The generalized manifold Galerkin (GMG) framework for structure-preserving model reduction on manifolds defines the reduced dynamics through the pullback τˇ=φ∗τ\check{\tau}=\varphi^{*}\tau of a nondegenerate (0,2)(0,2)-tensor field τ\tau along a smooth embedding φ\varphi of a low-dimensional reduced manifold into the full-order state manifold, and it is well posed only if τˇ\check{\tau} is again nondegenerate — Assumption 5.1 of the source framework. This assumption fails for natural embeddings (for instance the canonical linear embedding of R2n\mathbb{R}^{2n} into (R2N,ωcan)(\mathbb{R}^{2N},\omega_{\mathrm{can}}), the phase space of positions and momenta carrying the standard symplectic form), and while later work has settled special cases — nondegeneracy is automatic for Riemannian τ\tau, constructive symplectic-embedding routes exist for canonical Hamiltonian phase spaces, and genericity holds for constant port-Hamiltonian-type tensors with linear embedding structure — no characterization or construction is known for general nondegenerate tensor fields, notably the indefinite block tensor of Lagrangian manifold Galerkin, non-symmetric tensors, and nonconstant tensors on general manifolds. The problem asks for (a) a characterization of the pairs (τ,φ)(\tau,\varphi) with nondegenerate pullback — existence, genericity, and obstructions per structure class — and (b) constructive, certification-friendly criteria or algorithms producing embeddings that simultaneously guarantee nondegeneracy and approximate a given full-order solution set to prescribed accuracy, or impossibility results when the two requirements conflict.

Background

Repeated numerical solution of high-dimensional initial value problems — typically arising from semi-discretizing a partial differential equation (PDE) — is prohibitively expensive in many-query and real-time contexts. This is the domain of model order reduction (MOR): given a full-order model (FOM), construct a low-dimensional reduced-order model (ROM) that is cheap to evaluate and faithful to the FOM. Classical MOR restricts the state to a linear subspace; its achievable accuracy is governed by the Kolmogorov n-width of the solution set, that is, the best worst-case error attainable by n-dimensional linear subspaces. For transport- and wave-dominated problems these widths decay slowly, so linear subspaces fail; this motivates manifold MOR, in which the state is constrained to a low-dimensional submanifold φ(Mˇ)⊆M\varphi(\check{M})\subseteq M of the FOM state manifold MM, described by a smooth embedding φ:Mˇ→M\varphi:\check{M}\to M — a smooth injective map with injective differential that is a homeomorphism onto its image — together with a compatible smooth point reduction ϱ:M→Mˇ\varrho:M\to\check{M} satisfying ϱ∘φ=id\varrho\circ\varphi=\mathrm{id}.

The paper by Buchfink, Glas, Haasdonk, and Unger (arXiv:2312.01963; Physica D 468, 2024) introduced a differential-geometric framework unifying nonlinear projection and structure preservation within manifold MOR. Its central construction, the generalized manifold Galerkin (GMG) reduction, assumes that MM carries a nondegenerate (0,2)(0,2)-tensor field τ\tau: a smooth assignment of a bilinear form τ∣m\tau|_{m} on each tangent space TmMT_{m}M that is nondegenerate, i.e., invertible as a matrix in local coordinates. Important special cases are Riemannian metrics (symmetric, positive definite), symplectic forms (skew-symmetric, nondegenerate, and closed, meaning with vanishing exterior derivative; these underlie Hamiltonian dynamics), and the block tensor τLMG=(0τvτq0)\tau_{\mathrm{LMG}}=\begin{pmatrix}0&\tau_{v}\\ \tau_{q}&0\end{pmatrix} on the tangent bundle TQTQ of the configuration space of Lagrangian dynamics, where τv\tau_{v} is the Hessian of the Lagrangian with respect to the velocities and τq\tau_{q} is an additional freely chosen component (typically τq=τv\tau_{q}=\tau_{v}). Nondegeneracy of τ\tau identifies tangent vectors with covectors via the flat map ♭τ:X↦τ(X,⋅)\flat_{\tau}:X\mapsto\tau(X,\cdot), whose inverse is denoted ♯τ\sharp_{\tau}. The GMG reduction map is then

RGMG(m,v)  =  (ϱ(m),  ♯τˇ∘dφ∗∣ϱ(m)∘♭τ(v)),R^{\mathrm{GMG}}(m,v)\;=\;\bigl(\varrho(m),\;\sharp_{\check{\tau}}\circ d\varphi^{*}|_{\varrho(m)}\circ\flat_{\tau}(v)\bigr),

where dφ∗d\varphi^{*} is the pointwise dual (codifferential) of the differential dφd\varphi, mapping covectors on MM back to covectors on Mˇ\check{M}, and τˇ=φ∗τ\check{\tau}=\varphi^{*}\tau is the pullback tensor on Mˇ\check{M}, defined by (φ∗τ)∣mˇ(vˇ,wˇ)=τ∣φ(mˇ)(dφ vˇ,dφ wˇ)(\varphi^{*}\tau)|_{\check{m}}(\check{v},\check{w})=\tau|_{\varphi(\check{m})}(d\varphi\,\check{v},d\varphi\,\check{w}); in coordinates τˇ=Dφ⊤τ Dφ\check{\tau}=D\varphi^{\top}\tau\,D\varphi. Since the formula inverts τˇ\check{\tau}, the construction is well posed only under Assumption 5.1 of that paper: the embedding φ\varphi must be such that τˇ\check{\tau} is again nondegenerate on Mˇ\check{M}. Choosing τ\tau to be the Lagrangian block tensor or a symplectic form yields the Lagrangian and symplectic manifold Galerkin (LMG, SMG) structure-preserving reductions, for which the paper proves exact reproduction results — but always conditional on Assumption 5.1.

