Topological Flow Matching
Authors: Kacper Wyrwal
Ismail Ilkan Ceylan
Alexander Tong
Venue: ICLR 2026 Poster (2026)
Keywords: Flow Matching
Generative Models
Topological Deep Learning
Geometric Deep Learning
Graphs
Simplicial Complexes
Schrödinger Bridge
Optimal Transport
Primary area: generative models

Abstract: Flow matching is a powerful generative modeling framework, valued for its simplicity and strong empirical performance. However, its standard formulation treats signals on structured spaces---such as fMRI data on brain graphs---as points in Euclidean space, overlooking the rich topological features of their domains. To address this, we introduce \emph{topological flow matching}, a topology-aware generalization of flow matching. We interpret flow matching as a framework for solving a degenerate Schrödinger bridge problem and inject topological information by augmenting the reference process with a Laplacian-derived drift. This principled modification captures the structure of the underlying domain while preserving the desirable properties of flow matching: a stable, simulation-free objective and deterministic sample paths. As a result, our framework serves as a plug-and-play replacement for standard flow matching. We demonstrate its effectiveness on diverse structured datasets, including brain fMRIs, ocean currents, seismic events, and traffic flows.

Published as a conference paper ICLR 2026
TOPOLOGICAL FLOW MATCHING
Kacper Wyrwal" University of Oxford
Ismail Ilkan Ceylan TU Wien, AITHYRA
Alexander Tong AITHYRA
ABSTRACT
Flow matching is powerful generative modeling framework, valued for its simplicity and strong empirical performance. However, its standard formulation treats signals on struc- tured spaces ppaces--uch as fMRI data on brain graphs- as points in Euclidean space, overlook- ing the rich topological features of their domains. To address this, we introduce topolog- ical flow matching, topology-aware generalization of flow matching. We interpret flow matching as a framework for solving a degenerate Schrödinger bridge problem and inject topologiaal information by augmenting the reference process with Laplacian-deriver drift. This principled modification captures the structure of the underlying domain while preserving the desirable properties of flow matching: stable, simulation-free objective and deterministic sample paths. As a result, our framework serves as a drop-in replace- ment for standard flow matching. We demonstrate its effectiveness diverse structured datasets, including brain fMRIs, ocean currents, seismic events, and traffic flows.
INTRODUCTION
Many of the most valuable datasets in science and engineering do not consist of collections of independent points but are better viewed as signals defined structured domains- fMRI scans brain region graph, ocean current velocities on a mesh, or traffic flows on a road network. The underlying structure of these domains contains crucial information, and yet is often overlooked by standard generative models.
Flow Matching (FM) (Lipman ct al. 2023 Liu 2022 Albergo & Vanden Biiieennn--ndnnnnn 2023; Peluchetti| 2023) is a powerful generative modelling framework, achieving state-of-the-an art performance across modalities such as images (Esser et al. 2024). video (Polyak et al. 2025). and audio (Vyas et al., 2023), as well as in diverse scientific applications (Tong al. 2024 Klein et al. 2023). Its appeal lies in scalable, simulation- free training objective and deterministic sample paths. Despite these advantages, standard FM bears a key limitation: treats data as points in Euclidean space, neglecting the rich topological and geometric structure of hon-Euclidean domains. Prior work in geometric and topological deep learning has shown that respecting such underlying structure can yield substantial performance gains (Bronstein al., 2021 Papamarkou et al., 2024). Indeed, recent extensions of FM have taken steps in this direction, for example by adapting the framework generating points in Riemannian manifolds (Chen & Lipman| 2024) and discrete spaces (Gat et al. 2024). Yet, such ideas have not been utilized in FM for modeling signals over such spaces, e.g. fMRI signals on the nodes a brain graph or current velocities on the edges of a mesh discretizing the ocean.
To fill this gap, we introduce topological flow matching (TFM), principled generalization of FM that ex- ploits the topology of the signal domain. Our key insight is that the relation between FM and the Schrödinger bridge problem (SBP) can be leveraged to inject topological information by augmenting the reference pro- cess with a Laplacian-derive 2025). TFM has the distinct advantage of simulation- free objective and deterministic sample paths. This also makes TFM a seamless substitute for standard FM in applications on structured spaces.
drift. Compared to recent work on topological Schrödinger bridges (Yang)
"Correspondence to wyrwal www.kkwwwwpppa..c..cc... com.
Code available / http://////.ttooo/////////a////a

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2) Published as a conference paper ICLR 2026
Figure 1: Illustration of the Hodge Laplacian spectrum and its conresponding heat Gaussian process. Columns from the left: (1) sample from the normal distribution; (2) sample from the heat Gaussian pro- cess; (3) eigenfunction with zero eigenvalue; (4) eigenfunction with low frequency; (5) eigenfunction with high frequency. Top: a graph with node signals-low values shown in blue, high in red. Bottom: a 2- simplicial complex with edge signals, whose values edge proportional to arrow length.
Our contributions are threefold:
We introduce a principled way of incorporating topological information into FM, utilizing its connection with SBP and augmenting the reference process with Laplacian-derived drift.
We derive TFM. a topology-aware generalization of FM for modeling distributions over signals on finite graphs and simplicial complexes. TFM enjoys a stable, simulation-free objective, and deterministic sample paths, making it a drop-in replacement for standard FM.
We evaluate TFM on diverse datasets on structured mic events, and traffic flows slows-showing gains over FM and topological Schrödinger bridge matching.
including brain fMRI, ocean currents, seis-
2 BACKGROUND
Notation. For stochastic process X with law P and times la...., bm. we abbreviate P((Xto..... X, E.) as P tt....m.. Given a random variable Z and a value z, we write the conditional distribution IP( F For space Ω we denote the space of probability distributions on f by P(22). For two distributions Ho. E P(2) and random variables Xo ~ X1 distributions π E (( Q2) with marginals To = H4 and œ1 = ın-is denoted I[ (1).. We denote Brownian motion as W. For any vector or matrix, the i-th component is indicated by a superscript, e.g. X
f41, the set of couplings of Xo and Xı-that is,
2.1 SIGNALS ON STRUCTURED TOPOLOGICAL SPACES
Graphs. An undirected graph K consists of nodes Ko =n= {1,.... no) and edges =)=((ro r1] :o < t1 Ko} Denoting a = K the structure of graph is encoded by its edge-to-node incidence matrix Bi N The column of B corresponding to edge [Uo, 01] has a +1 in the u6-th row, -1 in the vi-th row, and 0 elsewhere. A node signal on graph K is function f: Ko + R. e.g. fMRI data nodes representing functional regions a brain graph, identified with its image f(o) E Ro An edge signal a function K₁ + R. e.g. traffic volumes edges in road network, identified with f(Ki) E R
The graph Laplacian Lg = B, is a positive semi-definite matrix acting on node signals. Its eigenvec- tors with non-zero eigenvalues analogous to sines and cosines for the classical Laplacian on [0, 1]. This analogy extends to dynamics: the graph heat equation f: - Loftt for ς > O, describes heat diffusion of an initial node signal fu. while the associated heat Gaussian process (0, exp(-xLo) is Gaussian distribution over node signals whose covariance reflects the graph structure, reducing to the standard Gaussian if the graph has no edges.
wave-like signals with frequencies proportional to their eigenvalues-
An edge (eu, is ordered by convention make the definition of the incidence matrix unambiguous. The same convention extends simplices in simplicial complexes and their boundary matrices.
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Simplicial complexes. Simplicial complexes model discrete structures more expressively than graphs. A k- simplicial complex K consists of nodes Ku, edges K1, triangles K₂ = [[eo, 11, U2] Up < 01 < U/ E Ko), 2y E Ku)- A k-simplicial complex can be identified with polyhedral subspace of IRd shown in Figure The structure of simplicial complex is encoded by the boundary mulrices B₁ E R XX2, which generalize the edge-to-node incidence matrix. The column of B₃ corresponding to k-simplex (vo...., Uk] has entries (-1}² in the rows indexed by its (k-1)-faces [uo..... U/-1, Uj+1...... Ug]], for each j = O,.... k; all other entries are In practice, K may represent existing real-world structure (c.g., a road network), be specified by domain experts (c.g., brain- region graph), be constructed from data (e.g., a eenearest-neighhor graph or a Vietoris-Rips complex), or arise from standard geometric constructions in synthetic experiments (e.g., triangulation of torus).
and so on up to k-simplices K = {leio,... Ukk : vo <
A k-simplex signal on K is a function f: K - R identified with f(K)) E R The Hodge Laplacian
L = B Ba + B++I
is positive semi-definng matrix, fully determined by the structure of K, which acts on k-simplex sig- nals, generalizing the graph Laplacian. Its eigenvectors with non-zero eigenvalues correspond to higher- dimensional wave-like signals e.g., to discrete vector fields for k = 1, illustrated in Figure 1 Its associated heat equation f. = ff diffuses signals both through (k 1) -)immplices via B B, and through (k + 1)- simplices via B B 1 and also admits structure-awant heat Gaussian process (O0 exp(-xLL3)
Laplacians and topology. Eigenvectors of L3 with non-zero eigenvalues are wave-like signals; those with zero eigenvalues reveal topological features-intuitiin intuitively, properties of K preserved under continuous de- formations like stretching or twisting, but not discontinuous ones like cutting or gluing For k = O, these are signals constant on connected components. for k- = 1, they loop around holes. In general, elements of ker L2 are signals circulating around "k-dimensional holes" called cohomology classey. Thus, components of f " in ker D3 can be viewed fundamentally aligned with the topological features of K.
