Riemannian Variational Flow Matching for Material and Protein Design
Authors: Olga Zaghen
Floor Eijkelboom
Alison Pouplin
Cong Liu
Max Welling
Jan-Willem van de Meent
Erik J Bekkers
Venue: ICLR 2026 Poster (2026)
Keywords: Flow matching
variational inference
riemannian manifolds
material generation
metal-organic framework
protein backbone generation
Primary area: generative models
TL;DR: We derive a variational objective for flow matching on manifolds with closed-form geodesics and test it on material and protein backbone generation.

Abstract: We present Riemannian Gaussian Variational Flow Matching (RG-VFM), a geometric extension of Variational Flow Matching (VFM) for generative modeling on manifolds. Motivated by the benefits of VFM, we derive a variational flow matching objective for manifolds with closed-form geodesics based on Riemannian Gaussian distributions. Crucially, in Euclidean space, predicting endpoints (VFM), velocities (FM), or noise (diffusion) is largely equivalent due to affine interpolations. However, on curved manifolds this equivalence breaks down. We formally analyze the relationship between our model and Riemannian Flow Matching (RFM), revealing that the RFM objective lacks a curvature-dependent penalty -- encoded via Jacobi fields -- that is naturally present in RG-VFM. Based on this relationship, we hypothesize that endpoint prediction provides a stronger learning signal by directly minimizing geodesic distances. Experiments on synthetic spherical and hyperbolic benchmarks, as well as real-world tasks in material and protein generation, demonstrate that RG-VFM more effectively captures manifold structure and improves downstream performance over Euclidean and velocity-based baselines.

Published as a conference paper at ICLR 2026
RIEMANNIAN VARIATIONAL FLOW MATCHING FOR MATERIAL AND PROTEIN DESIGN
Olga Zaghen' Floor Eijkelboom 1.2* Alison Pouplin 1* Cong Liu
Max Welling
Jan-Willen van de Meent Erik J. Bekkers AMLab, Bosch- Delta Lab, A14Science Lab, University of Amsterdam
Aalto University
o zaggenegguaa uva.n1
ABSTRACT
We present Riemannian Gaussian Variational Flow Matching (RG-VEM), a ge- ometric extension of Variational Flow Matching (VFM) for generative modeling manifolds. Motivated by the benefits of VFM, we derive variational flow matching objective for manifolds with closed-form geodesics based on Rieman- nian Gaussian distributions. Crucially, in Euclidean space, predicting endpoints (VFM), velocities (FM), or noise (diffusion) is largely equivalent due to affine interpolations. However, We formally analyze the relationship between our model and Riemannian Flow Matching (RFM), revealing that the RFM objective lacks curvature-dependent penalty encoded via Jacobi fields that is naturally present in RG-VFM. Based on this relationship. we hypothesize that endpoint prediction provides stronger learning signal by directly minimizing geodesic distances. Experiments on syn- thetic spherical and hyperbolic benchmarks, as well as real-world tasks in mate- rial and protein generation, demonstrate that RG-V CG-VFM more effectively captures manifold structure and improves downstream performance over Euclidean and velocity- based baselines.
curved manifolds this equivalence breaks down.
INTRODUCTION
Generative models play central role in machine learning, as they provide way to synthesize data and learn complex distributions. Diffusion models (Ho et al. 2020; Song et al. (2020) achieve state- of-the-art performance, but rely a fixed Gaussian noising process with predetermined variance schedules. As result, the reverse process is tied to this prescribed family of Gaussian marginals, and sampling requires numerical integration with diffusion speciffc samplers. In contrast, Continuous normalizing flows (CNFs) (Chen et al. 2018) directly learn the vector field of an ODE that transports a base distribution into the data distribution. In principle, this allows the transport path to be fully learned, but both training and sampling are computationally demanding since likelihood evaluation involves solving a high-dimensional ODE (Ben-Hamu et al. 2022, Rozen et al., 2021]; Grathwohl al. 2019). Flow Matching (FM) (Lipman al., (023) offers simulation-free alternative, as defines sam-le interpolants between the source and the target samples, and regresses the vector field to known conditional velocities.
Recent developments have extended flow matching in two key directions. Variational Flow Match- ing (VFM) (Eijkelboom al. 2024) reframes the problem posterior inference trajectories, providing a probabilistic perspective with flexible modeling choices. In parallel, Riemannian Flow Matching (RFM) (Chen & Lipman| 2024) has shown how incorporating -uclllidean geometry can improve modeling of distributions supported on manifolds.
VFM has demonstrated advantages over standard FM in discrete domains (e.g., CatFlow) and has been extended to mixed data modalities (Cuzmán-Cordeen Cordero et al. (225) as well as molecular gener- ation tasks (Eijkelboom et al.) 2025 Sakalyan et al.). key strength of the variational formulation is its flexibility: problem-specific constraints can be incorporated directly into the objective. For
$ These authors contributed equally.

---

Published as a conference paper at ICLR 2026
instance, censored flow matching for sea-ice forecasting enforces physical bounds such as non- negative ice thickness through the variational loss (Finn et al. 2025). Similarly, when the target distributions live Riemannian manifold, the objective can be formulated to respect the geome- try of the data support, which is the setting we study in this paper.
The geometric extension is particularly relevant for biological and chemical domains where intrinsic geometric structure governs the data. Recently, generative models have been extensively applied
material discovery (Jiao et al., 2023; Fu et al., 2023; Kim et al. 2024) and to the generation of large biomolecules such as protein backbones (Guo et al. 2025 Yue et al., [222 Yim et al., 2023a b). These applications highlight that data often live on heterogeneous manifolds: Euclidean space for atomic coordinates, rotation groups SO(3) for orientations, and other structured domains. Early works in crystal generation, such as Jiao et al. (2023), focus purely on Euclidean parameters without explicitly modeling rotational degrees of freedom. In contrast, recent methods for metal- organic frameworks (MOFs) and proteins (Yim et al. 2023a, Yue et al.] 2025; Kim et al., [024]] Guo et al. 2025) adopt mixed approach where Euclidean parameters (e.g., positions) modeled with standard FM while non-Euclidean parameters (e.g., rotations) are modeled with RFM. These methods lack fully variational treatment across both parameter types, and address this gap by demonstrating the benefits of our geometric variational approach on these applications.
When extending from Euclidean space general Riemannian manifolds, it becomes unclear how different generative objectives relate. In Euclidean space, training a generative model by predicting an endpoint (VFM), a velocity (FM), noise (diffusion) is largely equivalent up to affine transfor- mations, since these quantities noise, score, velocity field, and endpoints - are interchangeable parameterizations of the same training signal (Vuong et al.] 2025 Lipman al. 2023 Eijkelboom et al. 2024). On curved manifolds, however, this near-equivalence breaks down: tangent spaces vary across points and curvature introduces higher-order deviations, preventing any explicit closed-form relation between the velocity-based and endpoint- based perspectives. This raises the questions: if the endpoint and velocity-based perspectives are no longer equivalent on curved manifolds, how do they differ, and which one is preferable?
In this paper, we develop Riemannian Gaussian Variational Flow Matching (FM) which ex- tends VFM to Riemannian manifolds with closed-form metrics, thereby bridging the variational and geometric extensions of flow matching. Our contributions are threefold:
We define variational flow matching objective for general geometries, extending endpoint-based training to manifolds.
We formally analyze its properties, establishing how RG- VFM relates to RFM and showing that the gap between them encodes curvature through Jacobi fields.
We demonstrate that variationalizing existing geometric generative models in material and protein design consistently improves performance, highlighting the practical advantages of endpoint-based training.
BACKGROUND
Flow Matching. Modern generative modeling interprets sampling from a target distribution Pr as transporting base distribution Po by learning dynamics. Typically, po is a standard Gaussian, and the transformation follows time-dependent mapping yo: (0, 1] x R
IR where yo is the identity and 11 pushes Po onto P1. E.g., normalizing flows (Chen et al. 2018) use an ODE governed by some time-dependent velocity field u, Though likelihood training is possible through the change of variables formula, solving an ODE during training is expensive.
Flow Matching (FM) (Lipman et al. 2023 Liu et al. 2023: Albergo et al. 2023) bypasses this by defining an interpolation between noise and data, and directly learning the associated velocity field in supervieed manner. Though the goal learn the intractable objective
(1) this can be made computationally feasible by reformulating u, with a conditional velocity field (i.e. assumed dynamics towards a given Iı, time derivative of the interpolation), giving rise to Conditional Flow Matching (CFM):
(FM(O) = E., [1)()) of - (=)
LCPM(0 Ry [[ue(2] (1) (x)))
2

