The practicable simulation-free nonlinear moment matching (NLMM) algorithm of Varona, Gebhart, Suk and Lohmann (2019) made Astolfi's steady-state (center-manifold) framework for nonlinear model order reduction implementable: the nonlinear invariant mapping that solves the invariance partial differential equation is replaced by a linear projection whose columns are obtained by solving full-order nonlinear algebraic equations at collocation instants, so that neither a simulation of the full model nor a numerical treatment of the PDE is needed. The price is the loss of essentially all guarantees: it is not known which classes of signal generators adequately characterize a given nonlinear system, the linear projection provably cannot satisfy the invariance equation in general, no a priori or a posteriori error bounds exist for the resulting reduced models, and neither the hyperparameters of the construction (collocation times, number of generators and shifts, tangential directions, deflation rank) nor the cost of up to r⋅Kr\cdot K full-order nonlinear solves has been analyzed. The construction of a theoretically grounded simulation-free nonlinear moment matching method for general nonlinear systems is open.

Background

Model order reduction (MOR) seeks low-dimensional surrogates of large-scale dynamical systems x˙=f(x,u)\dot{x}=f(x,u), y=h(x)y=h(x), x∈Rnx\in\mathbb{R}^n (the full-order model, FOM), so that a reduced-order model (ROM) of dimension r≪nr\ll n can be simulated, controlled, or optimized at a small fraction of the cost. For linear time-invariant (LTI) systems, interpolatory methods are a standard simulation-free route: the transfer function G(s)=C(sI−A)−1BG(s)=C(sI-A)^{-1}B is approximated by matching its value and derivatives (the 'moments' G(σi)G(\sigma_i), G′(σi)G'(\sigma_i), ...) at chosen interpolation points σi\sigma_i, which reduces to solving Sylvester equations (linear matrix equations of the form AX+XB=CAX+XB=C) and building projection matrices from Krylov subspaces (nested subspaces spanned by powers of AA applied to input directions).

Astolfi (2010) reinterpreted moment matching in the time domain so that it extends to nonlinear systems. The FOM is driven by an autonomous signal generator ω˙=s(ω)\dot{\omega}=s(\omega), u=ℓ(ω)u=\ell(\omega), a low-dimensional system whose trajectories are observable (the input distinguishes all generator states) and locally Poisson-stable (each trajectory neither decays to an equilibrium nor diverges, so that a persistent steady-state response is well defined). For the interconnected system, the moment of the FOM at (s,ℓ)(s,\ell) is the steady-state output map h∘νh\circ\nu, where ν\nu maps generator states to FOM states on the attracting invariant manifold (a center manifold) and solves the Sylvester-like invariance partial differential equation, equation (18) of the source paper,

∂ν∂ω(ω) s(ω)  =  f(ν(ω), ℓ(ω)).\frac{\partial\nu}{\partial\omega}(\omega)\,s(\omega) \;=\; f\big(\nu(\omega),\ \ell(\omega)\big).

A ROM (of any structure) achieves nonlinear moment matching if its own steady-state output under the same generator coincides with h∘νh\circ\nu; it then reproduces the FOM's asymptotic output behavior exactly for the input class generated by (s,ℓ)(s,\ell). The theory is generator-agnostic, and it was developed by Ionescu and Astolfi (2016), in the Scarciotti-Astolfi monograph (2017), and in a data-driven variant (Scarciotti and Astolfi 2017). The practical obstacle is that ν\nu solves a nonlinear, state-dependent PDE in high dimension, so exact nonlinear moment matching is unattainable for large-scale systems.