2.2 FLOW MATCHING
Let (4: l41 E ((Rd)) be two boundary distributions. Flow matching (FM) (Lipman et al. 2023 Peluchetti
2023). also known as rectified flows (Liu 2022), or stochastic interpolants (Albergo et al. 2025) learns a
time-dependent vector field u: [0, 1)x Rx R such that the law IP of the process X driven by the flow ODE
X, (X.): Xo Po
(1)
satisfies P1 f Samples from μio can be transformed into samples from μn by integrating the ODE, which
is used for generation by choosing simple Po like (0, Ia). FM constructs u from conditional vector fields
n² driving the conditional process (X Z 2), for chosen conditioning variable Z n, as the average:
u (x) 2~~ (1====))xx) (x)].
E
Sampling from x(- | X, = x) is generally intractable, which makes computation of u via Equation impossible, and also prevents direct minimization of the FM loss
(f) U.-Unif(0.1 X-P. [X])(X) - un (x)
E
To overcome this, we need three operations: (1) evaluation of (x)). (2) sampling X, PP, (3) sampling Z ~ n. If we can perform them efficiently, u can be learned by minimizing the conditional FM loss
E (CF = t-Unii((,1). Z~z, X-P?
|| 7 (X) - u (x)]
Foormally, K is not topological space, since it is not equipped with topology (a collection of open subsets, closed under unions and finite intersections), Nevertheless, ous way, since any two polyhedral subspaces that K identified with topologically equivalent, homeomorphtc.
study its topological features in meaningful and unambigu-
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VoLcFy (0). An especially effective variant of FM, called conditional flow matching (CFM) (Lipman et al. 2023). is given by choice of coupling и E [[ M1) and
guaranteed by the identity (O
= (Xo, X1);
20 (x) = 21 30-
this case, (X, Xu = zo. x1 x1) follows the straight line (1 am + 1. i.e. P -(1 +1) Two variants of CFM are particularly notable: I-CFM, which uses the independent coupling Mo Pr, and OT-CFM, which uses the optimal transport (OT) coupling n solving the exact OT problem
min
E
II - Xool"] -
s.t. E (1)o
OT-CFM yields straighter sample paths than I-CFM, which can boost performance (Tong et al. 2024 Poola- dian al. 2023). While powerful, this formulation's 's connection to the Schrödinger bridge problem, which we introduce next, provides the key to efficiently embedding topological structure in TFM.
2.3 THE SCHRÖDINGER BRIDGE PROBLEM
The Schrödinger bridge problem (SBP) Léonard 2014 with boundary distributions No. μn E P(RR) and reference law over paths C(0,, 1]; R with Ho @ p4 < Poi. is the minimization problem
min D D P)),), s.t. Q 1]: Q << P. Qo1 E II(po, p1).
Intuitively, its solution is the most likely posterior evolution of system, given prior belief P, an initial observation μo. and a final observation μr. It has a unique solution Q given as the mixture
@+ (F) f p (F)Qor (dæo; dz.).
implying Q p Moreover, if Qi < @ Qn is the solution of the entropic OT problem
min E [c] (KL(C 40 & n1), s.t. Qoi E [l(Ho, (1))
(5)
with the transport cost c log duaS#1 dPai
We focus on the SBP with reference law of the diffission SDE
dX, = (X,) dt + (xX),) dW, Xo ~ no.
(6)
with given drift b,: Rd Rd and noise o, : Rd + dxd Heuristically, for small s > 0, we have
X ~ X₁ (X,)s + (X,)e,
E (0,,00,saa))
P. =;
x:) has the Lebesgue density (a, y), the conditional process (X Xo = xo, X1 = x1) follows the diffusion bridge SDE
dX. (X.) u. go.31 (X.)) di o. (X.) dw. uy "0.00 (x) (x)o (r) log Pt, 21).
(7)
Proposition shows that the SBP solution follows an SDE in terms of conditional controls =o.
Proposition Let be the law of a diffusion process X and define the marginal control u by
uu (x)x) = (Xo.Xi (-X,==)
Xo.X (x)
(8)
The solution Q" to the Schrödinger bridge problem with reference law P is the law of the process
dx, = (X.) + u.(xx)) or (X,, dW,, Xo ~ no-
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3 INCORPORATING TOPOLOGY INTO FLOW MATCHING
this section, we connect CFM to the diffusion SBP with zero drift. This motivates the use of topological drift, which we justify via spectral analysis as topology-awas smoothness bias.
3.1 FLOW MATCHING SOLVES A DEGENERATE SCHRÖDINGER BRIDGE PROBLEM
CFM arises as a way of solving the SBP with trivial drift b=0 and constant noise E 1R in the limit a -> 0. It is instructive, as a blueprint for TFM, to see this connection in terms of the three key CFM components.
1. Conditional vector field
20,21 2t 20 If b- 0 and Ø E R4, the diffusion bridge SDE simplifies to
dx2 = 30. (Xi) + o dw,, 30 (x)
21 σ. Xo z0-
1-t
As - 0, the dynamics reduce the ODE X. in 20.21 (Xt). Its unique solution is the straight line
X, = (1- + 1.1. Thus, 2o.22 ut (X,,) = 21 - To, which recovers the CFM conditional vector field.
2. Conditional path P In the zero-noise limit, (X, Xu Tu. X1 z1) becomes deterministic. Thus, i 0, its law Pr converges to the CFM conditional law 6(1 t(
Coupling (Xo, X1) n. If = and a E R4, the entropic OT problem Equation (5) simplifies to
min
||x
Xool?] xXol + o² (Qoillll @ p1), s.t. Qui E (1).
(10)
A - this converges the exact OT problem in Equation Léonard 2014). Therefore, Qn coincides with the coupling nt used in OT-CFM. I-CFM can be seen as the independent approximation Q6 = @
Proposition now shows that the marginal vector field of OT-CFM is the drift of the SBP solution
dX, un (X.) dt + dW,, Xn ~ pi.
In the limit a O, these dynamics converge to the flow ODE in Equation (T). Thus, OT-CFM can be viewed solving the zero-noise limit of the diffusion Schrödinger bridge problem by learning the drift of this limiting ODE. This is a formal viewpoint, since an SBP with o is generally not sell-posed") However, the optimal drift of the SBP does converge under 00 solving the Benamon-Brenier OT problem
min L( (114(())) 2 (dz) dt, s.t. O.P, + V (Pru) = 0, Po = Ho:
P M11
Intuitively, this drift is the minimum-energy vector field transporting Pu to μr in Euclidean space.
3.2 TOPOLOGICAL REFERENCE PROCESS
Topological diffusion. CFM can be seen as solving the zero-noise limit of a drift-free SBP. Since the reference process plays the role of prior, complex K, by augmenting it with topology-awan drift (Yngg 2025):
bias the SBP solution respect topology of simplicial
by (X,, = (L,)) -
(11)
For instance, let b= 000, and fil 520- In this case, P is a Dirac delta on the constant path t ra. Any solution ! must satisfy Q P. Since P is point mass, this implies == IP and, consequently, Q1 = ===i= Therefore, unless
/41 Ezg- Q cannot simultaneously satisfy the necessary condition Q = mi. Thus, the SBP has solution.
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Table 1: Flow ODE, conditional vector fields, and bridge process of CFM and TFM.