---

Published as a conference paper at ICLR 2026
Minimizing eq. (2) provides unbiased estimate of VeLFM, allowing efficient per-sample training. As FM can seen as regressing directly onto the derivative of an interpolant between source and target in self-supervised manner, it provides unifying framework: by choosing different interpo- lations, dynamics, or conditioning structures, it can be adapted to various data types and constraints.
Riemannian Flow Matching. Riemannian Flow Matching (RFM) (Chen & Lipman, 2024
tends FM to Riemannian manifolds. Given a smooth Riemannian manifold M with closed-fon geodesics and metric g, RFM learns vector field uy:
(R) = Ex,,, [l]](x) -
with log,((z) (z1) denoting the Riemannian log map, which returns the initial velocity vector of the geodesic connecting x to 21 (more details on Riemannian manifolds are in section C.I).
Unlike Euclidean Flow Matching, RFM respects the curvature and geodesics of the underlying space M. Through geodesic or spectral distances, enables simulation-free training when manifold op- erations are available, and can utilize approximate distances when closed-form geodesics are in- tractable, maintaining theoretical guarantees while enabling efficient generative modeling.
Variational Flow Matching. Variational Flow Matching (VFM) (Eijkelboom et al. (2224) refor- mulates FM by introducing a variational distribution cf(<1 z)) to approximate the unknown pos- terior Pa pa z), where the learned velocity vy is expressed as the expectation of the conditional velocity under this variational approximation over trajectories. Then, the VFM objective is to mini- mize the KL divergence between joint distributions, i.e.:
(Q) E, KL (p) |I (x1, = [og de (a1 [)] + const.
(4)
When uz œ is linear in e.g. a straight-ling line interpolation the expectation depends only on marginal distributions, implying this objective reduces series of D univariate tasks:
LVPM(() E,
Σ log of x)) e.g. LvrM(O) E,,,, [[]((()) 21 [|]
(5)
in case of is Gaussian, relating VFM directly back FM (see Eijkelboom al. (0224) for details). For sampling with the standard flow matching case of linear interpolation, the vector field reduces the first moment of the variational approximation:
21
E
B (ca (m) [zi] -
(x(x)) 1-t
o (x) - (a)l)))
(6)
1-t
1 1-t
A key feature of VFM is its flexibility in choosing dh as different choices allow adaptation to various geometries and data types, improving efficiency and expressiveness.
RIEMANNIAN GAUSSIAN VARIATIONAL FLOW MATCHING
The geometric generalization of the VFM framework stems from the observation that the posterior probability (x .:) implicitly encodes the geometry of the distribution's 's support. For example, in CatFlow (Eijkelboom et al. 2024), defining of [x1 x)) as a categorical distribution ensures that the velocities point towards the probability simplex. This raises the question of whether other geometric information about the support of can be similarly encoded in oe (x1 x).
To investigate this, consider the case where Pe(a1 2) is defined as Gaussian distri- bution with its support on a general manifold M == supp(p) rather than being restricted to Fuclidean space. In this setting, the Rieman- nian Gaussian distribution naturally arises as a generalization of the Gaussian to a Rieman- nian manifold. We refer to velocity-inferring methods (CFM and RFM) as vanilla models and endpoint-inferinng methods (VFM and RG- VFM) as variational models.
Po
P1
Variational
P0
P1
generalizazion
CFM
VFM
Geometric
Geometric generiizzatin
generalization
RFM
RG-VEM
Variational
perspective
Pb
Figure 1: Overview of the models relevant for
framework. The square represents Fuclidean space, while the sphere represents general M.
The advantages of the variational perspective in geometric setting are twofold:
3

---

Published as a conference paper at ICLR 2026
Jacobi Field
Riemannian Intrinsic: RFM RG-VEM
Euclidean:
Riemannian Extiinsc::
GFM VFM
RG- RG-VFM
Wee
Mine
b
Figure 2: a: Representation of shooting family of geodesics on S with corresponding Jacobi field. b: Visualization of predictions (color-coded to match the name colors) each model relevant to our framework, for target distribution p1 supported the sphere S²
Flexibility on the support of the distribution: the prior po can be defined either on M (in- trinsic) in the ambient Euclidean space (extrinsic), while vanilla RFM only supports the intrinsic viewpoint. The extrinsic framework maintains the simplicity and efficiency of a linear flow in Euclidean space, avoiding the need for the manifold's exponential and loga- rithmic maps, while encoding more geometric information than purely Euclidean methods (section 3.1). Note that, while intrinsic and extrinsic traditionally refer the manifold's internal geometry versus its embedding, we use these terms to distinguish whether points lie on the manifold or in the ambient space, rather than coordinate choices. For example, our intrinsic framework can be expressed using ambient coordinates.
Supervision on the endpoints, rather than on the velocities, by minimizing their geodesic distance on the manifold, which in practice leads more effective learning of the signal. We show this in section 4 by reformulating the objective through Jacobi fields.
3.1 THE RIEMANNIAN GAUSSIAN VFM OBJECTIVE
To extend VFM to the geometric case, one first needs to define a relevant variational posterior with support over the manifold. In contrast to Euclidean settings, we need to take particular care to ensure the distribution is properly defined on the manifold. Let M be a Riemannian manifold with metric g: the Riemannian Gaussian (RG) distribution (Pennec) (2066) is defined as the maximum-entropyy distribution specified by its mean value and covariance, formally
Ngiim (z a. (1) C exp (- disst((,, 1))
26
where 24 E M (with μ the mean). o > geodesic distance determined by The constant C' depends on both and μl, and it normalizes the distribution over M. A more detailed geometric explanation can be found in section C.2.
scale parameter, and distg (5ngg(2,))) p) denotes the
We define the Riemannian Gaussian VFM objective by using the Riemannian Gaussian as our vari- ational approximation, i.e.
LRG-VIM(P Er,ay,z [- log Nuiim (a) (x).))),,)))) (x)))
In the Euclidean Gaussian VFM case, this setting reduces to straightforward mean squared error optimization, so it is natural to wonder whether a similar simplification holds here. In fact, such a simplification exists under two assumptions: (1) the manifold is homogeneous that is, any point can be transformed into any other by a distance-preservinng symmetry (a formal definition is provided in section C.T); and (2) we have access closed-form expression for its geodesics. Notably, these requirements are not too restrictive, as most manifolds used in deep learning satisfy them, including S". H!" T", and SO(n.. Formally, the following holds (see section D.I for details):
Proposition 3.1. Let M be homogeneous manifold with closed-form geodesics. Then, the RG- VFM objective reduces to
LRG-VFM E [|] log,,
( E,,r1. |dist (2 (f
(9)
where log denotes the logarithmic map the manifold and dist. the geodesic distance.
4

