Flow-Induced Spaces

Updated 3 December 2025

Flow-induced spaces are rigorously defined constructions where dynamical flows create invariant structures that capture topological, analytic, and geometric properties.
They encompass quotient spaces, combinatorial complexes, smoothed metrics via analytic PDE flows, and induced function spaces in neural network models.
This framework unifies discrete and continuous invariants, offering insights into Morse–Smale systems, porous media channelization, and synthetic spacetimes in analogue gravity.

Flow-induced spaces are a broad, mathematically rigorous class of constructions in which a dynamical flow induces intrinsically meaningful spaces—ranging from quotient and leaf spaces encoding the topological or dynamical invariants of a system, to analytic constructions where the flow acts to smooth metrics or define canonical function spaces. The term as used in current research encompasses (1) topological quotients or combinatorial complexes associated to continuous flows, (2) metric and geometric structures evolved via analytic PDE flows, (3) synthetic analogues designed via controlled physical flows, and (4) function spaces induced by the flow limit of deep neural network architectures. Each of these provides a distinct, flow-induced notion of "space" that serves as a natural invariant, model, or analytic domain for understanding the underlying system.

1. Dynamical Flows and Quotient-Orbit Constructions

Continuous flows $v \colon \mathbb{R} \times X \to X$ on a topological space induce canonical quotient invariants, such as the abstract weak orbit space $X/[v]$ (Yokoyama, 2020). For each point $x \in X$ , the weak orbit-class $\check O(x)$ is defined using orbit closure, and asymptotic sets $\alpha(x), \omega(x)$ . The resulting equivalence relation partitions $X$ into classes whose quotient topology yields the abstract weak orbit space. This space sharpens classical Morse and Reeb graphs: for compact manifolds, it unifies Morse decompositions and Reeb graphs of Hamiltonian flows, providing a finer topological and stratified description. Under mild conditions (Morse–Smale, generic Hamiltonian flows on closed surfaces), the abstract weak orbit space is finite and complete, and encapsulates gradient, recurrent, and chaotic regimes in a single preorder-stratified object. For higher-dimensional Morse–Smale flows, open questions remain on the finiteness of such spaces.

2. Combinatorial and Geometric Structures: Loom Spaces and Veering Triangulations

Loom spaces provide a combinatorial embodiment of flow-induced spaces, generalizing both the leaf spaces of pseudo-Anosov flows and the link spaces underlying veering triangulations (Schleimer et al., 2021). Formally, a loom space $L \cong \mathbb{R}^2$ is equipped with two transverse non-singular foliations. Its local structure is governed by two combinatorial axioms: the cusp–axiom (every cusp-side is continually extendible) and the tet–axiom (every rectangle lies in a tetrahedron rectangle). Skeletal rectangles (cusps, edges, faces, tetrahedra) generate a 2-complex mirroring the flow's transverse combinatorics.

From any loom space, one canonically constructs an ideal triangulation $V(L)$ of a non-compact 3-manifold: vertices are cusps, edges/faces/tetrahedra correspond to the respective rectangles. V(L) is a locally veering triangulation whose realization is homeomorphic to $\mathbb{R}^3$ . This construction is functorial, unifying all known veering triangulations and encompassing pseudo-Anosov leaf-spaces and link-spaces in a single combinatorial framework.

3. Analytic Flows and the Emergence of Flow-Induced Spaces in Geometry

On a smooth or metric measure space, analytic flows (e.g., the Gigli–Mantegazza flow) induce a time-dependent family of metrics, yielding a continuum of flow-induced geometric spaces (Bandara et al., 2015). Starting from a rough metric $g_0$ on a compact manifold $M$ , the GM flow leverages the heat kernel $p_t(x, y)$ to define an evolving Riemannian metric $g_t$ via a divergence-form PDE. The construction is robust under low-regularity and accommodates conical or Lipschitz singularities.

These flow-induced metrics smooth out initial singularities (outside the singular set) and propagate regularity according to heat kernel estimates. In the context of RCD( $K,N$ ) spaces (metric-measure spaces with Ricci curvature lower bound and dimension upper bound), the GM distance coincides with the length metric induced by $g_t$ for admissible points. This provides a synthetic theory of Ricci flow in the non-smooth setting, canonically smoothing the geometry and unifying diverse examples, from smooth manifolds to stratified singular spaces.