Model
Flow ODE
u0 (X,)
(X: Xo 2u, X1 21)
CFM
X. (X.) = (X.)
J1 20 t E, (2) mi (2o)) mi(xo) + Ex, Ej, (x) mi (zo))
(1 (--t + txı
TFM
for a polynomial function of Lg by the Cayley-Hamilton theorem (Yang) 2025). The resulting topological reference process
achieving tractability and flexibility, as (Lx) can approximate any analytic
dX2 H2 (,{ X + 04 + a dW,
serves the starting point for derivation of TFM in Section
To motivate this choice, we focus on the case (L =n for v: > O, as it connects to the heat equation
and the heat kernel, providing a clear interpretation; however, other choices like H, (n.² - L1 /utn/22 for K > O, V 1/2 could be useful for their connection the Matérn kernel (Borovitskiy al. 2021 In the zero-noise limit the reference process becomes the heat equation X, = L, with diffusion rate k. Furthermore, let UR = (M,,...., (,,,) ) be the eigenvectors and D₁ = diag Ao .... An.)) the eigenvalues of Lg. so that L UR DuU k. Under the change of coordinates Y = U X, the heat equation is diagonalized
X. -K + Yi = -KAY ViE (O....,..yJ n2).
Its solution is given by Y = exp(-RAd))(- Thus, the eigenfunctions with non-zero eigenvalues decay exponentially quickly at rate proportional to their frequency, while the eigenfunctions with zero eigenval- ues, corresponding to topological features (cf. Section 2 stay constant. Therefore, our proposed drift can - denoising the signal-while preserving
be seen as bias dampening high-frequency signal components aligned with the structural features of K, where bias strength is proportional to k.
Topological initial distribution. For generative tasks, we can also inject topological information in the initial distribution μo, by setting it to the heat Gaussian process (0, exp(-x Setting K = 0 recovers the standard Gaussian maassiannddisegaadiin disregarding the structure of K by disallowing heat flow between adjacent simplices.
4 TOPOLOGICAL FLOW MATCHING
CFM operates in Euclidean space, overlooking the topology of structured domains like graphs and simpli- cial complexes. Since OT-CFM learns the solution of the degenerate drift-free SBP, we propose to make it topology-aware, by augmenting the reference process with the topological drift from Equation [T.. Retrac- ing the derivation of OT-CFM from SBP, this time for the topological SBP
dx, [H, (,, + or) + dW.. X,- Mo:
yields topological flow matching (TFM) principled, topology- aware extension of FM. Although inspired by topological Schrödinger bridge matching (TSBM) (Yang 2025), in stark contrast to TFM enjoys the
key advantages of standard FM: scalable, simulation- free training and deterministic sample paths.
To see why the SBP perspective useful, let us notice that, if we augment the flow ODE with the topological
drift, yielding the topological flow ODE
X, (Lx) + a, + (X,,),
(12)
a sensible choice of "x,"((,,) is not clear. For instance, un 20.01 (Xt) = (x) - -0) (H, + ax) yields the same conditional path as CFM, which ignores the topology:
X (H, (Lx) + a.) + u, 20 (X,)
To-
(13)
The degenerate SBP, however, has a unique solution, which induces a principled choice of , (X,) as
stated in Proposition As in the Euclidean case, while the zero-noise limit of the topological SBP is formal,
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its optimal drift u, converges in the limit - to the minimizer of the dynamic optimal control problem
min S.' (dxx) dt, s.t. O.P, + (Lx( a + uz)) = O, Po = pox
PP A1
Intuitively, this u, the minimum-energy corrective force needed to steer from 140 pa when the system, left undisturbed, evolves according to the heat equation.
Notation. We denote (Z) = E X, = z)), Eg == o ( Xx X, = x), with the short- hands my mijo- Σe, == E.o We write Ps,t for the solution to L.,t (1) 2 Inn-
Proposition (Conditional control u, 20:01) Let X, be the topological reference process. The bridge (Xe Xo = X1 = z1) follows the SDE dX, [Lx)L,)X, + a + 2,
20.21 (X,) dt + a dW,, with
Ut
(X.) Dr, 2
m1je (X:)).
(14)
t1,1 ST. mi (xoo)). For In the limit + O, ut 20.01 (Xx) becomes independent of X ui 2 (Xt)
(L(),) Lk we have the simple formulas in spectral coordinates
31 y0
= =
(ut 20 (Y+))²
2n exp( (xp(-xx((--)) (vi - exp(-RXA)ub)
else.
exp(2xA')
TFM learns the topological flow ODE the same way as CFM learns the standard flow ODE: by choosing coupling (Xo, X1) π and minimizing the CFM loss
E
l Xo. (X) u. (x)]13]
t~Unif(0,1) (Xo.X1)~<, Xap,Na.X1
This is simulation- free objective, since the bridge (Xe Xo = Fo, X1 = 21) is deterministic.
Proposition 3 (Conditional path 1P20=*1, ). Let X, be the topological reference process. The bridge (Xe Xo To, X1 z1) has the mean
m, 2o.21 = m=( +
(0((11-m((_))) m2()))).
As + 0, the bridge law concentrates on the mean Peo.ar 5m. For H, HL()s) = -x we have
tyi + +1 t)yg
== X2 =
(m Vo.Vi
sinh(x>" yō + sinh(r<2)
sinh(ax''), sinh((())) V1
else.
3
3
3
3
3
3
3
3
Figure 2: Conditional paths from noise to data on the MNIST dataset at intervals 1/9. showing the
smoothing effect of the topological reference process. Top: CFM. Bottom: TFM k k= 0.5, Mo =N
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Table Mean I-Wasserstein distance. 1 standard deviation, of CFM and TFM on real-world datasets from Yang 2025) compared against the best performing TSBM variant.
Method
Earthquakes
Traffic flows
Brain IMRI
Single-cell
Ocean currents
I-CFM OT-CFM
8.37
1.72_0.01
10.71
0.02210.001
1.95
8.2544.08
1.59_0.01
11.30=0.01
0.019
2.00
1-TFM OT-TFM
0.03
1.27:10011
6.33
0.018 : 001
1.87.00.0
5.53
1.27:10.00
5.8610.01
0.019
1..0
TSBM (best)
7.69
9.92_0.02
7.51
0.140
6.89_0.00
Any coupling yields valid TFM variant. However, only the OT-TFM variant with π = Qi solves the de- generate topological SBP. Fortunately, Qo can be computed as efficiently as in the Euclidean case, since the transport cost e corresponding the topological SBP has simple formula given in Proposition
Proposition (Coupling (Xo, ~ (6 For the topological reference process X, the entropic OT problem in Equation (5)
equivalent to
min
E
[c] + o² Dxu (Qoilloo @ p1), s.I. Qoi M(po
(15)
(Noo
and, in the limit a + O, converges to exact OT problem with the same transport cost c. given by
x1) (x1 mm((2)) ((ero))TEE) "T m2(
With H((Lx)) Lk we have the formuul c(yo, 81) = =-i=1 cs (Vó: vi). with
(vi - y6)² 2nA²
x²= or =0
c vi) 1-exp(-2xX) (vi exp(-n2))6) else
(16)
Thus, TFM enjoys the scalable, simulation-free formulas mirroring CFM, summarized in Table 1 Any variant of TFM, such as I-TFM with क = Ho @ μn, exploits topological information via the flow ODE, the conditional vector fields, and the conditional path, while OT-TFM also uses it in the coupling Q01-
5 EXPERIMENTS
We developed TFM to improve performance over CFM generative tasks topological signals by exploiting the structure of their domains. To examine whether this goal has been achieved, we compare TFM--pecifically its heat flow variant with the topological drift Lx X CFM on diverse real- world datasets on graphs and 2-simplicial complexes, compiled by Yang (0025) to evaluate TSBM. This also lets us compare TFM against TSBM, showing whether TFM can serve as a scalable, simulation-free alternative. To ensure fair comparison, we use the data, models, and experimental setup given by Yang (2025) wherever possible. We use fixed = 2.0, though a choice tailored to given experiment can improve performance (cf. Section F3 We also investigate if the smoothing effect TFM has conditional paths (cf. Figure [
can aid image generation, by viewing images as node signals on the grid, using λ == 0.01. The experiments are divided into: generation, where the initial distribution is simple and the final distribution is a data distribution; and matching. where the initial and final distributions are different data distributions.
Generation. We first tackle modeling of earthquake magnitudes around the globe, using data between 1990 and 2017 from the IRIS dataset. To capture regions of seismic activity, we discretize the Earth's 's sur- face mesh and construct a (O-naarest-neighbors graph from the nodes closest to earthquake events (cf.
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6.50
6.25
6.00
5.73
5.50
0.50
0.25
0.00
Figure 3: Left: Road network in the traffic flow experiment with a node signal. Right: Simplicial complex built from the road network, with trian- gles shown in green, and an edge signal.
Figure 4: Top:
Historical earthquake events overlaid with mesh discretizing the globe. Bor- tom: Graph used in the earthquake experiment with an example node signal.
Figure 4). For each year, events are mapped to nodes and their magnitudes are averaged to create a target signal. Secondly, we consider traffic flow modeling using the PeMSD4 dataset, which contains node signals from 307 measuring stations. Following [hen et al. (2022), signals are lifted to the edges and the road network graph to a 2-simplicial complex (ef. Figure B From Table 2 we find that TFM significantly out- performs CFM and TSBM. Interestingly, OT-TFM is outperformed by TFM on the earthquake experiment.