---

Published as a conference paper at ICLR 2026
Algorithm RG-VEM intrinsic
Algorithm 2 RG-VFM extrinsic
Require: base p E M, target e M.
Require: base p IK', target M.
Initialize parameters of μ4:
Initialize parameters 0 of j4r
# Training Phase
# Training Phase
while not converged do sample ~ (0, 1). 20 p1 ~
while not converged do sample / (/(0, 1), 30 ~ P.
compute geodesic interpolation:
compute linear interpolation:
Et = exPxo (L- oo (#1))
21 f.
-) 20
((0) E,,,, distt (21, M4 (2;; 0))
(()) Er ( pe(a;; 0))
opt (errss(ee(((())
0 = opt imi zer step({(0))
end while
end while
# Generation Phase
# Generation Phase
sample noise zo
sample noise 2o p
21 so LODE [0. 1], mo: 10sx, (ua (x):88)
=olvve ODE (0., 2o zo. de (m
1-1
Minimizing this loss is equivalent to computing the Fréchet mean of the distribution, that is: μ² arg min Ex, (ditt(()..p))) averaged over the generative steps and samples In other words, the point 440 minimizes the expected squared geodesic distance to the target (Frécher, 1948). Intuitively, this can be viewed as generalization of the mean squared error from the Eu- clidean setting to Riemannian framework. We obtain this result by assuming that (x(x)) is constant. Nonetheless, this term could for example be set to (x) = 1 to achieve time normalization, as done in our material and protein generation experiments (sections 5.2 and 5.3).
The RG-VFM objective (eq. (9)) minimizes the geodesic distance on .M between predicted and target endpoints, so it only needs to capture the local geometry around p1. This allows for flexible choice of po support, leading to two plausible model variants when M is embedded RR:
RG-VEM-R": the prior Po is Euclidean with M & supp(ps) = 1n and conditional velocities use linear interpolation in the ambient Euclidean space R";
2. RG-VEM-M: the prior intrinsic (supp(Pu) S M) with conditional velocities defined via geodesic interpolation on tangent spaces. Here, no embedding of M in IR" is required.
The extrinsic variant RG-VFM-E" thus learns simple linear flow while retaining geometry-ave aware loss, whereas the intrinsic variant RG- VFM-M mirrors the RFM setup but differs in its loss defi- nition (algorithms and 2). Indeed, Vanilla RFM also requires supp(Pe) = .M because its vector fields depend on the manifold's s intrinsic geometry. Because of these different frameworks, direct comparison is only meaningful between RG-VFM-2 and RFM, comparison present in the next section. The choice between intrinsic and extrinsic versions represents a trade-off: the extrinsic version can only be used in an ambient space R of sufficiently large dimension embed the man- ifold without degeneracy. In such cases, linear interpolation simplifies implementation and reduces computational costs by requiring only the geodesic distance, rather than logarithm and exponential maps at every step.
RG-VEM VS RFM: A COMPARISON BASED ON JACOBI FIELDS
In this section, we refer to RG-VFM-M simply as RG-VFM for brevity. Given a sampled inter- mediate point at timestep I, our variational loss LRG. measures the geodesic distance between target and predicted endpoints on the manifold. In contrast, the vanilla loss CRFM in Riemannian flow matching compares target and predicted velocities in the tangent space at that point.
In Euclidean space, these two formulations coincide since the difference between the endpoints is directly proportional to the difference between their initial velocities. But in curved space, this equivalence breaks down: curvature influences how geodesics separate from one another.
In this section, examine how small changes in initial velocities affect geodesic endpoints by constructing families of related geodesics. In differential geometry, those variations are described by Jacobi fields, which characterize how geodesics spread apart Riemannian manifold. We
5

---

Published as a conference paper at ICLR 2026
use this framework to establish the connection between the vanilla and variational loss functions. Specifically, we (1) define a Jacobi field formulation of the RFM and RG- losses in section 4.1, (2) derive the relation between these field- field-specific instances (proposition 4.2). and (3) eventually establish the corresponding loss relationships in proposition
4.1 JACOBI FIELD FORMULATION OF THE FLOW MATCHING OBJECTIVES
We consider a smooth family of geodesics (7a} all starting from the same point 7o(0) = xn E M.
and determined by an initial velocity of the form 7 (0) = of == v + sw, with v. w E T M,
where su represents the perturbation level. A schematic representation is in fig. 2 (a).
[0, 1] which indexes the perturbation of its initial velocity, and T E |0, 1J. the parameter along one geodesic that links the initial point % (0) = to the endpoint (1) = z². For convenience, we denote (s,,T) := (s(T) the two-parameter map which simultaneously describes the entire family of perturbed geodesies. Definition 4.1 (Jacobi field at a vanishing starting point). The family of geodesics defined as:
Each geodesic (s, r) - (T( is parametrized by two variables:
als, T) == T - exP22o (T + sw)),
with E [0, 1] and 7 [0, 1J. u, E Mt is smooth family of shooting geodesics with (0 To. (0) = v and 1/2 (0) = v + w.
For each fixed E [0, 1]. there exists vector field, called Jacobi field,
J(T) == (ea((s)))sss
along the geodesic (T(T) == T) satisfying the ODE equation: D + R(J. = 0, with R the Riemannian curvature tensor of the manifold. In particular, this Jacobi field is uniquely defined by the initial conditions and at T 0 one has the initial conditions: J(0) = 0, and J+J((0) w.
Borrowing the notations from (Chen & Lipman 2024). denote the target velocity w0 ue(x | J,). the predicted velocity e² = vy (Ir, and their respective endpoints (1) = x1 and (1)
14 (2). The losses be formulated in the Jacobi field framework with the following:
Proposition 4.1 (Loss functions as evaluation of Jacobi fields). Consider a Jacobi field (() := defined in definition 4.1. We denote Lurm the loss function of the (vanilla) Rie- mannian Flow Matching (Chen & Lipmant 2024), and LRG-VFM VPM the loss function for our proposed Riemannian Variational Flow Matching. Then the following equalities hold:
0,a(s, T)
(R)M(()= Ee an [lun(1 mi of (x) E.,,y [I),((0)
(10)
LRG.VEM E [[logg, (1 (z)) [2] R. [1[())
(11)
4.2 RELATION BETWEEN RG-VFM AND RFM OBJECTIVES
Now that we have expressed the losses through the Jacobi fields, we observe that CRFM is first- order approximation of LnG VFM through the following proposition:
Proposition 4.2. (() is linear approximation of J(1).
The proof essentially consists of deriving the Taylor expansion of JT)), centered at T = 0 and evalu- ated at 1, and identifying (0) as the linear term. By truncating at the linear approximation, curvature information is absent from ( but remains implicitly encoded in (11. This distinc- tion directly affects the relationship between the RFM and RG-VFM losses: while they coincide in Euclidean space, their difference in curved spaces is generally nonzero and curvature-dependent.
Euclidean case. In Euclidean space, the Taylor expansion reduces to the linear term: J(r) ((0) + (O) which, for r = 1 and J(0) 0, leads to J(1) J(). As consequence,
E 2 [] (O)
E [1]
(12)
which confirms that LCFM and LVPM be reduced to another, with proper normalization terms.
More generally, the two losses differ by a curvature vurvature-dependent term on non-fla flat manifolds, as shown in the following result as direct consequence of proposition 4.1 and proposition 4.2