Varona, Gebhart, Suk and Lohmann (2019) proposed a practicable compromise (NLMM), a special case of the general nonlinear Petrov-Galerkin projection (a projection in which the equations are enforced against test mappings chosen independently of the state ansatz) in which the state ansatz x=ν(xr)x=\nu(x_r) is replaced by the linear ansatz x=Vx~rx=V\tilde{x}_r. This turns the invariance PDE into algebraic equations which, being underdetermined (nn equations for n⋅rn\cdot r unknowns), are solved column-wise at KK collocation instants tk∗t_k^* of the generator trajectories, with tangential directions rir_i (the input directions along which the individual interpolation conditions are imposed), shifts σi\sigma_i (interpolation points), and a deflation rank rdefr_{\mathrm{def}} (the rank retained after removing redundant basis columns) controlling the final basis. The offline phase solves up to r⋅Kr\cdot K nonlinear algebraic systems of full dimension nn (for instance by Newton iterations with initial guesses from linearized models) instead of simulating the FOM or discretizing the PDE; the method is demonstrated on the FitzHugh-Nagumo benchmark, a standard two-variable nonlinear excitable-system model. The authors are explicit about what is lost: the linear projection does not enforce the invariance equation; the linear/exponential generators inherited from the linear theory have a validity for characterizing nonlinear systems that is 'questionable and not completely clear yet'; and no prescription or error analysis is given for the hyperparameters. Four later lines cover restricted fragments of these gaps. For quadratic-bilinear systems (polynomial systems with linear and quadratic terms in the state), Khattak, Ahmad, Feng and Benner (2022) extended a posteriori error bounds (bounds computable from the reduced solution, after the fact) and greedy shift selection from the linear case. Huang and Astolfi (2025) decomposed the invariance PDE for multi-input multi-output (MIMO) polynomial nonlinear systems into recursively solvable Sylvester equations, yielding analytically parametrized ROMs with a stated precision of degree κ\kappa. Padhi and Gugercin (2026) proved that quadratic-manifold projections (state approximations of the form x≈V1xr+V2(xr⊗xr)x\approx V_1 x_r + V_2(x_r\otimes x_r), i.e. linear plus quadratic functions of the reduced coordinates) can match the nonlinear moments of an LTI FOM exactly, with closed-form projection matrices again obtained from Sylvester equations. Doebeli, Astolfi, Kalise, Moreschini, Scarciotti and Simard (2024, revised 2026) went furthest toward the general case: they compute a polynomial (monomial-basis) approximation of the invariance-equation solution itself by a Galerkin residual method with Newton iteration on the expansion coefficients - a simulation-free nonlinear ansatz demonstrated on general nonlinear systems up to state dimension of order 1000, with both linear and nonlinear signal generators - but its accuracy and residual are assessed by numerical experiment only, with no theorems or error bounds. None of these lines delivers provable guarantees - error bounds, residual control, or selection principles - for general nonlinear FOMs, for which the questions raised by the NLMM construction remain unanswered.

Problem Statement

For general nonlinear systems x˙=f(x,u)\dot{x}=f(x,u), y=h(x)y=h(x) of large state dimension nn, establish the theoretical foundations of the simulation-free nonlinear moment matching construction of Varona, Gebhart, Suk and Lohmann (2019). Two interlocking components jointly serve one objective - a reliable simulation-free, system-theoretic nonlinear model reduction method whose construction rests on a justified choice of signal generator and on a projection ansatz that provably respects the invariance equation:

(1) Signal-generator theory. Characterize which classes of signal generators (nonlinear, linear/exponential, constant, or problem-specific trajectories) adequately characterize a given nonlinear system for moment matching; in particular settle whether linear/exponential generators can adequately represent the steady-state behavior of general nonlinear systems, and give a principled, computable selection criterion in place of heuristic choice.

(2) Nonlinear projection ansatzes. Design and analyze computable nonlinear mappings ν(⋅)≠V⋅\nu(\cdot)\neq V\cdot that satisfy the invariance equation ∂ν∂ωs(ω)=f(ν(ω),ℓ(ω))\frac{\partial\nu}{\partial\omega}s(\omega)=f(\nu(\omega),\ell(\omega)) exactly or with a provably controlled residual, remain simulation-free in their construction, and are tractable for large nn - replacing the current linear projection, which cannot satisfy the invariance equation in general.

Quantitative output-error bounds for the reduced models, prescriptions or bounds-driven selection rules for the hyperparameters (collocation times, number of generators and shifts, tangential directions, deflation rank), and analyses of the offline cost of the up-to-r⋅Kr\cdot K full-order nonlinear solves and its scaling in nn are background for the two components above rather than conditions of this problem.

For general nonlinear systems, component (1) asks for a precise formulation of what it means for a class of signal generators to characterize a nonlinear system, an identified adequate class, and a computable selection criterion; component (2) asks for a computable nonlinear ansatz that satisfies the invariance equation exactly or with a provably controlled residual, remains simulation-free, and is tractable for large nn. Neither component is settled by partial results: a counterexample against a single candidate class - for instance, an explicit system on which a linear/exponential generator provably fails to characterize the steady-state behavior - is informative partial progress on component (1), but it does not settle that component, since other generator classes may still be adequate and selectable; and results restricted to linear, quadratic-bilinear, or polynomial systems resolve only the corresponding subquestions.

References

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