We also test image generation on the CIFAR-10 dataset, treat- ing images as node signals on a 32-by-32-by-3 (width. height, channels) grid. From Table 3 we see that I-TFM and OT-TFM achieve only small mean and median improvement over I- CFM and OT-CFM respectively, with a relatively high variance between runs. The advantage of TFM is thus much larger on the earthquake and traffic experiment, where the signal domains have complex structure, than in image generation, where the do- main is a regular grid. This suggests that, while TFM shows some promise for image generation, it gains significant advan- tage by capturing topological features of complex domains.
Table 3: The mean, median, and standard deviation of FID scores on CIFAR- generation over 10 independent runs.
Method
Mean
Median
SD
I-CFM OT-CFM
3.7005
3.7061
0.0462
3,8238
3.8308
0.0615
I-TFM
3.6972
3.6795
0.0821
OT-TFM
3.8107
3.8046
0.0771
Matching, We first consider edge signals representing water currents on a 2-simplicial complex discretiz- ing the North Atlantic Ocean, extracted by Chen & Meila (202I) from data collected by NOAA Atlantic Oceanographic and Meteorological Laboratory. As the initial distribution we use an edge GP fitted to data (Yang et al. 2024). and for the final distribution we choose a synthetic curl-free Gaussian process. The target samples exhibit only sinks and sources, while the initial samples form realistic patterns (cf. Figure 5 Sec- ondly, we study the diffcrentiation of cells in an embryoid body across 5 time steps, using data from Moon jet al. (2019): Tong et al. (2020). Using data from every time step, construct n-nearest-neighhors graph capturing the global structure of the temporal cell evolution. Then, as the initial and final distributions we use a normalized indicator function over the data the first and last timestep. Lastly, we consider the graph of 360 functional regions of the brain, with edges weighted by connection strength. We use 1,190 fMRI signals from the Human Connectome Project, decomposed into fluettuunua aligned and tmee-eersisten liberal components, serving as initial and final distributions respectively. From Table 2 we see that TFM outper- forms CFM on all experiments most of all on the brain graph, which has a complex weighted structure. TFM also consistently outperforms TSBM demonstrating the advantage of a simulation-free framework.
We do not include the GTSB result, provided by Yang 2022 for the occan current data, since do not assume knowledge of the boundary distributions that GTSB necessitates.

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Day 00-43 Day
Figure 6: Single-cell data in the PHATE coordinates.
Figure 7: Initial (left) and final (right) brain fMRI signals.
Table 4: Summary of topological datasets. A; signal order
(0=node, ==dgge). N: number of data points, oc if synthetic.
Earthquake
Traffic
IMRI
Cell
Ocean
Task
Generation
Generation
Matching Graph
Matching Graph
Matching
K
Graph
Mesh
Mesh
Figure 5: Ocean experiment data. Top: Ocean mesh (subset). Middle: Initial sample. Bottom: Final sample.
k
0
1
0
TkR
576
340
360
18K
20K
N
28
17K
2K
6K
30
6 CONCLUSION
We introduced topological flow matching, a topology-aware generalization of flow matching for generative modeling over signals on structured spaces. We proposed way to incorporate topological information into FM by utilizing its relation to the Schrödinger bridge problem and augmenting the reference process with Laplacian-derived drift. This addition can be seen prior which smooths signal components that do not correspond to topological features captured by the Laplacian kernel. Furthermore, we proved that TFM retains stable, scalable, and simulation-free objective well as deterministic sample paths, which makes a drop-in replacement for standard FM. We demonstrated performance improvements over FM and topological Schrödinger bridge matching with the greatest advantage on the most complex structures. While we focused on the natural choice of topological drift based the heat equation, future work could further explore different reference processes, including Matérn-like drifts or drifts with a learned diffusion rate k. In addition, a drift with a time-dependent Laplacian could enable matching tasks between signals two different spaces. Finally, variants of TFM for signals manifolds could be examined for applications c.g. in geostatisties and robotics. Altogether, TFM provides a principled extension of FM that improves performance on structured spaces and opens new avenues for both theoretical exploration and practical applications.
variety of datasets graphs and simplicial complexes,
ACKNOWLEDGEMMENTS
KW thanks Maosheng Yang for providing additional theoretical and experimental details of (Yang 2025
ETHICS STATEMENT
Our work is primarily focused on thcoretical algorithmic development for faster and more accurate gener- ative models for topological datatypes, with reduced focus experimental implementation. However, recommend that future users of our work exercise appropriate caution when applying it to domains that may involve sensitive considerations.
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REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our work, we provide detailed information regarding our experimental setup, theoretical claims and datasets. An implementation of topological flow matching and the code to reproduce all experiments will be made available upon publication. The theoretical claims and derivations underlying TFM are presented in Section A All datasets used in our experiments are publicly available at their cited source in Section
Further implementation details are available in Section
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APPENDIX
A PROOFS
A.1 PROOF OF PROPOSITION
This proposition is direct corollary of Proposition of generator matching (GM) (Holderieth et al. 2025 which states that for any Markov process with law P,, the generator L, of X is given as the mixture of generators L of (X X1 = x1)
(r(())= =
E X1-,
1 f(x)].
(17)
Now, general diffusion SDE
dX,
dW,
(18)
has the generator
( - ( (Ha())) (()))
where (x) (z)), (z), the first inner product is the vector inner product, and the second inner product is the Frobenius matrix inner product. It follows that the bridge X Xo = ze. X1 = 21 with the dynamics
dx,- (X,, ur (Xx) dt (X,)
(19)
No. x000"(())) (z) (x) V log Pt, (x. has the generator and ut
h (b)(xr) ut 70.=1 ( VS(x)) + (()
Moreover, noting the SBP solution by X Q", we have that _ is also the generator of X* Xx To, Xi 21, because (Q*) = 1 Additionally, since the generator does not depend on the initial distribution, L also the generator of X* X 21. Taking expectations, follows from Equation (7
that the SBP solution has the generator
fx)(x) = (x(x)) + x;-01(1x;=a)
(u(()) + (Far(t), v>((s),
and so follows the SDE
dx; = [bx(x;) + ux(x)) + ox(x)) dW,
with
ue (X.)
E x1 (11;;)
"
x
(x:)
Since neither side of the equation actually depends on Xo. we can take expectation over Q6 Xx, obtaining
up (Xi) X-~ XF), X Xx1
E
ut
xg.X7
(xx)
E
Ut
x7,
(x;)
(Xg.x)) (-1x;)
Renaming X to X finishes the proof.
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A.2 PROOF PROPOSITIONS
2 AND
General topological reference process. reference process takes the form
To simplify notation, let A, = Hz(L,), so that the topological
dX, (A,X, + az) dt + o dW,
(20)
This is an example of a linear Gaussian SDE. It is well-known that a linear Gaussian SDE can be expressed for any 0 r <
= Dr,x, + ff Dana, ds [ Pas
where & s, is the solution of the ODE
① Dett = A, 2a.s 20. Inn.
It follows that the conditional process (Xt X, x) is Gaussian process with the mean (z)(()) and covariance Eg
m4j = =[[X, X, xzy] 2r + ff as, ds,
where
Du u,s du.
This yields Gaussian transition probabilities (x, x1) = N[x X, = 211, Var(X X, = =x)) N (1)))) Σ1,11... Thus, the conditional control ui 20.21 of the corresponding bridge SDE takes the form
u 20
(X,) -o² No (x); E1.11e)
o² , (11 (X))) tb,1 5-1 1,1| (x) (x1)((,))
Moreover, conditional Gaussian process formulas show that the (X, Xo Fu, X1 x1) is also Gaussian process with the mean m,
z0:21 and covariance 21
Zo.21 my
E[X, Xo Zn, X₁ 21 mz
Ex, Ex m, (zo))
=m((xo) + E5,11 E - m. (2o)), 21 21 = Cov(X,, Xx, |xo xo, X1 21)
E,,t Ex,1 ET,1
02 =0((is,t  E1,, 1 E1,1
and m, zo.2r stay constant, while where m. == myjo and Ex,t = Ee,(0- Taking the limit a 0, these ut 20.21 the law of the bridge (Xe Xo Tp, X1 z1) converges to
(X, X, -oo Xi x11) ~ pp m
Finally,
the fact that X, X, Fog X1 1 is deterministic, further simplify 20 u Fo. (X,) .. (x1 - mm (m mo.mi )) = o [ m. (xo)).
yielding a stable conditional control independent X This proves the general formulas in Propositions and]
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Heat topological reference process. The spectral coordinates U X diagonalize the topologiaal reference process and hence the formulas for ui and (X; Xo Fo, X1 = z1), which we denote uy Vo.M/ and Y; Yo yo. Y1 = y/1 follows then by simple algebraic manipulation that for (x))) -X,, we have the spectral formulas:
s.1 exp(=xXX s)),
m1 (y)* exp(- (o(--R)")
/,' exp( (1x--2x2 - dt)
== or A
E
1 1-expí 22x)" 2xd
else,
(21)
vi= Vú
k= 0 X
(u. No.8h (Yi))*
2nd exp(-=x1(113) (ui -xxp(=XX) w6))
else,
1-exp(-2xAt)
Lyi + (1 1)v1
or X =
(m yf
(inh(xx((1-)) sinb((=22))
vó +
sinh(w22t)
sind(
else
which was to shown.