---

this setting, we can restate the result of proposition 4.3 in terms of the constant sectional curvature K. The formulation and proof are given in corollary D.I, and in the experimental Published as a conference paper at ICLR 2026
Proposition 4.3 (Difference of loss functions curvature term). Consider Jacobi field [T)
(,((s)))._o as defined in definition 4.1 and the equivalences shown in proposition 4.1 The differ-
ence between LRG-VEM and LRFM encodes the manifold curvature through:
(RG-VFM(() (R) LL D,) v) Enigher
(13)
cervoture-dependent term
where the leading- ooder curvature functional is:
1 (VV.1110,,10,00000,,.0000 (14) D, = - (I) ( and Enigher = 0(]D)-]())]113
(15)
with the Riemannian curvature tensor and v 5/o the reference geodesic velocity. The higher- order term Enigher encodes curvature variation along geodesics through covariant derivatives of R. In terms of the RFM loss terms, uu( 21) and (() (x) ue(x x1).
Geometric interpretation. The curvature functional C captures how the manifold's geometry af- fects the loss comparison, encoding the first- and second-order effects of curvature on geodesic deviation. Thus, RG-VFM implicitly captures the full geometric structure through the exact Jacobi field J(1), while RFM uses only the linear approximation (O). This lack of curvature infor- mation results in weaker, less precise supervision in directing the flow toward the actual endpoint, leading in practice to RG- learning the signal more effectively than RFM. Special cases are:
In Euclidean space, R implies both C -0 and Enigher
O. This leads LRG VFM
LVFm = LCFM = CRFM as expected from eq. (12).
In spaces of constant curvature (e.g. hyperspheres or hyperbolic spaces) RR 0. In
section we focus primarily on manifolds that fall within this category.
alternative to RFM for learning velocity field manifold, providing a variational formulation whose objective fully captures higher-order curvature effects, unlike RFM. This results in generally different objectives on curved manifolds. In Euclidean space, however, the RFM objective reduces to CFM, while RG- CFM reduces to VFM (assuming a Euclidean Gaussian posterior rather than Riemannian), and the CFM and VFM objectives become equivalent under appropriate normalization. These relations are schematized in fig. | and fig. 2(b),, and their schematic algorithms can be compared in section D.2 terms of computational costs, extrinsic RG- VFM has the same complexity as VFM during both training and sampling. The only difference between the two methods is that VFM compares endpoints using Euclidean distance, while we use geodesic distance. Since we assume geodesic distance to be in closed-form, this introduces no additional computational overhead compared to VFM. Similarly, implicit RG-VFM maintains the same complexity as RFM, with the main difference being that velocity computation happens during sampling rather than during training (see algorithms to 8).
In summary, we introduced RG-VFM
EXPERIMENTS
Goal of the experiments. The goal of our experiments is twofold. First, we aim to observe the practical implications of proposition 4.3 by studying the behavior of vanilla and variational models, both Euclidean and Riemannian, in a controlled synthetic setting with a visually precise target dis- tribution. Second, we conduct real-world experiments on MOF and protein backbone generation, motivated by a gap in the literature.
Motivation for material and protein generation and common pattern. Existing works on pro- tein and material generation often based on diffusion- and flow-based models with structural losses inspired by Yim et al. (2023b|a), a). such as Yue et al. (2025); : Kim et al. (2024); Guo et al. (2025) follow common pattern. Their generation scheme is split between Euclidean and Eucliden parameters, where Euclidean parameters are learned through process that effectively corresponds to variational flow matching, since the model predicts endpoints, minimizes MSE with the target,
7

---

Published as a conference paper at ICLR 2026
CFM
VFM
RG-VEM-R
RFM
RG-VEM-M
Ground truth
Figure 3: Comparison of the spherical checkerboard distribution generated with CFM, VFM, RFM and our methods RG- RGM-R and RG- V..M...M.
and uses these predictions to compute velocity fields during integration. Non-Euclidean parameters instead employ partially but not fully variational form of Riemannian FM: endpoints are still pre- dicted, but the loss minimizes the squared distance between ground-truth and predicted velocities, with the latter obtained via the logarithm map of the manifold. This reveals roomfor improvement, as full alignment of the loss components would suggest minimizing the geodesic distance between predicted and target data points in the non-Euclident case. Our method directly explores this op- tion. Furthermore, prior works report that endpoint learning improved empirical performance, and we interpret our approach, together with Eijkelboom et al. (2024), as providing complementary the- oretical justification for this choice. In this setting, we choose to variationalize the losses of two models from distinet applications: MOFFlow (Kim et al. 2024) for MOF generation and ReQFlow (Yue et al. 2025) for protein backbone generation.
5.1
CURVATURE EFFECTS IN SYNTHETIC DATA
Dataset and experimental setup. Inspired by the planar checkerboard benchmark in generative modeling (Grathwohl et al. 2018), we introduce two curved checkerboard distributions as our target Pr. whose support is either on the hypersphere S R the upper-sheet hyperboloid HH2, R which we define in section E. The noisy distribution po is defined differently for each model: for CFM, VFM, and RG- CGGVVFMRR" Po the standard normal distribution in R33 while for RG- VFM-M and RFM, it is obtained by wrapping the standard normal distribution on either S or H²1-
We conduct two sets of experiments: we (1) compare the extrinsic models in their ability to capture the correct geometry assessed by the distance of the generated samples to the data manifold and (2) evaluate vanilla versus variational models in reproducing the target distribution. For this last point, the evaluation is based on metrics such as Coverage (% of generated points falling within the desired checkerboard area) and Classifier 2-Sample Tests (C2ST) metric (Lopez-Paz & Oquab, 2016 Dalmasso et al., 2020; Lueckmann et al. 2021). The C2ST technique utilizes a neural network classifier to separate true samples from generated ones, where a score of 0.5 indicates the distribu- tions indistinguishable to the classifier, while scores approaching 1 suggest the distributions are easily separable. Additional experimental details are provided in section E.
Results. We observe that (1) Riemannian models better capture manifold geometry by generating points with minimal distance to the manifold compared to Euclidean ones (see Distance columns in table [). and (2) variational models produce sharper and less blurred distributions than vanilla models, with RG-VFM-R3 and RG-VFM-M showing the best visual performance in fig. 3 This is reflected in table I in Coverage metric results, where variational models particularly Riemannian ones achieve the highest values. For C2ST, no consistent pattern emerges between spherical and hyperbolic cases, except that in both cases standard VFM demonstrates the strongest performance. In essence, emphasizing endpoint accuracy enables variational models capture fine details of the target distribution's 's shape, and additional geometric awareness of RG-VEM further enhances the result. We tested setting (x())) = versus (x)) = 1 1-6 t in eq. (8) during training and found negli- gible differences, so we report only the results obtained with (I( 1. Additionally, preliminary findings suggest that using L' loss (which corresponds to using a Riemannian Laplace instead of a Riemannian Gaussian distribution in eq. (8) instead of 2 may enhance performance, particularly in hyperbolic spaces, and we start exploring this option in section E.4
5.2
MOF GENERATION WITH MOFFLOW
Dataset and experimental setup: from MOFFlow to V-MOFFlow. MOFFlow (Kim et al.) 2024) is flow- generative model for MOF structures operating on rigid building blocks. A
8

