Space by the Laplacian, Time by Koopman

A sheaf network reads space with a Laplacian, but a spacetime problem such as flow matching or simulation needs a temporal operator too. Let the Koopman operator be the temporal half, glued to the spatial laplacian into a single spacetime generator, with the uncoupled case a special instance of a genuinely mixing one.

Space is a Laplacian; time is missing

A graph or sheaf network propagates with the Laplacian . This operator encodes the base’s geometry and enforces consistency across space, one snapshot at a time. It is the only operator a plain sheaf or graph network carries, a model of place with no model of dynamics.

Time is a Koopman action

Model the temporal evolution of a field’s observables by the koopman-operator over a totally ordered time , continuous or discrete alike, as or the powers . Because Koopman is linear, the temporal half is a linear operator even when the underlying dynamics is nonlinear. This is the same kind of linearisation the spatial Laplacian already is, so both halves of a spacetime problem can be operators of the same character, one per axis.

One spacetime generator, and the mixing case is primary

Fields live on . The general object is a single spacetime generator on observables over spacetime, coupling place and dynamics. Its separable form

has space and time acting independently, and this is the special, non-mixing case, what results only when transport does not depend on position. The primary object is instead the coupled generator, where the temporal action varies over the base, as in advection whose velocity is a field on the graph or diffusion whose rate is spatially modulated. There space and time do not separate, and the tensor-sum generator is its degeneration. In the separable case the joint spectrum factorises as (Laplacian modes) (Koopman modes), the harmonic skeleton of space times the Koopman skeleton of time. Coupling deforms that product, and how it deforms is the content.

The interaction is the crux

Separable is almost content-free, being a per-frame graph network plus independent linear time-stepping, each mode of decaying on its own as under heat flow. Everything of interest is the failure to separate, that is, how the time action moves energy across the Laplacian spectrum instead of within each eigenspace. This mode coupling is measured by the non-commutation of the spatial and temporal parts. The archetype is advection–diffusion. Diffusion is , advection is a Koopman transport whose velocity is a field on the base, and their ratio (a Péclet regime) is the whole behaviour. Categorically the coupled generator is a crossed product of the spatial observable algebra by the time action, or a connection on the spacetime sheaf, whereas the separable case is the trivial action, the direct product. So the object to study is not and apart but the transport that mixes them; the interaction is the crux.

Why flow matching and simulation want this

A flow-matching path is a curve of distributions in time; its marginals evolve by a continuity / Fokker–Planck equation whose generator is the transfer operator, with Koopman its adjoint (state-predicate-duality). So the temporal structure flow matching already fits is a Koopman generator, and convex flow matching’s velocity field is its symbol. Pairing that temporal generator with the spatial Laplacian gives a simulation surrogate that message-passes in space and advances linearly in time, a spacetime sheaf whose temporal restriction is Koopman rather than a network that sees only single frames.

Given now, learned next

Here the generator (or the operator ) is given, a known generator or a fixed dictionary of observables. The data-driven version, where Koopman is learned through DMD or a latent-linear autoencoder that makes the dynamics linear in a learned observable space, is the natural sequel. It drops into the same spacetime generator with replaced by its estimate, so the base construction here is what the learned model approximates.

Open

• What is the right coupled , a connection and parallel transport on a spacetime sheaf, or a crossed product of the spatial observable algebra by the time action?
• When does it separate? Is separability a symmetry (a space–time factorisation) with a cohomological obstruction measuring how far the dynamics is position-dependent?
• Is there a spacetime Hodge/Koopman decomposition of the joint operator, and is its harmonic part the conserved or steady structure of the simulation?

References

• Koopman, “Hamiltonian Systems and Transformation in Hilbert Space,” PNAS (1931)
• Hansen & Ghrist, “Toward a Spectral Theory of Cellular Sheaves” (2019)
• Bodnar, Di Giovanni, Chamberlain, Lio, Bronstein, “Neural Sheaf Diffusion” (2022)
• Schmid, “Dynamic Mode Decomposition of Numerical and Experimental Data,” J. Fluid Mech. (2010)
• Lusch, Kutz & Brunton, “Deep Learning for Universal Linear Embeddings of Nonlinear Dynamics,” Nature Communications (2018)