Nonlinear Reduced Order Modeling

• Nonlinear reduced order modeling is a suite of methods that constructs low-dimensional surrogate models to accurately capture the intrinsic nonlinear dynamics in high-dimensional systems.
• Techniques span operator-theoretic lifting, data-driven manifold learning, and structure-preserving reductions to handle slow-decaying Kolmogorov n-width scenarios.
• These approaches significantly reduce computational cost—often achieving 200×–600× speedups—while maintaining accuracy and preserving essential physical invariants.

Nonlinear reduced order modeling (ROM) refers to a collection of methodologies for systematically constructing low-dimensional surrogate models that accurately capture the essential dynamics and structure of high-dimensional nonlinear systems, especially those governed by nonlinear partial differential equations (PDEs) or nonlinear dynamical systems. While linear-subspace ROMs (e.g., POD-Galerkin) rely on a fixed linear basis, nonlinear ROMs utilize a nonlinear reduced representation—such as lifted spaces, dynamically evolving or data-driven manifolds, or structure-preserving reductions—to approximate the solution space more efficiently when the Kolmogorov $n$-width decays slowly (as in advection- or convection-dominated phenomena). These methods achieve a significant reduction in computational cost without sacrificing the intrinsic nonlinear characteristics of the original system.

1. Classical and Operator-Theoretic Nonlinear Reduction

Nonlinear ROMs can be categorized into analytically driven and data-driven frameworks.

Operator-theoretic approaches: Koopman-based ROMs

A prominent class is the Koopman-based nonlinear reduced order modeling strategy, which seeks a finite-dimensional lifting of the original nonlinear state-space to a space of observables where the evolution is approximately linear. Given a discrete-time nonlinear process

$$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$

a set of nonlinear lifting functions $\Psi_{KG}(x)$ is selected (via, e.g., Kalman-based GSINDy sparse regression) from a candidate library of monomials, trigonometric, radial basis, and other nonlinear features. The lifted state

$$\Psi(x) = \begin{bmatrix} x \ \Psi_{KG}(x) \end{bmatrix}\in\mathbb{R}^N$$

(where typically $N\gg n$) satisfies an approximate linear evolution

$z_{k+1} = A z_k + B u_k, \quad z_k = \Psi(x_k),$

where $A\in\mathbb{R}^{N\times N}$ and $B\in\mathbb{R}^{N\times m}$ are determined by least-squares fit to data. This high-dimensional linear predictor then undergoes further dimensionality reduction using proper orthogonal decomposition (POD), projecting the lifted coordinates onto dominant POD modes: $$q_k = \Phi_r^T (z_k-\bar z)\in\mathbb{R}^r, \qquad z_k\approx \bar z + \Phi_r q_k,$$ with $r\ll N$ and $$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$0 the leading eigenvectors of the empirical covariance of the lifted snapshots $$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$1. The projected evolution is

$$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$2

and system output/state reconstruction is given by $$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$3, with $$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$4 selecting the original state entries. This approach achieves substantial reduction—e.g., reducing a 16-dimensional Koopman predictor to $$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$5 dimensions—while preserving predictive accuracy in nonlinear process systems (Zhang et al., 2024).

Variational and invariant-manifold reductions

In dispersive nonlinear PDEs (e.g., the nonlinear Schrödinger equation), classical nonlinear ROMs include the reduced-Lagrangian approach—parametrizing the solution by a small number of time-dependent variables in a variational ansatz—and the "reduced-order nonlinear solutions" (RONS) methodology, which projects the full evolution onto the tangent space of a finite-dimensional ansatz manifold (Anderson et al., 2022). The RONS framework is robust to absence of variational structure and systematically yields an ODE system for the ansatz parameters, even for modified or nonvariational PDEs. While the reduced-Lagrangian approach may fail to capture group velocity or translational effects in general frames, RONS recovers correct dynamical evolution even when the Lagrangian formulation is degenerate.

2. Data-driven Nonlinear Manifolds and Machine Learning

Recent developments leverage advances in representation learning to construct data-driven, nonlinear low-dimensional manifolds on which the reduced system's dynamics are modeled explicitly or implicitly.