---

Published as a conference paper at ICLR 2026
Table 1: Results of synthetic experiments. Clipping applied in the variational setting stabilize sampling. The distance between generated points and the ground-truth manifold is computed only for extrinsic models since intrinsic ones generate directly on the manifold. Abbreviations: Eucl. Euclidean, Riem. Riemannian, Ext. extrinsic, Int. intrinsic, Van. vanilla, Var. variational.
Sphere
Hyperboloid
Coverage
C2STL
Distance
Coverage |
C2ST
Distance
Fucl/Fxxt/Van (CFM)
64.97
58.36 + 1,56
0.012 + 0.099
69.05
$7.38 + 1,30
0.008 * 0.339
Eucl//Ett/Var. (VFM)
79.08
56,33 = 0.48
0.044 # 0.045
75.89
$7.03 # 0,59
0.061 = 0.140
Riemm/Ext/Vaan (Ours)
83.10
56.58 = 0.28
0.010 = 0.035
78.84
63.55 + 0.35
0.021 0.056
Riem./Int./V (RFM)
66.83
57.99 0.58
60.75
61.66 0.92
Riem./lnt//aa (Ours) 84.21
59.72 + 0.87
68.38
59.73 0.31
MOF is represented as S (B, q, T, C), where B denotes building blocks, and the model learns their roto-translations (g, ) and lattice parameter& L. The conditional normalizing flow (g.q.77 T, [)) uses re-parameterized training objective predicting clean data (01 Tı, {1) from intermediate structure ((*), The Euclidean loss minimizes endpoint L distance following VFM, while the rotational part computes conditional velocities from predictions and minimizes squared distance to ground-truth velocities, in RFM. Our contribution makes MOFFlow fully variational by applying our method to its rotational component. A detailed explanation with loss equations is in section F.
We evaluate the resulting model, Variational- (V-MOFFlow) against the original MOF- Flow and DifECSP (Jiao et al., (2023) on MOF structure prediction using the large-scale dataset of Boyd al. (2019), where structures are decomposed into building blocks and split into train valnvaaiiaiii/esstttttt set. We follow the experimental setup of Kim et al. (2024), and performance measured by match rate (MR) and RMSE between original structures and generated samples.
Results structure prediction. We report results in table 2 Our model outperforms all competi- tors except for MR stol 1.0, which Kim al.) (2024) consider too lenient for practical use. This validates our theoretical findings that RG-VFM loss guides training more effectively than RVM. We report additional analyses and experimental details in section
Table 2: Structure prediction accuracy. We report results for DifICSP and MOFFlow with TimeBatch implementation from Kim et al. (2024), and reproduce MOFFlow and evaluate V-MOFFlow with Batch implementation. "stol" is the site-tolerance for matching criteria.
stol = 0.5
stol 1.0
# of samples
MR (%)f
RMSEL
MR (%)
RMSEL
0.09
0.3961
23.12
0.8294
DiffCSP
0.34
0.3848
38.94
0.7937
31.69
0.2820
87.46
0.5183
MOFFlow (Paper results)
44.75
0.2694
100.0
0.4645
30.40
0.2832
83.50
0.5255
MOFFlow (Reproduced)
46.97
0.2717
95.82
0,4603
33.52
0.2789
89.08
0.5096
V-MOFFlow (Ours)
50.14
0.2629
97.18
0.4384
5.3 PROTEIN BACKBONE GENERATION WITH V-REQFLOW
Dataset and experimental setup: from QFlow ReQFlow to V-QFlow and V-ReQFlow. QFlow Yue et al.] 2025) is a flow- model for protein backbone generation. Unlike previ- ous methods (Yim et al. 2023b Bose et al. 2023) that represent SO(3) elements with rotation matrices, QFlow uses quaternions, which provide improved training stability. Building this foun- dation, ReQFlow (Yue et al. 2025) further enhances QFlow by incorporating rectified flow with re- re-paired samples and noise, inspired by Liu et al. (2023), improving the designability of generated protein backbone structures. Similar to MOF structure generation, protein backbone structures are represented sequences of (3)(3) elements (z, N where 4 SO(3) defines the frame the