A.3 PROOF OF PROPOSITION 4
We can use the formulas found in Section A.2 find the transport cost log
duoBip when P is the law
dPoi
of process with the linear Gaussian SDE dynamics (cf. Equation (20p)
General topological reference process. To this end, we first notice that if in has Lebesgue density p. then the transport cost factorizes as
dyo @ fl (wo, x1) = log
du
log
(x1) = log p(x1) log Po. P T1).
(22)
dlPoi
dIP (. Xo mo)
Since Bo., [log p ((x))] En llog p(X,)), ).. the first summand does not change the minimizer of the associated entropic OT problem. Hence, in solving this problem, it is equivalent to consider the cost
c(ro, 21) log Po,1 x1)= log.V((1: m, (zo), 21,1) (I1 m, (fo)) 27. m. (xo)), (23)
where we drop the constant log(2x:)1.1) since it too does not affect the minimizer. Furthermore, can extract out
c(zg,21) = (a1 - mi (zo)) (
(I1 - mi (zo)) = -= - m, (zo))
E7, (zo)) - my (xo))
(24)
Multiplying the associated entropic OT problem by a², we get the equivalent problem
min
E Goo
[X, m, (Xx)) ET, m. (Xo)) +o DKL. (Qo IIpo @ p1),
s.t. Qo E 11 41).
(25)
Since the cost is now no longer dependent σ, can pass to the zero noise limit obtaining the exact OT problem
min
E Go
[(X, (Xo)) Tx 1 (Xi mX()))
s.t. Qoi E Il(po,
(26)
what was to be shown.
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Heat topological reference process. In the spectral coordinates, the covariance of X is diagonal: hence, the probability density (Yo, Y1) factorizes as I Po.1 (Y6, Y1). Consequently, the cost factorizes e(yo: V1) = D ca vi) with
cu vi) = (ET,))
mi (36))
Finally, in the case H((A,) = RX', plugging in the expression from Equation 21 simplifies the transport cost even further. Most simply, if either = 0 or Lx = 0, we get
cu vi) (ui -0))² /1 (vi -²))=²²²
(27)
Otherwise, we still get an efficient formula
2xd²
Ca vi) = 1 (ui -exp(- exp((nx\'jb))
(28)
which to be proven.
B ADDITIONAL RELATED WORK
B.1 TOPOLOGICAL SCHRÖDINGER BRIDGE MATCHING
Absence of a Topological Flow-Matching Framework in Yang (2025). To the best of our knowledge, Yang (2025) does not present a topological flow matching framework. The closest relevant result is Corol- lary E.2, which provides the probability- -flow ODE associated with the solution of the topological SBP. However, the drift this ODE is expressed terms of two auxiliary random variables defined via a coupled system of heat equations, making it computationally intractable and therefore unsuitable basis for flow matching.
Distinction Between TFM and TSBM. TFM differs fundamentally from TSBM along the same axis that distinguishes flow matching from Schrödinger bridge matching:
Simulation-free vs. simulation-based training. TFM inherits the simulation- training paradígm of FM: its objective requires only deterministic evaluations of the vector field, and no stochastic sample path simulation involved. In contrast, TSBM inherits the simulation-based nature of SBM and relies on stochastic path sampling during training. This yields substantial em- pirical benefits for TFM: faster training, increased numerical stability, and more direct scalability to high-dimensional settings.
Deterministic vs. stochastic sample paths. TFM produces deterministic sample paths governed by the learned flow ODE, while TSBM yields stochastic paths arising from diffusion processes. This is a qualitative difference intrinsic to the two formulations.
In summary, TFM retains the key properties of FM (scalable, simulation ffree, deterministic), whereas TSBM retains those of SBM (simulation- based, stochastic). The methodological gap between TFM and TSBM therefore as substantial as the gap between FM and SBM.
Topology-Aware Initialisation. For generative modelling, we additionally propose topology-aware ini- tial distribution, which can further improve fidelity of CFM and TFM. Empirical evidence for this design choice is provided in Section F2
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B.2 OTHER ARCHITECTURES AND FRAMEWORKS
In this work, we consider a topological form of flow matching which generates signals with respect to a specific topological space. A large literature exists on graph signal processing (lsuñi et al. (2055 and geometric deep learning (Bronstein et al. 2222)) In this work could potentially benefit from the vast literature on topological deep learning architectures (Yang et al. 2025
simple architectures, but note that TFM
Battiloro et al. 2024; Goh al. 2022), or graph neural networks (in the case of simple 1-simplices) (Kipi
& Welling 2017).
This literature is in contrast to that of generating topologies (Papamarkou et al. 2024 which is related field with many potential synergies, but is not directly applicable to our setting.
C SKETCH OF EXTENSIONS
We sketch how TFM extends to vector-valued signals, countably infinite simplicial complexes, and compact Riemannian manifolds. For the latter two cases, the guiding principle is that once the reference Brownian motion is defined in the correct infinitediimessinnal space and mild regularity holds, the linear Gaussian reference dynamics are well posed, bridges are Gaussian, and the formulas from Section carry carry over in the sense of operator calculus. However, since the Laplacian eigendecomposition is infinite. it must be finitely truncated handle in practice. This eliminates most functional analytic technicalities, since the setting becomes fiite-dimensional.
Vector-valued signals. To model distributions over vector-valued k-simplex signals X,: K₁ + Rd. we can simply assume that the topology acts only spatially and not across output dimensions. This simply means that we identify the signal X, with an element of Rend and apply TFM with the block-diagonal Laplacian:
L
L & Ta-
Signals on infinite simplicial complexes Let K be a countably infinite, but locally finite, simplicial com- plex and fix k. We can model k-simplex signals in the Hilbert space H 1² (²(K))) A direct sum of ii.d. I-dimensional Brownian motions does not produce -valued paths: Σ, [[Weee'' 12 has infinite expected value, so paths "blow up" immediately. To remedy this, we can use Q-Brownian motion, which simply down-weights figh-frequenc directions so the expectation stays finite. Specifically, choosing an orthonor- mal basis (e:) of H and nonnegative weights (q:) with E. 4: < we can define W.º as the convergent series
Wº Σ 1=1
where Wi are independent 1-dimensional Brownian motions. With such a choice the reference process
dX2, = [H+(Lx)), + +ay dt o aw
is well-posed, and the bridges remain Gaussian. All expressions for conditional controls, conditional paths, and transport costs from $ carry over essentially verbatim. The zero-noise limit yields TFM on infinite simplicial complexes.
Functions and differential forms compact Riemannian manifolds The compact Riemannian man- ifold setting mirrors the simplicial one and can be treated exactly the same as the infinite simplicial com- plex setting under proper identifications. On compact Riemannian manifold M, we replace the space of k-simplex signals R with the space of k-forms A (.² and the Hodge Laplacian L2 with the Laplace- de Rham operator Ak. Because L A (.M) has countable basis (c;), again drive the reference SDE
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Table Summary statistics (mean = std across 5 seeds) for all models under the identity (I) and optimal- transport (OT) couplings.
Model
1-Wasserstein distance
2- Wassersttei Distance
I-CFM
11.84
8.41
I-TFM
40.94
6.42
I-TAN
11.2440009
7.97 =0.0
OT-CFM
11.7710.08
10.06
OT-TFM
8.97
6.41 10.04
OT-TAN
11.22:00.9
7.95=0.00
with a Q-Brownian motion, this time on L'A The cigenfunctions of A have an analogous inter- pretation as wave-like signals for non-negative eigenvalues and signals circulating around "k-dimensional holes"; hence, the motivation for the topological reference process remains the same. Thus, under the iden- tification (R L&) ++ (I.2A( ) the construction of TFM for signals M can proceed exactly for infinite simplicial complexes. An extension to functions on non-compact manifolds may be facilitated by taking functional flow matching (Kerrigan et al. 2024), or functional rectified flow (Zhang Scott 2026) as a starting point, possibly aided by the literature on the probability-flow ODE in function spaces (Na et al.
2025
D FURTHER MOTIVATION OF THE TOPOLOGICAL REFERENCE PROCESS
This section expands motivation for the reference process used in TFM by comparison to alternative flow matching formulations based on a Q-Brownian motion that depends on the topology of the signal domain K. We provide a quantitative and qualitative comparison of these methods on a synthetic experiment, where K is triangulated torus.