4. Flow-Induced Channelization and Morphogenetic Spaces in Porous Media

In physical and engineering systems, flow-induced spaces manifest as spatial patterns carved by the interplay of fluid flows and material dynamics. Channelization in porous media is governed by the coupled erosion–deposition processes subject to Darcy–scale pressure gradients (Mahadevan et al., 2010). The minimal multiphase continuum model considers the evolution of immobile solids, mobile grains, and liquid phases through coupled mass-balance and Darcy equations. Erosion is triggered when local stress exceeds a heterogeneous, nonlocal threshold; deposition restores immobile sites above a packing fraction.

System dynamics are controlled by dimensionless parameters such as Π₁ (deposition/erosion rate ratio), Π₂ (advection-to-erosion timescale), and the variance of the critical stress field. Above a threshold Π₂^c, initial heterogeneities grow, leading to self-organized channelized "spaces" that preferentially conduct fluid. Theoretical and numerical results provide explicit handles for engineering or suppressing channelization, with implications for applications such as filtration, angiogenesis, and hydrological resource management.

5. Flow-Induced Spaces in Neural Networks: Function Space Perspective

In machine learning, flow-induced spaces define the natural function spaces associated to deep residual neural networks (E et al., 2019). For residual nets, the flow-limit—where layer-wise increments scale to a time-continuous ODE—induces the flow-induced spaces $D_p$ . A function $f \in D_p$ admits a flow-parametrization via such an ODE, with a seminorm determined by an associated linear ODE for a cost vector. Direct and inverse approximation theorems rigorously quantify the efficiency and representational benefits of depth, showing $O(1/L)$ rates for residual nets vs. $O(1/m)$ in width for Barron space (two-layer) nets. The inclusion $D_p \supset \mathcal{B}$ (Barron space) is strict. The Rademacher complexity for bounded subsets of $D_p$ matches optimal rates for high-dimensional approximation, with a dependence only logarithmic in input dimension, reflecting the absence of a curse of dimensionality for flow-induced spaces.

6. Synthetic and Experimental Flow-Induced Spacetimes

In analogue gravity, flow-induced "space-times" are engineered via controlled physical flows that replicate effective metrics (Bossard et al., 28 Aug 2024). In the shallow-water regime, surface waves on a variable-depth channel experience an effective 2+1D Lorentzian metric, with horizons at locations where the Froude number $Fr(x) = v(x)/c(x)$ crosses unity. Flow profiles, obstacle geometry, and filling factors determine the resulting phase diagram, separating subcritical, transcritical, and supercritical regions. Dispersive effects further enrich the classification, introducing group and phase velocity thresholds. High-precision subpixel detection of the free-surface enables full reconstruction of the flow-induced metric, enabling laboratory realization and systematic exploration of synthetic spacetimes and their associated "flow-induced" geometry.

7. Unification, Open Questions, and Future Directions

Flow-induced spaces thus arise in a spectrum of mathematical and applied contexts—from dynamical and combinatorial invariants, through geometric flows, to synthetic physical and analytic function spaces. The combinatorial equivalence between loom spaces and locally veering triangulations, the categorical structure of flow-induced function spaces, and the extendability of the analytic flow constructions to general metric measure settings provide a unifying framework that synthesizes previous constructions and raises new questions (Schleimer et al., 2021, E et al., 2019, Bandara et al., 2015).

Key open problems include:

Equivalence of categories: Is the functor from loom spaces to triangulations an equivalence? Does a canonical inverse exist?
Finiteness and stratification: For Morse–Smale flows in higher dimensions, does the flow-induced quotient space always remain finite?
Physical realization and geometric control: How general is the correspondence between engineered flow profiles and target synthetic geometries in laboratory setups?
Analytic regularity and convergence: To what extent do the flow-induced metrics provide canonical regularizations for singular or non-smooth spaces in general curvature-dimension conditions?
Function space characterization: What is the precise inclusiveness hierarchy between flow-induced spaces for various network architectures, and what additional approximation or generalization benefits can be leveraged?

Flow-induced spaces represent a convergence of topological, geometric, analytic, and combinatorial models, providing both a foundational and versatile theoretical setting for studying dynamics, geometry, and representation under continuous or discrete flows.

Key references:

(Yokoyama, 2020) Quotient spaces and topological invariants of flows (Schleimer et al., 2021) From loom spaces to veering triangulations (Bandara et al., 2015) Geometric singularities and a flow tangent to the Ricci flow (Mahadevan et al., 2010) Flow-induced channelization in a porous medium (E et al., 2019) The Barron Space and the Flow-induced Function Spaces for Neural Network Models (Bossard et al., 28 Aug 2024) On the art of designing effective space-times with free surface flows in Analogue Gravity