Nonlinear encoding and manifold learning

A class of approaches utilizes autoencoders—shallow or deep neural networks trained on simulation data—to learn nonlinear mappings between the high-dimensional physical state and a latent space of moderate dimension. For instance, a shallow autoencoder

$$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$6

with decoder $$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$7 provides a nonlinear manifold $$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$8, where $$x_{k+1} = f(x_k, u_k), \qquad x_k\in\mathbb{R}^n,\; u_k\in\mathbb{R}^m,$$9 are latent variables. The reduced dynamics are formulated via least-squares Petrov–Galerkin projection of the discretized (e.g., backward-Euler) residual onto the tangent space of the manifold, with Gauss–Newton optimization performed in the latent coordinates (Kim et al., 2020).

Manifold learning techniques such as ISOMAP are sometimes applied to high-fidelity snapshot data, yielding intrinsically low-dimensional embeddings that more accurately capture nonlinear phenomena such as shocks and discontinuities in compressible flows, which are not efficiently represented by linear subspaces (Mufti et al., 2024).

Physics-based neural ROMs

Smooth neural field ROMs (SNF-ROM) replace the linear subspace approximation with an implicit, coordinate-based neural field,

$\Psi_{KG}(x)$0

with $\Psi_{KG}(x)$1 a sine-activated MLP mapping spatial coordinates and latent state vectors to field values. The entire time and parameter dependence is encoded in a smooth, low-dimensional latent trajectory $\Psi_{KG}(x)$2 through a regularized MLP. Evolution of $\Psi_{KG}(x)$3 is obtained by Galerkin projection and automatic differentiation, ensuring that the reduced trajectories remain on a smooth nonlinear manifold and that the physics-based residuals and their Jacobians are accurately computed (Puri et al., 2024).

Latent dynamics models (LDMs) generalize the approach by requiring that the time evolution of the latent variables $\Psi_{KG}(x)$4 be itself a neural ODE, parameterized by convolutional and affine mechanisms to retain spatial coherence and parametric dependence (Farenga et al., 2024). The training process aims for time-continuous generalization and stability with rigorous error bounds.

3. Structure-Preserving Methods and Intrusive Reductions

Preservation of the geometric and algebraic structure of the governing equations—especially symplectic, Poisson, or other invariant properties—is central in many nonlinear ROM frameworks.

Hamiltonian and Poisson-structure ROMs

For systems with Hamiltonian or Poisson structure, nonlinear model reduction methodology enforces the preservation of invariants (energy, momentum, Casimirs) through careful construction of reduced bases and nonlinear closures. For instance, the ROM of the Ablowitz–Ladik equation is derived by Galerkin projection onto a reduced POD basis, with non-canonical Hamiltonian structure invariant under the projection. The quadratic nonlinearity in the Poisson matrix is efficiently approximated via DEIM and tensor techniques, maintaining the conservation properties and enabling robust, long-time integration (Uzunca et al., 2022).

Structure-preserving discretizations, such as the midpoint or exponential midpoint rule, ensure that continuous invariants are retained at the reduced-order level. These approaches also generalize to parametric and geometrically nonlinear solid mechanics (e.g., using modal derivatives and parametric reduced tensors) (Saccani et al., 8 Dec 2025).

Nonlinear Normal Modes and Invariant Manifolds

Nonlinear normal modes (NNMs) generalize the modal decomposition of linear systems to nonlinear settings by describing dynamics on invariant manifolds parameterized by a small set of master coordinates. Modern approaches employ machine learning (e.g., autoencoders and LSTM latent regression) to discover NNM coordinates from data and forecast system evolution under general forcing (Simpson et al., 2020). While not always yielding analytic low-dimensional ODEs, this data-driven NNM-based reduction robustly captures strongly nonlinear behavior with few degrees of freedom.

Invariant-manifold approaches also target nonlinear correlations in temporal POD coefficients, using sparse polynomial regression ("slaving maps" and SINDy) to uncover a minimal set of active driving modes and express the remaining as nonlinear functions of these (Callaham et al., 2021). This often results in low-dimensional reduced dynamics (e.g., coupled Stuart–Landau oscillators) that stabilize and compress the predictive ROM.