Alternative Topology-Aware Reference Processes. Topology awareness in flow matching could also be introduced by modifying the Brownian component of the reference SDE prior to taking the zero-noise limit. natural candidate is a Q-Brownian motion,
dX, 122//
where depends the Laplacian (e.g., Q L). To investigate this idea, we analyze general reference process,
dx, a Q, 1/2// where Qi may vary in time and a E R4. Conditioned on endpoints Xo = Do and X1 = 21. the process follows the bridge SDE
(29)
Σg,t = S Qr dr.
dx, = Q. (I) - X,) + too dw,
(30)
Taking the zero-noise limit + 0, we obtain the deterministic bridge
X, = 21 + Er, Eg.1 (21 zo), dX, Q, Eg.1 (x1 2o).
Consequences for Flow Matching. If Q is time-homogeneous (Q+ Qn). then
-ali-Qo( aa(L Qu ds) = I,
Q =
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1nee jo =0)
4.0
1he jy (
3.0
LCFM (meat)
3.5
LCEM (meac)
Standard Deviation
Standard Deviation
LTFM (meat)
LTEM (meat)
2.5
LTAN (meat)
ETAN (neeat)
1.5
2.0
ad
0.3
0.0
0,0
0
100
200
300
400
500
0
200
400
600
800
1000
Curl-Pree Dimension Index
Divergence-Frees Dimeusion Index
Figure 8: Standard deviation of the predicted distributions by the I-CFM. I-TFM, and I-TAN models in spectral coordinates. Left plot shows the results for coordinates corresponding to curl-free eigenvectors. Right plot shows the results for coordinates corresponding to divergence- free eigenvectors.
so the resulting vector field is exactly æ - zo, recovering conditional flow matching (CFM). Thus, constant- topology-aware noising does not produce different FM method.
When Qs is time-dependent, the zero-noise limit yields a genuinely different bridge. However, choosing a meaningful time-inhomogee time-inhmmoeencous Qi is nontrivial, in contrast to the physically motivated process
dX,- = -cLX, dt + a dW,
which corresponds heat diffusion perturbed by Brownian noise.
A Time-Lnhomogennous Alternative. For comparison, we consider
Q: = exp((=xL))
the Laplacian eigenhasis this yields the conditional drift
R exp(-xxX) expl xx>*)
(ví v6), X >>> 0,
(o =
yd.
=== 0.
This expression resembles the TFM vector field but lacks the topology-depended rescaling of vó via exp(-xX²) that TFM introduces. For generation tasks from noise (æo) to data (z1), this mechanism ef- fectively denoises low-frequency components before high-frequency ones.
Synthetic Comparison on Triangulated Torus.
To further motivate our approach and compare against this topology-aware noising (TAN) alternative, we conducted controlled synthetic experiment on a trian- gulated torus. The task is to match distributions over edge signals: the initial distribution is divergence-free (no sources or sinks), and the target distribution curl-free (no "swirls"), Specifically, the initial distribution is modge-compositional Matérn Gaussian process (Yang al. 2024) with smoothness parameter 2.5. We make this choice, instead of choosing the heat Gaussian process, to prevent making the task by-design easy for TFM and TAN.
We find that TFM outperforms both CFM and TAN in terms of the Wasserstein distance, as reported in Ta- 5 Furthermore, consider the standard deviation in the spectral coordinates of the final distribution
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Tuee pi (E-0)
4.0
Tree pl: (C00)
3.0
OECIM (mean)
3.5
01-CM (mean)
Stundard Deviation
2.5
OTC md
Standard Deviation
CTM
OF-IYM (men)
01-1FM (mear)
Of aid
2.5
OT-TFM al at
OF-AAN (mean)
01-IAN (mean)
1.5
GT-TAN al
2.0
OT-TAN al ai
0.3
0.0
0.0
0
100
200
300
400
500
200
400
600
800
1000
Curl-Free Dimension
Divergence Free Index
Figure 9: Standard deviation of the predicted distributions by the OT-CFM, OT-TFM, and OT-TAN models spectral coordinates. Left plot shows the results for coordinates corresponding curl-free eigenvectors. Right plot shows the results for coordinates corresponding to divergence- free eigenvectors.
modelled by each method. Specifically, by dividing the plots into the spectral dimensions corresponding to the curl-free and divergence-fres components, we can understand how well these FM variants interpolate these qualitative differences the boundary distributions. As shown in Figures X and TFM dampens the curl-free features of the initial samples, introducing divergence- freggence-freeeee flow in the final samples. In particular, TAN and CFM struggle to interpolate between the high high-frequency components of the boundary distributions. These differences qualitatively visible in sample visualizations in Figure 10
E ADDITIONAL EXPERIMENTAL DETAILS
E. TOPOLOGICAL SCHRÖDINGER BRIDGE MATCHING EXPERIMENT SUITE
For the earthquake, traffic flow, brain fMRI, single-cell differentiation, and ocean currents experiments we replicate the experimental setup described extensively Yang (2025). We summarize key aspects here, and provide additional details unspecified by Yang (2025) and ones that pertain to TFM specifically.
Topological drift. In all experiments we use the heat equation topological drift with x 2.0, where the value of a was set after initial testing on a synthetic dataset, though values from 0.5 to 4.0 performed similarly well. Depending on the size the graph, we approximated the eigendecomposition
L3 with eigenpairs with the lowest eigenvalues: for earthquakes we take the full spectrum, for traffic flow the full spectrum, for brain the full spectrum, for single-cell differentiation m = 256, and for ocean currents 500.
Training. To learn the conditional control we trained a residual neural network, as well as a graph neural network and simplicial neural network, depending the signal domain, implemented by Yang (2025). Each training run consisted of up to 100 epochs with 25,600 samples each, stopping early after 10 epochs of improvement greater than 1 1% in terms of W-Wasserstein distance ation experiments and ocean current experiments, where at least one distribution analytic, we approximate the optimal transport plan with batch-wise optimal transport (Tong et al. 2024).
withheld validation set. For gener-
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(a) Samples from the CFM (left), TFM (middle), and TAN (right) models.
(b) Samples from the true initial (left) and final (right) distributions.
Figure 10: True and predicted samples in the synthetic matching experiment the triangulated torus.
Evaluation. In generation tasks, evaluation was done by sampling 512 points Xo from the initial distribu- tion μuo obtaining predicted samples X, by simulating the flow ODE started X₂ and computing the 1- and Wasserstein distances between their distribution and the test set. Averaging the result over 16 independent samples to obtain final metric. For the ocean experiment we also resample 512 points from the target distribution 16 times. For the single-cell and brain datasets we simply compute the Wasserstein distances exactly between the empirical distribution of the withheld target point and the initial points transported along the flow ODE.
compute the metrics in the same way, except
E.2 IMAGE GENERATION ON CIFAR-10 10
For image generation the CIFAR- 10 dataset we used the experimental setup of Tong et al. (2024). Because it is designed for CFM, we simply replaced its corresponding components according to Table After initial testing of к {0.001, 0.01, 0.1, 1.0), chose the best-performing TFM variant with the Laplacian, which can be done efficiently due to the product structure
0.01. We
perform full cigendecomposition of the grid. All other setup pertaining to training and testing is done according to Tong al. (2024). In particular, we use a UNet (Ronneberger et al. 2015) for learning of the conditional control and compute the FID scores using clean-fiss (Parmar al.
2022
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Published as conference paper ICLR 2026
E.3 ADDITIONAL RESULTS
OT-IFM
1/TFM
OT CPM
Wasserstein distance
Figure 11: Test performance the earthquake magnitude generation experiment, in terms of W-Wasserstein distance measured over 10 independent runs. Orange bars show median value, green dashed bars show the mean, boxes show interquartile range, and outliers are shown as circles.
0T-TFM
OT-CPM
1.PM
+
3.5
4.0
4.5
5.0
5.5
6.0
2-Wassostein distance
Figure 12: Test performance on the earthquake magnitude generation experiment, in terms of 2- Wasserstein distance measured over 10 independent runs. Orange bars show median value, green dashed bars show the mean, boxes show interquartile range, and outliers are shown as circles.
OT-TFM
4
1TFM [H
CFM
1.5
Wassersteen distance
Figure 13: Test performance of topological and Euclidean flow matching models on the traffic flow genera- tion experiment, in terms of I-Wasserstein distance measured over 10 independent runs. Orange bars show median value, green dashed bars show the mean, boxes show interquartile range, and outliers are shown as circles.
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Published as conference paper ICLR 2026
(Т-TFM
HBH
[HH
OT-CFM
1.FM
1.05
1.10
1.15
1,20
1.20
1.30
1.35
1.40
Wassenstrin distanee
Figure 14: Test performance of topological and Euclidean flow matching models on the traffic flow genera- tion experiment, terms of 2- Wassersstein distance measured over 10 independent runs. Orange bars show median value, green dashed bars show the mean, boxes show interquartile range, and outliers are shown as circles.