---

Published as a conference paper at ICLR 2026
a-carbon of each amino acid, and R represents the zero-mean coordinate of the a-carbon. For further details, see Yim et al. (2023b). The goal is to learn a conditional flow Pa(Q, N) where Q {q.) V and T [t,) N. with N denoting the number of residues in the desired backbone struc- ture. Like MOFFlow, both QFlow and ReQFlow employ a re-parametrized training objective that predicts endpoints from which vector fields are reconstructed. We apply our method by variational- izing the rotational component of their loss function, similarly to section F.1 while maintaining all other implementation details identical isolate the benefits of our variational objective.
We tested Variational-QFlovv (V-QFlow) and Variational- -itaaioonalReeFFlww (V-ReQFlow) on filtered Protein Data Bank (Berman al., 2000) dataset with 23366 protein structure with lengths ranging from 60 to 512. The filtering pipeline follows Yue et al. (2025). )) For evaluation metrics, we follow Yue et al. (2025), using designability, diversity and novelty concretely evaluate the quality of the generated protein backbone structures. We trained our V-QFlow with 4 NVIDIA-H1 GPUs for around 260 epochs. For V-ReQFlow, we further finetuned it on our rectified dataset for 10 epochs.
we observe that V-QFlow and V-ReQFlow surpass their vanilla counterparts on both designability and folding RMSD, emphasizing the effectiveness of applying variational objectives when learning probability paths on manifolds.
Results in protein backbone structure generation. From table 3
Table 3: Performance comparison with baseline models protein backbone gener- ation on PDB dataset. 50 samples are generated and evaluated for each length in {50, 100, 150, 200, 250, 300). For both ReQFlow and V-ReQFlow, we generate the rectified dataset with 20 samples for each length in [60, 512]. We filter the generated samples following the proce- dures in the repo provided by Yue al. (2025). Samples used for evaluation generated by flow models trained with 10 epochs on the rectified dataset for both ReQFlow and V-ReQFlow.
Efficiency
Designability
Diversity
Novelty
Step
Fraction
scRMSDJ
TM1
TM
RFDiffusion
50
0.904
1.102
0.382
0.527
Genie2
1000
0.908
1.132
0.370
0.475
FoldFlow2
50
0.952
1,083 1.300 308
0.373
0.527
FrameFlow
500
0.872
1.380=1.390
0.346
0,562
QFlow (Reproduced)
500
0,924
1,252
0.357
0,641
QFlow (Paper results)
500
0,936
1.163 1.163=0.988
0..988
0.356
0.635
V-QFlow (Ours)
500
0.968
0.923 70.787
0.387
0.647
ReQFlow (Reproduced)
500
0.964
0.939 =0.572
0,400
0.630
ReQFlow (Paper results)
500
0.972
1,071 +0.482
0.377
0.645
V-ReQFlow (Ours)
500
0.980
0.961 83.83
0.408
0.644
CONCLUSION
We introduce Riemannian Gaussian Variational Flow Matching. which extends VFM general manifolds through Riemannian Gaussian distributions, unifying RFM and VFM under a common probabilistic framework. Through reformulation of their objectives using Jacobi vector fields, we demonstrate that RG-VFM captures richer curvature-dependen information compared to stan- dard RFM. our expeximmemmn experiments, we validate that this theoretical advantage translates to more precise supervision and better learned signals: (1) for synthetic spherical and hyperbolic checkerboard dis- tributions, enhanced curvature awareness leads to improved sharpness in learned distributions, and (2) for real-world protein backbone and material generation tasks, applying our variational perspec- tive through a simple modification to the rotational component of existing flow matching losses consistently improves generation quality metrics. A current limitation is that our method is defined for simple geometries with closed-form geodesics. However, most practical tasks involve manifolds with explicit exponential and logarithmic maps, and we believe this framework can be straight- forwardly extended to more complex geometries. These results establish RG- VGM as a promising approach for modeling distributions complex geometries with minimal implementation overhead.
10

---

Published as a conference paper at ICLR 2026
ETHICS STATEMENT
This work aims to advance machine learning and AI for science. Material and protein genera- tion hold great promise for driving scientific discovery and tackling global challenges in medicine, sustainability, and biotechnology. At the same time, the technology raises ethical considerations, cluding the need for appropriate regulatory oversight as it matures, In terms of readiness, this work remains at an early stage, focusing on foundational computational methods rather than immediate applications, and therefore presents no direct benefits or risks at this time.
REPRODUCIBILITY STATEMENT
To ensure reproducibility and completeness, all required notation, mathematical background, defi- nitions, and proofs of mathematical statements provided in sections B to D implementation details are included in section 5 and sections
Experimental and
and
7 ACKNOWLEDGMENTS
This project was funded by the CaLiForNIA project (Marie Sklodowska-Cun Actions Doctoral Network 2022), the Bosch Center for Artificial Intelligence, and the Finnish Center for Artificial In- telligence (FCAI). This publication is also part the project SIGN with file number VI. Vidi.233.22 of the research programme Vidi which is (partly) financed by the Dutch Research Council (NWO) under the grant https://doi.coo waw.oiiiorrr//o/// wetg/10.6.. (0..1686/PKKGZZ/
The author C.L. was financially supported Top Sector Life Sciences & Health (LSH-TKI), project number LSHM22023, which realizes a public-private partnership (PPP) between the University of Amsterdam and Janssen Vaccines & Prevention B.V. We thank SURF (www.surf.nl) for the support in using the National Supercomputer Snellius.
by
O.Z. would like thank David Wessels for his great help and advice on the coding and engineering aspects, Nayoung Kim for the valuable conversations about MOFFlow that got her interested in the real-world applications considered here, and Kiyoung Seong for making these conversations happen. O.Z. and A.P. are grateful to their co-authors for the amazing collaboration. Additionally, the authors are indebted to Tullio Levi-Civita for giving the Jacobi equation its modern geometric formulation in terms of curvature.
REFERENCES
Michael Albergo, Nicholas M Boffi, and Eric Vanden-Fijnnden. Stochastic interpolants: unifying framework for flows and diffusions. arXiv preprint 2txXXx:2303308797 2023.
Heli Ben-Hamu, Samuel Cohen, Joey Bose, Brandon Amos, Maximillian Nickel, Aditya Grover, Ricky T. Q Chen, and Yaron Lipman. Matching normalizing flows and probability paths on manifolds. In Proceedings of the 39th International Conference Machine Learning. volume 162 of Proceedings of Machine Learning Research, pp- 1749-1763, PMLR, 17-23 Jul 2022.
Helen M. Berman, John Westbrook, Zukang Feng, Gary Gilliland, N. Bhat, Helge Weissig, Ilya N. Shindyalov, and Philip E. Bourne. The protein data bank. Nucleic Acids Research, 28(1):235- 242, 01 2000. ISSN 0305-1048. doi: 10.1093/nar/28.1 1.235. URL httph:tt///// http:://dooooooooo
1093/mar /28 2.
Avishek Joey Bose, Tara Akhound-Sadegh Guillaume Huguet, Kilian Fatras, Jarrid Rector-Brooks, Cheng-Hao Liu, Andrei Cristian Nica, Maksym Korablyov, Michael Bronstein, and Alexan- der Tong. Se (3)-stochastic flow matching for protein backbone generation. arXiv preprint arXiv:2310.02391, 2023.
Peter G Boyd, Arunraj Chidambaram, Enrique Gareía-Díez, Christopher P Ireland, Thomas D Daff, Richard Bounds, Andrzej Gladysiak, Pascal Schouwink, Seyed Mohamad Moosavi, M Mercedes Maroto- Valer, al. Data-driven design of metal-organic frameworks for wet flue gas co2 capture. Nature, (76(7786):233-2566 2019.
Ricky TQ Chen and Yaron Lipman. Flow matching on general geometries. In The Twelfth Interna- fional Conference on Learning Representations, 2024.