4. Hyper-Reduction and Implementation Strategies

Despite their intrinsic dimensional compressiveness, the main computational bottleneck in nonlinear ROMs is evaluation of nonlinear operators at each time step or Newton iteration. Hyper-reduction strategies address this challenge:

• Empirical interpolation (EIM/DEIM): These methods approximate nonlinear terms by projecting them onto a small set of sampled components (or spatial locations) selected for their ability to reconstruct the full operator from reduced information. Tensor acceleration techniques further improve online efficiency, as in the Hamiltonian and Burgers' ROMs (Uzunca et al., 2022, Kim et al., 2020).
• Component-wise hyperreduction: Component-based problems are addressed by constructing reduced bases and quadrature rules for each archetype component, enabling systems with many parameters and topological variation to be efficiently assembled and solved. Online adaptive selection of hyperreduction fidelity is guided by posteriori error bounds (e.g., Brezzi-Rappaz-Raviart) for robust and parameter-flexible simulations (Ebrahimi et al., 3 Jan 2025).
• Nonintrusive frameworks: For industrial-scale nonlinear structural mechanics, nonintrusive ROMs combine snapshot-based data extraction from commercial solvers, in-house offline reduction, and distributed-memory online solution, using operator compression and domain decomposition to scale to millions of degrees of freedom (Casenave et al., 2018).

5. Predictive Control and Robust Model Reduction

Nonlinear ROMs enable deployment of advanced feedback and optimal control strategies otherwise infeasible in high-dimensional full models.

• Reduced-order predictive control: Example: Reduced-order Koopman MPC exploits linear control synthesis on the reduced lifted state, with robustification (tube-based MPC) providing constraint satisfaction and disturbance rejection. In chemical process benchmarks, the robust reduced-order controller achieves closed-loop tracking and regulation performance that matches or exceeds full-order MPC at less than half the computational effort (Zhang et al., 2024).
• Slow-manifold-based reduction for control: For general nonlinear dynamical systems with a timescale separation, model reduction on the slow manifold (intersection of unstable/stable manifolds), expressed in Koopman isostable/phase coordinates, yields compact control-oriented ROMs that are directly interpreted for feedback design. This facilitates simple control synthesis in complex biological and oscillatory regimes, such as neuron silencing or circadian rhythm manipulation (Wilson, 15 Jul 2025).

6. Comparative Performance and Challenges

Extensive numerical experiments across a range of nonlinear PDEs, dynamical systems, and industrial test cases confirm the superior accuracy, stability, and computational efficiency of nonlinear ROMs in regimes where linear-subspace approaches fail (e.g., advection-dominated transport, shocks, geometric nonlinearity).

• Accuracy: Nonlinear ROMs consistently achieve lower relative errors (e.g., 3.5% in compressible flow with shocks (Mufti et al., 2024), ∼1% in thermal-structure parametric analysis (Ebrahimi et al., 3 Jan 2025)), and maintain long-term invariants where required.
• Computational speed-up: Hyper-reduction and tensor acceleration yield up to 200×–600× speedups, enabling multi-query and real-time scenarios (Puri et al., 2024, Tantaroudas et al., 16 Mar 2026).
• Limits and extensions: Offline training cost and interpretability remain challenges, especially for data-driven neural ROMs. Maintaining structure and ensuring generalization out-of-distribution are ongoing research directions, as are manifold adaptation and stability for highly nonlinear and high-codimensional dynamics.

7. Synthesis and Outlook

Nonlinear reduced order modeling unifies advanced analytical, numerical, and data-driven strategies for drastic model order reduction in systems with significant nonlinearities. By explicitly modeling the intrinsic nonlinear structure of solution manifolds—through operator-theoretic lifting, data-driven nonlinear embeddings, invariant manifolds, or carefully selected nonlinear features—these models yield high-fidelity, physically consistent, and computationally efficient surrogates. Current state-of-the-art algorithms blend rigorous mathematical reduction (e.g., projection, Galerkin, structure preservation) with machine learning and optimization techniques to extend applicability to complex, multi-scale, parameter-rich, and real-time applications across physics, engineering, and biology. Future developments will likely focus on further automation of nonlinear feature selection, enhancing error certification and interpretability, and integrating nonintrusive, black-box, and hyper-reduction techniques for industrial-scale nonlinear simulation and control.