OT-TFM
1/FF
CPM
Wasserstein distance
Figure 15: Test performance on the ocean current matching experiment, terms of 1-Wasserstein distance measured over 10 independent runs. Orange bars show median value, green dashed bars show the mean, boxes show interquartile range, and outliers are shown as circles.
QT-TFM-
PTFM
OT-CFM-
ECFM
1.35
1.40
1.45 2-Waserstein discance
1.55
Figure 16: Test performance the ocean current matching experiment, in terms of 2- 2-Wasserstein distance measured over 10 independent runs. Orange bars show median value, green dashed bars show the mean, boxes show interquartile range, and outliers are shown circles.
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Published as conference paper ICLR 2026
0Т-TFM HOH
OT-CFM
1.PM
10
distaner
Figure 17: Test performance on the brain fMRI matching experiment, in terms of 1-Wasserstein distance measured over 10 independent runs. Orange bars show median value, green dashed bars show the mean, boxes show interquartile range, and outliers are shown as circles.
OT-TFM
1-TPM
+0-
1-PPM
Wassenstein distance
Figure 18: Test performance on the brain fMRI matching experiment, in terms of 2- 2-sseserntein distance measured over 10 independent Orange bars show median value, green dashed bars show the mean, boxes show interquartile range, and outliers are shown as circles.
OT-TFM
OT CTM
FTTM
3.50
3.45
3.70
3.75
3.80
3.85
3.00
3.95
FID
Figure 19: FID on the CIFAR-10 image generation experiment, computed over 10 independent runs. Orange bars show median value, green dashed bars show the mean, boxes show interquartile range, and outliers are shown circles.
F SUPPLEMENTARY EXPERIMENTS AND ABLATIONS
F.1 EFFECT OF COORDINATE FRAME ON CFM PERFORMANCE
One way to attempt an incorporation of topological information into the neural network uf, is to predict the vector field in spectral coordinates Y U X. Algebraically, the CFM conditional vector field in spectral
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Table 6: Mean W-Wasserstein distance, +1 standard deviation, CFM in spectral and standard coordinates real-world datasets from Yang 2025).
Method (coordinates)
Earthquakes
Traffic
Brain
Single Cell
Ocean Currents
I-CFM (Spectral)
8.37 10.05
1.72+0.01
11.71 20.02
0.02240.00
1.95
1-CFM (Standard)
8.2940.05
1.7610.01
11.8640.23
0.02010.001
1.9320.02
OT-CFM (Spectral)
8.25+0.06
1.59+0.01
11.30=0.01
0.019
2.00
OT-CFM (Standard)
8.26+0.07
1.47+0.18
11.50=0.05
0.019+0.001
1.9840.02
coordinates takes the same form as in standard coordinates
(X,) I1 Zo Uki1 Ukyo Ux(11 yo) U Mu (Yi).
Our empirical results reported in Table 6 suggest that there is no meaningful difference in performance between performing CFM in standard coordinates compared with spectral coordinates on the datasets from Yang 2025).
F.2 EFFECT OF THE INITIAL DISTRIBUTION ON PERFORMANCE
Using the heat Gaussian processes exp(-kLL) can boost performance of CFM and TFM in the generation experiments with earthquake magnitudes and traffic data. The performance gain is significant for CFM and relatively small for TFM. We report the results in Table 7
Table 7:
W-Wasserstein distance for TFM and CFM compared across the normal and heat Gaussian process (GP) initial distribution the traffic flows and earthquake magnitudes experiments.
Dataset
Initial Distribution
I-TFM
OT-TFM
I-CFM
OT-CFM
Normal
1.30
1.28+0.01
1.72+0.01
1.59+0.01
Traffic
Heat GP
1.27
1.27
1.47
1.45
Normal
5.35
5.4940.10
8.37
8.2540.06
Earthquakes
Heat GP
4.93
5.53
7.02+0.07
7.39
F.3 EFFECT OF K ON PERFORMANCE
The parameter in the topological drift L X, dt may be further tuned improve performance of TFM. This is shown for range of values across the earthquake magnitude, traffic flow, brain fMRI, single-cell differentiation, and ocean current experiments in Figures 20 to 24 Except for the single-cell experiment, appears that the 1- and 2- 2-Wasserstein distance is approximately convex in K.
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20
I-TPM
I-TPM
20.0
OT-TFM
OT-TPM
I-Wassersteil Distance
2- Wasserstein Distance
17.5
15.0
14
12
10
10.0
7.5
5.0
4.0
5.0
1.5
2.0
Figure 20: Test performance of I-TFM and OT-TFM across range of x choices the earthquake magnitude generation experiment over independent runs. Bars show median value, boxes show interquartile range, and outliers are shown circles.
OTFTFM
1.06
OT-TFM
1.30
I-TMM
1-TFM
1-Wasserstein Distance
Distance
1.04
1.25
1.02
Wasserstein
1.20
0.98
1.15
0.96
1.10
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0 18.0
20.0
2.0 4.0
6.0
S.0
12.0 14.0 16.0 18.0 20.0 10.0
K
Figure 21: Test performance of 1-TFM and OT-TFM across range of choices on the traffic generation experiment over independent runs. Bars show median value, boxes show interquartile range, and outliers are shown as circles.
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Published as conference paper ICLR 2026
I-TFM
I-TFM OT-TFM
8.0
OT-TFM
L-Wasserstenl Distance
2 Wassersten Distance
7.5
5.5
7.0
5.0
6.5
6.0
5.5
1.0
1.5
2.0
2.5
3.0
4.0
5.0
6.0
7.0
8.0
1.0
1.5
2.0
2.5
3.0
4.0
5.0
6.0
7.0
8.0
K
Figure 22: Test performance of 1-TFM and OT-TFM across range of к choices the brain fMRI matching experiment over 5 independent runs. Bars show median value, boxes show interquartile range, and outliers are shown circles.
1-Wasserseeh Distance
0.0325
FIFMM OT-TFM
2- Wassersseei Distance
0.11
OT-TFM
ITEM
0.0300
0.10
0.09
0.08
0.0225
0.07
0.0200
0.06
0.0175
0.05
0.04
0.0150
20
2.5
3.0
4.0
0.0
10.0
25.0
(00.)
10060.00
1.0
1.5
2.0
2.6
3.0
4.0
5.0
10.0
25.0
100.0
250.0
560
1000..0
1000000
K
Figure 23: Test performance of I-TFM and OT-TFM across range of choices the single cell differ- entiation matching experiment over independent runs. Bars show median value, boxes show interquartile range, and outliers are shown as circles.
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Published as conference paper ICLR 2026
OT-TFM I-TFM
OT-TFM
I-Wasserstein Distance
2.4
2. Wasserstein Distance
1.8
I-TFM
2.2
1.6
1.4
1.6
1.4
1.2
2.0
25.0
0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
500.0
1000.0
2.0
25.0
50.0
100.0
150.0
200.0
250.0 a
300.0 o
350.0 o
500.0
1000
Figure 24: Test performance I-TFM and OT-TEM across a range λ choices on the ocean current matching experiment over 5 independent runs. Bars show median value, boxes show interquartile range, and outliers are shown as circles.