---

Published as a conference paper at ICLR 2026
Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018.
Niccolo Dalmasso, Ann Lee, Rafael Izbicki, Taylor Pospisil, Ilmun Kim, and Chieh-An Lin. Valida- tion of approximate likelihood and emulator models for computationally intensive simulations. In International Conference Artificial Intelligence and Statistics, pp. 3349-3361. PMLR, 2020.
Floor Eijkelboom, Grigory Bartosh, Christian Andersson Naesseth, Max Welling, and Jan- Man-Willem van de Meent. Variational flow matching for graph generation. arXiv preprini arXiv:2406.0484. 04843, 2024.
Floor Fijkelboom, Heiko Zimmermann, Sharvaree Vadgama, Erik J Bekkers, Max Welling, Chris- tian A Naesseth, and Jan- willei van de Meent. Controlled generation with equivariant variational flow matching. arXiv preprint wrXiv2550.188300 2025.
Tobias Sebastian Finn, Marc Bocquet, Pierre Rampal, Charlotte Durand, Flavia Porro, Alban Farchi, and Alberto Carrassi. Generative ai models enable efficient and physically consistent sea-ice simulations. arXiv preprint arXiv::508. /4984, 2025.
Maurice Fréchet. Les éléments aléatoires de nature quelconque dans un espace distancié. In Annales de I' I'instiiut Henri Poincaré, volume 10, pp. 215-310, 1948.
Xiang Fu, Tian Xie, Andrew Rosen, Tommi Jaakkola, and Jake Smith. Mofdiff: Coarse-grained diffusion for metal-organic framework design. arXiv preprint arXiv:2310.10753 2023.
Will Grathwohl, Ricky TQ Chen, Jesse Bettencourt, Ilya Sutskever, and David Duvenaud. Ffjord: Free-form continuous dynamics for scalable reversible generative models. arXiv preprint arXiv:1810.01167, 2018.
Will Grathwohl, Ricky T. Q. Chen, Jesse Bettencourt, and David Duvenaud. Scalable reversible generative models with free-form continuous dynamics. International Conference on Learning Representations, 2019.
Hongyu Guo, Yoshua Bengio, and Shengchao Liu. Assembleftow: Rigid flow matching with in- ertial frames for molecular assembly. In The Thirteenth International Conference on Learning Representations, 2025.
Andrés Guzmán-Cordena Floor Eijkelboom, and Jan-Willem de Meent. Exponential family variational flow matching for tabular data generation. arXiv preprint arXiv:2506. 0594000060050 2025.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-688 2020.
Rui Jiao, Wenbing Huang, Peijia Lin, Jiaqi Han, Pin Chen, Yutong Lu, and Yang Liu. Crystal structure prediction by joint equivariant diffusion. Advances in Neural Information Processing Systems, 16:11444417497 2023.
Nayoung Kim, Seongsu Kim, Minsu Kim, Jinkyoo Park, and Sungsoo Ahn. Mofflow: Flow match- ing for structure prediction of megal-organic frameworks. arXiv preprint a1Xiv:2410.17270, 2024.
John M Lee. Introduction to Riemannian manifolds, volume 2. Springer, 2018.
Yaron Lipman, Ricky Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference on Learning Repre- sentations, 2023.
Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning generate and transfer data with rectified flow. The Eleventh International Conference on Learning Repre- sentations, 2023.
David Lopez-Paz and Maxime Oquab. Revisiting classifier two-sample tests. arXiv preprint arXiv: 06.455.555 2016.
12

---

Published as a conference paper at ICLR 2026
Jan-Matthis Lueckmann, Jan Boelts, David Greenberg, Pedro Goncalves, and Jakob Macke. Bench- marking simulation- inference. In International conference on artificial intelligence and statistics, pp. 343-351. PMLR, 2021.
Maria G Monera, Montesinos-Amilibia and Esther Sanabria-Codesal. The taylor expansion of the exponential map and geometric applications. Revista de la Real Academia de Ciencias Exaclas, Fisicas y Naturales. Serie A. Matematicas, 108:881-906, 2014.
Xavier Pennec. Intrinsic statisties on riemannian manifolds: Basic tools for geometric measure- ments. Journal of Mathematical Imaging and Vision, 25:127-154, 2006.
Noam Rozen, Aditya Grover, Maximilian Nickel, and Yaron Lipman. Moser flow: Divergence- based generative modeling on manifolds. Advances in Neural Information Processing Systems, 14.17669-17680. 2021.
Kristiyan Sakalyan, Alessandro Palma, Filippo Guerranti, Fabian Theis, and Stephan Günnemann. Modeling microenvironment trajectories on spatial transcriptomics with nicheflow. In ICML 2025 Generative Al and Biology (GenBio) Workshop.
Yang Song, Jascha Sohl-Dickstein, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In Interna- tional Conference on Learning Representations, 2020.
An B Vuong, Michael McCann, Javier E Santos, and Yen Ting Lin. Are we really learning the score function? reinterpreting diffusion models through wasserstein gradient flow matching. arXiv preprint arXiv:2509.00336 2025.
Jason Yim, Andrew Campbell, Andrew YK Foong, Michael Gastegger, José Jiménez-Luma Sarah Lewis, Victor Garcia Satorras, Bastiaan Veeling, Regina Barzilay, Tommi Jaakkola, al. Fast protein backbone generation with se (3) flow matching. arXiv preprint arXiv:2310.05297,, 20233
Jason Yim, Brian L Trippe. Valentin De Bortoli, Emile Mathieu, Arnaud Doucet, Regina Barzilay, and Tommi Jaakkola. Se (3) diffusion model with application to protein backbone generation. arXiv preprint arXiv:2302.0222 022777 2023b.
Angxiao Yue, Zichong Wang, and Hongteng Xu. Reqflow: rectified quaternion flow for efficient and high-quality protein backbone generation. arXiv preprint 1rXiv 2025.
13

---

Published as a conference paper at ICLR 2026
A DISCLOSURE OF LLM USAGE
We declare that the use of LLMs for writing this paper was limited to general- writing assistance. Specifically, used them only polish the wording of text sections and in no way to generate the research ideas or technical results and proofs presented in this paper.
B NOTATIONS
this section, we report the notations that are used the paper and rest of the appendix, summarized in 4
Symbol
Name
Type
Description
M
manifold
object tensor
Smooth Riemannian manifold (M, g). Riemannian metrie; (+,>> x(++) and
g
metric
Vg(+.).
p
base point
point
Fixed point in M: normal coordinates taken at p.
T.M
tangent space exponential map
vector space
Tangent space at pt all v, w, 6, u, live here.
expp
map
exp, TμM = U M. diffcomorphism
small ball U. Inner product (M, g).
inner product distance curvature tensor
scalar
distg (s))
scalar
Riemannian distance on (M,g).
tensor
(1, 3)-tensor R(X, Y) = VxVyZ
Vx Vx.r
Levi-Civiie connection
operator
Metric, torsion-fret free connection; D; denotes variant derivative along curve.
sectional curvature
scalar
Constant curvature space forms; sphere of radius r, K = 1/7".
0, 20
tangent vectors
vectors
Elements Tę.M; initial velocities of the two geodesics.
parameters
scalars
is the geodesic time (small); [-e,e] parametrizes the variation within the family of geodesics.
g?
2-sphere
object
g? = (a E R (x,x)) 1]
2-hyperboloid
object
I == (ar E Rd (I, -1, 0)
(x(T)
geodesic
curve
(.(T) == exp, geodesic with initial ve- locity at
a(s,T)
ruled surface
2-parameter map
a(s,7) := (T); two-parameter family used
variations.
(())
Jacobi field
vector field
J(T) := 0, along Y: J(0) = 0. (D, =
0(-)
remainder
notation
Big-O with constants uniform for v, w in fixed small ball in T,M.
Table 4: Notations of objects mentioned in this paper.
C GEOMETRIC BACKGROUND
C.I RIEMANNIAN MANIFOLDS
In this section, we provide a comprehensive introduction to Riemannian manifolds, establishing all necessary definitions from first principles.
Basic definitions. manifold M is a mathematical structure that appears curved globally but looks flat when viewed locally. Formally, d-dimensional manifold be covered by coordinate charts, where each chart provides local parameterization. For any point p E M, there exists a neighborhood that can be mapped smoothly to open subset of Rd via coordinate charts.
14

