Projection-tree reduced order modeling for fast N-body computations
• (Bai et al., 2021): Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators
• (Barnett et al., 2022): The Schwarz alternating method for the seamless coupling of nonlinear reduced order models and full order models