The assumption is delicate. If τ\tau is a Riemannian metric, φ∗τ\varphi^{*}\tau is automatically positive definite, hence nondegenerate. For other structure classes this fails: Example 5.14 of the source shows that even the natural linear embedding of R2n\mathbb{R}^{2n} into (R2N,ωcan)(\mathbb{R}^{2N},\omega_{\mathrm{can}}) — the phase space of positions and momenta carrying the canonical symplectic form ωcan=∑i=1Ndqi∧dpi\omega_{\mathrm{can}}=\sum_{i=1}^{N}dq^{i}\wedge dp^{i} — yields an identically degenerate pullback of the canonical symplectic form; for symplectic τ\tau, a nondegenerate pullback is equivalent to φ\varphi being a symplectic embedding (a diffeomorphism onto its image preserving the symplectic form, which also forces the reduced dimension to be even); and the block tensor τLMG\tau_{\mathrm{LMG}} is indefinite even for mechanical systems with positive definite velocity Hessian, so its pullback nondegeneracy is likewise an open condition rather than an automatic one. The paper observes the failure mode but provides no characterization of the pairs (τ,φ)(\tau,\varphi) for which Assumption 5.1 holds, and no constructive criteria linking nondegeneracy to the primary purpose of φ\varphi — approximating the FOM's solution manifold. Subsequent work has settled special cases (cotangent-lift and symplectic-autoencoder constructions for canonical Hamiltonian phase spaces — the cotangent lift being the classical construction that extends a configuration-space embedding to the cotangent bundles of positions and momenta and is automatically symplectic; genericity for port-Hamiltonian tensor structures with linear embedding maps, port-Hamiltonian systems being a dissipative, open-system generalization of Hamiltonian systems), while the question for general nondegenerate (0,2)(0,2)-tensor fields, and in combination with approximation requirements, remains open.

Problem Statement

Let MM be an NN-dimensional smooth manifold endowed with a nondegenerate (0,2)(0,2)-tensor field τ\tau, and let φ:Mˇ→M\varphi:\check{M}\to M be a smooth embedding of an nn-dimensional reduced manifold with n<Nn<N, so that the pullback τˇ=φ∗τ\check{\tau}=\varphi^{*}\tau is the (0,2)(0,2)-tensor field τˇ∣mˇ=Dφ∣mˇ⊤ τ∣φ(mˇ) Dφ∣mˇ\check{\tau}|_{\check{m}}=D\varphi|_{\check{m}}^{\top}\,\tau|_{\varphi(\check{m})}\,D\varphi|_{\check{m}} on Mˇ\check{M}.

(a) Characterize the pairs (τ,φ)(\tau,\varphi) for which τˇ=φ∗τ\check{\tau}=\varphi^{*}\tau is everywhere nondegenerate on Mˇ\check{M}: for which tensor structures τ\tau — indefinite symmetric, non-symmetric, nonconstant on a general manifold, or structure-induced as in the Lagrangian block tensor τLMG\tau_{\mathrm{LMG}} or port-Hamiltonian systems — and for which reduced dimensions nn such embeddings exist; is nondegeneracy generic among embeddings (for instance, holding on an open dense or residual set of embeddings in a Whitney CkC^{k} topology); and what sharp obstructions beyond pointwise linear algebra (such as the even-dimension constraint for symplectic τ\tau) apply?

(b) Given a compact set S⊂MS\subset M (for instance an attractor or the set of solution snapshots of a full-order system) and a tolerance ε>0\varepsilon>0, with the reduced manifold Mˇ\check{M} and its dimension nn fixed as part of the instance data, give constructive criteria or an algorithm producing an embedding φ\varphi, together with a compatible smooth point reduction ϱ\varrho with ϱ∘φ=id\varrho\circ\varphi=\mathrm{id}, such that φ∗τ\varphi^{*}\tau is nondegenerate on Mˇ\check{M} and dist⁡(S,φ(Mˇ))≤ε\operatorname{dist}(S,\varphi(\check{M}))\le\varepsilon — where dist⁡(S,φ(Mˇ)):=sup⁡s∈Sinf⁡mˇ∈Mˇd(s,φ(mˇ))\operatorname{dist}(S,\varphi(\check{M})):=\sup_{s\in S}\inf_{\check{m}\in\check{M}}d\bigl(s,\varphi(\check{m})\bigr) is the one-sided distance induced by a fixed Riemannian distance dd on MM (the Euclidean distance in the typical case M=RNM=\mathbb{R}^{N}) — or prove that no such embedding exists when the structure of τ\tau is incompatible with the geometry of SS or with the prescribed dimension nn (for instance, symplectic τ\tau with nn odd).

The problem is posed at the generality of the GMG framework, which admits arbitrary nondegenerate (0,2)(0,2)-tensor fields. The cases already settled in the literature do not by themselves resolve it: τ\tau Riemannian (nondegeneracy automatic), τ\tau a canonical symplectic form on a cotangent bundle — the manifold of positions and momenta — with cotangent-lift or symplectic-autoencoder embeddings, and constant port-Hamiltonian-type tensors with linear embedding structure (genericity). An answer that only re-derives these known cases leaves the problem open; a complete answer either covers general τ\tau and general manifolds, or establishes sharp boundaries — including incompatibility results — on the approximation quality achievable under enforced nondegeneracy. Parts (a) and (b) are separate demands, each resolving on its own terms: a characterization of the pairs (τ,φ)(\tau,\varphi) as in (a) answers (a) and leaves (b) open, and constructive criteria or an impossibility result as in (b) answers (b) and leaves (a) open; a complete resolution of the problem requires both, while a sharp boundary or obstruction established for a given structure class resolves the corresponding part for that class.

References

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