29


Reviews:
[{"review_id": "7x5LhWTACq", "reviewer": "Reviewer_kNj3", "summary": "This paper proposes a flow matching approach to generative modelling over nodes, edges, triangles, and so on via the use of Hodge-Laplacian-based linear conditional vector fields. They theoretically build up to this through a Schrodinger bridge formulation and eigendecomposition analysis of the Hodge-Laplacian. They offer two variants of their approach – I-TFM (independent coupling) and OT-TFM (mini-batch OT coupling). Overall, they demonstrate empirical improvements over prior work: modestly for image generation and more significantly for generation on irregular topologies.", "questions": "1. Can you elaborate a bit more on why we had to go through the Schrodinger bridge problem formulation to end up with the result? Why can't you remain in the deterministic setting of flow matching, or make use of the probability flow ODE?\n2. Why is I sometimes better than OT and vice versa?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Good results: on images, and better on irregular grids.\n2. Good theoretical build-up.\n3. Two variants: I-TFM and OT-TFM.\n4. Generality of Hodge-Laplacian bodes well for future extensions.", "weaknesses": "1. In section 2.1 in the Graphs paragraph, the subscript notation was a bit confusing for me. After reading the Simplicial complexes paragraph, I understood. But I think the flow could be improved.\n2. Didn't see an ablation of different prior choices, e.g., standard Gaussian vs. heat Gaussian.\n3. No experiments for $k \\geq 1$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6Qx7m17v0D", "reviewer": "Reviewer_8d47", "summary": "This paper extends Flow matching to the  general topological structure on which the Hodge Laplacian is well defined. The paper presents the flow matching as the variance0  limit of the Schrodinger bridge, and extends it to the Simplicial complex in a computable form by changing the reference process that acts as the regularizing anchor from which the solution flow can not diverge too greatly. The reference process is changed from the usual Wiener process to a process with Laplacian drift. \nMost importantly , the paper's proposed loss objective maintains that the solution of their topological analogue of the Schrodinger bridge problem can be computed with the coupling that can be optimized with the loss of the closed form. \nThe method is shown competitive on various datasets.", "questions": "1. My greatest question is posed in the part 2 of the weaknesses. \n\n2. Would you please mention more about the hyperparameter kappa?  This shall be quite important as it desribes the speed of the diffusion. The choice of this parameter may also relate to question1.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 4, "presentation": 1, "contribution": 4, "strengths": "This paper not only tackles an important problem of extending FM to general geometry (not necessarily Riemannian), it showcases a beautiful analogue between the propositions and the classical results. The way that the paper is narratin/organizing the classical works are very instructive and that on its own is a contribution to the community. \nThe method itself seems impactful when polished with appropriate engineering techniques, showing much potential for future applications.\nThe idea of changing the reference process to establish the unifying method with consistency across different topology in a scalable way, is also an approach that I as a researcher as waiting for a long time as well, because the  \"wiener reference process\" has long been adopted as a convention that does not always match the data-domain of interest.", "weaknesses": "I believe that the paper suffers much from the presentation. \n\n\n1. From place to places, the conditions of the theoretical results are \"embedded\" in the paragraphs, for instance, in the first subsection of 3.1 (1. Conditional vector field ....  The diffusion briged in Eq 7 simplifies to)... shall, at least in my understanding, read as \"when $b=0$ and $\\sigma_t=\\sigma_0$, the diffusion bridge SDE in equation 7 simplifies to\".  Indeed, this is stated at the head of the 3.1, but it is separated by the section header and it took a while for me to make the connection.  I got lost several places through out, I would very much appreciate if the preambles of the theoretical statements are more complete and self contained inside each \"thm environment\". \n\n2. Also, the reason of using the reference process of the proposed form seem to lack strong enough motivation.  I buy that this choice leads to very clean results and introduces a topological heat equation flavor, but why this choice, more precisely?   Using a refernece process \"with drift\" seems like a wild card that I have never seen out of finance. The effort  of the justification seems to be made in 3.2 but it did not click with me----why mix Wiener with (11) instead of completely replacing it with, say, a  corresponding stochastic analogue of heat equation in the topological space (maybe a one driven with an ensemble of random walk)?  Is additive mix with standard wiener process a logical choice?  What extreme case is this choice supposed to resolve? \n Because the paper is presenting a method that I believe, is intended to merit the applied community of wide range, I believe that the audience deserves further elaboration.   Correspondence table of comparison against standard Schrodinger may also be helpful. \n\n\n3. In a related note with 2, a demonstrative extreme case experiments produced by synthetic data might have been helpful. \n\n\n4.  Figure 2 did not appear too demonstrative to me, what advantageous aspect of the method is this figure supposed to convey?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "llYawfzK9r", "reviewer": "Reviewer_Dei1", "summary": "This work proposes topological flow matching model, which is derived by first solving a Schrodinger Bridge Problem with drift, and then taking the noise scale to 0. The proposed model is designed to model data with a graph structure. The math is carefully derived and experiments are conducted on several topological datasets with the topological Schrodinger Bridge Matching as the only baseline.", "questions": "1. how should I interpret the SBP when $\\sigma \\to 0$? It seems like the reference process then becomes an ODE? What if you just solve the SBP with zero-noise? Would it still be a well-posed optimal transport problem? If so, will it lead to the same vector field on line 246?\n\n2. What motivates the particular form in (11) and $H_t$ as a polynomial? \n\n3. The form of (11), the change of coordinate interpretation (around line 220), and the infinite dimensional generalization (line 781-795) in the appendix altogether present significant resemblance in form with some recent works on diffusion/flow models on functional space, like [1, 2, 3, 5]. Do you think there is any potential connections? Maybe the functional models can be deduced from the SBP perspective or your proposed model can be considered as a special instance of functional model on non-Euclidean space?\n\n4. Isn't line 134 also called rectified flow [4]?\n\n\nReferences:\n\n[1] Zhang, J., & Scott, C. (2025). Flow Straight and Fast in Hilbert Space: Functional Rectified Flow. arXiv. https://arxiv.org/abs/2509.10384\n\n[2] Kerrigan, G., Migliorini, G., & Smyth, P. (2023). Functional Flow Matching. arXiv. https://arxiv.org/abs/2305.17209\n\n[3] Na, K., Lee, J., Yun, S.-Y., & Lim, S. (2025). Probability-Flow ODE in Infinite-Dimensional Function Spaces. arXiv. https://arxiv.org/abs/2503.10219\n\n[4]Liu, X., Gong, C., & Liu, Q. (2022). Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow. arXiv. https://arxiv.org/abs/2209.03003\n\n[5]Franzese, G., Corallo, G., Rossi, S., Heinonen, M., Filippone, M., & Michiardi, P. (2023). Continuous-Time Functional Diffusion Processes. arXiv. https://arxiv.org/abs/2303.00800", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "1. This work proposes a novel generative framework for data embedded in graphs. \n2. The proposed model can be trained simulation free and samples can be generated by solving ODE.\n3. This work reveals an interesting connection between Schrodinger Bridge and flow models, i.e., a flow model can be obtained by solving a SBP with diminishing diffusion noise.", "weaknesses": "First, I think the overall presentation is unsatisfying to the extent that I'm not able to appreciate some of the authors' major claims. Especially,  \n\n* The word \"topology\" is not precisely defined in the paper. I could only guess from the context that it is related to the graphical structure of the data, which does not seem to be a standard definition for me (For me, the standard definition of topology refers to a topological space in analysis).\n  \n* It is not clear to me where does the matrix $L_k$ come from and how the data $X_1$ is related to the matrix $L_k$. From the context, I can only infer that $L_k$ is determined by some domain experts or auxiliary knowledge and is independent from data $X_1$.\n\n* As a consequence of 1 and 2, it is not clear to me on how the flow ODE ignores the topology while the proposed method retains it around line 247, which seems to be an important advantage of the proposed method.\n\nOn the other hand, I think the empirical validation is insufficient. While this work claims standard FM in Euclidean space ignores the rich graph structure on data, it does not empirically compare the proposed method against generative models in Euclidean space. \n\nAdditionally, the equation on line 220 seems to suggest that we can diagonalize the flow with a change of coordinates. If so, what is the advantage of the proposed method against  1. performing change of coordinates 2. then applying generative models in Euclidean space. 3. transforming the generated data back to the original coordinates when sampling.\n\nFor the ACs and the authors to better understand my position, I'm familiar with SDE, diffusion, and flow models and I have some exposure to Schrodinger Bridge literature. But I have no working knowledge on data with a graph structure. \n\nI'm happy to increase the score if the authors could help me better appreciate their work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "85MJvbaXOF", "reviewer": "Reviewer_g1EL", "summary": "The authors then derived topological flow matching based on the following two ideas or framework:\n1. the unifying framework between the popular flow matching and Schrodinger bridge matching methods in terms of optimal transport (OT).\n2. a piece of recent work generalizing the classical schrodinger bridge matching to the topological setting ---- where a brownian motion reference process is generalized by a topological diffusion driven reference process.", "questions": "1. I quickly skimmed the recent work by Yang 2025 on topological bridge matching, and noticed there is also a section about topological flow matching. I suggest authors to address the main contributions, difference or improvement of this work compared to the previous work. \n2. In experiments, how do you optimize the diffusion coefficient $\\kappa$? Could you elaborate experiment findings, in terms of, e.g., topological diffusion? Moreover, could we measure the training efficiency of flow matching compared to bridge matching? Without this, it's hard to see the advantage of flow matching besides improving the numbers. This is why I think the presentation can be improved by reducing some classical results in the main text so you have space to discuss your results and method.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 1, "strengths": "This paper studied the flow matching method in the topological setting and the experiments are diverse.", "weaknesses": "1. In my opinion, both methods, flow matching and bridge matching, have their pros and cons, where the former is more straightforward and efficient in terms of training, whereas the latter is more detailed in terms of modeling the intermediate distributions. Intermediate states modeling and learning are quite important in many engineering and science problems, e.g., single-cell dynamics, or the other trajectory related problems in this paper. For this reason, I think the paper can be improved by considering trajectory related tasks in topological settings, otherwise, it's hard to capture the advantage of flow matching besides improving the numbers. \n2. I think the paper presentation can be improved by removing some previous results like flow matching and classical bridge matching and  elaborating the main topological related results, so you have more space to explain these results, instead of just layout tons of equations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)