---

Published as a conference paper at ICLR 2026
The tangent space Tp.M a point p .M represents the collection of all possible directions move from p while staying on the manifold. This vector space encodes the local linear approxima- tion to the manifold at and maintains the same dimensionality the ambient manifold.
Riemannian metric. A Riemannian metric g on M is a smoothly varying collection of inner products, one for each tangent space. Specifically, for each point M, the metric defines an inner product > on the tangent space T.M. This inner product must be:
Bilinear: (ar + bw, u)g a(v, u)g + Ku, u)g for tangent vectors v, w, u TpM and scalars a, b,
Symmetric: (U, w)x (w,
Positive definite: (v, U) for all non-zero TyM.
manifold M equipped with Riemannian metric g called Riemannian manifold and denoted by (M,g).
The metric enables to measure lengths of tangent vectors and angles between them. For tangent vectors v, M, their lengths are ||o| (u, and | ((w, w)g. respectively.
Geodesics are the natural generalization of straight lines curved spaces. On Rie- mannian manifold, a geodesic (T(T) is a curve that maintains constant "speed" and "direction" in the sense defined by the Riemannian metric. Mathematically, geodesics are characterized by having vanishing covariant acceleration.
Geodesics.
These curves play a fundamental role they represent paths of extremal length between nearby points. Given any point M and initial tangent vector u T.MM, there exists unique geodesic originating at p with initial direction v.
Distance function. The Riemannian metric induces a natural distance function on the manifold. The Riemannian distance distg(p: (p, q) between two points p, q M is defined as the infimum of the lengths of all piecewise smooth curves connecting and q:.
[ dr dt 7
(pst 9) inf
(16)
where the infimum is taken over all piecewise smooth curves y [0,1] 1] MM with 7(0) p and 7(1) q. Under appropriate completeness conditions, this distance is achieved by geodesics.
Exponential map. The exponential map exp, Tp-M - M provides canonical way translate between the linear tangent space and the curved manifold. For a tangent vector v Tp,,, the exponential map is defined
exp,, = To (1)
(17)
where (T) represents the geodesic initiating at p with velocity evaluated at parameter value T 1. This construction allows us to "walk" along geodesics to reach new points on the manifold.
In sufficiently small neighborhoods around any point p, the exponential map establishes smooth bijection between a region the tangent space and a region on the manifold.
Logarithmic map. The logarithmic map log, .M - T..M the (local) inverse of the exponen- tial map. For point g e M sufficiently close p, the logarithmic map returns the tangent vector v E T such that (x), = q.
In regions where the exponential map a diffeomorphism, we have (exp,,,))) = and (log, = q. The logarithmic map essentially tells us which direction and how far to travel in the tangent space to reach given nearby point the manifold.
In this work, we consider complete, connected, and smooth Riemannian manifolds (M, g), ensuring that geodesics be extended indefinitely and that the exponential map is well-defined globally.
15

---

Published as a conference paper at ICLR 2026
Tangent bundle. By collecting all tangent spaces across the manifold, we obtain the tangent bun- dle:
TM
U (p) x TMM PE.M
(18)
The tangent bundle is itself smooth manifold of dimension 2d, where d is the dimension of M.. Each element of TM can be written as (p, v) where E M is a point on the manifold and v E Tp.M a tangent vector at that point.
Vector fields. A vector field on M is a smooth section of the tangent bundle, i.e., a smooth map M TM such that u(p) T,M for each point p M. In local coordinates, vector field can be expressed as = Σ.= u' pxr where the coefficient functions u' are smooth. We specifically consider demeeddenddntt vector fields [ue which are smooth families of vector fields parameterized by time t. The Riemannian metric g extends naturally to define pointwise inner products between vector fields: (u, (p) (u(p), u<p))> for any two vector fields u and w.
Homogeneous Manifold. A Riemannian manifold M is homogeneous if its isometry group acts transitively on M, i.c. for any two points n, y E M, there exists an isometry f M - M such that f(x) y.
C.2 RIEMANNIAN GAUSSIAN DISTRIBUTIONS
We describe the construction of the Riemannian Gaussian (RG) distribution, which generalizes the familiar Gaussian distribution the setting of Riemannian manifold. The definition of the Rie- mannian Gaussian is specific instance of the Normal law presented in Pennec (2006):
call Normal law on the manifold M the maxi- mum-entropps distribution specified by its mean value and covariance. Assuming no continuity or differentiabilitty constraint on the cut locus ( and a symmetric domain D(t), the probability density Sunction of the Normal law with mean 3 and concentration matrix I is
Definition C.1 (Normal law Pennec (2006)..
N(ar)(() = k expl
Tzy
(19)
where the normalisation constant ls and the covariance are related by
I expl (
k-
dM(y).
Σ = Ty
exp
xy
d.(y)).
(20)
By simply defining the concentration matrix I' as the chosen metric and is a fixed variance parameter, we obtain the following definition.
G where G is the metric tensor associated with
Definition C.2 (Riemannian Gaussian). Let M be Riemannian manifold endowed with the metric tensor g. The RG distribution is defined by
1 exp( dist (2, 20²
NRiem o, p1)
(21)
where M is point the manifold, μ M plays the role of the mean, and is scale parameter controlling the spread of the distribution. Here, distg(z, μ) denotes the geodesic distance between and μ determined by the metric g, and C. is normalization constant chosen so that the total probability integrates to over M:
distg(z,112 dM,. 202
c= exp
(22)
The measure d.My represents the Riemannian volume element, which in local coordinates takes the form
dM, / g(=)dz,
(23)
with dz being the standard Lebesgue measure in the coordinate chart and () is the Riemannian metric tensor at the point z. This formulation ensures that the probability density is adapted to the geometric structure of the manifold.
16

---

Published as a conference paper at ICLR 2026
Observation. In the special case where M R2 and the metric is Euclidean (i.e., g(2) D, the geodesic distance reduces to the usual Euclidean distance, and the RG distribution becomes the standard multivariate Gaussian with covariance matrix I On more general manifolds, however, the curvature and topology are taken into account through the geodesic distance and the volume element, leading natural extension of the Gaussian concept. This construction be applied to spaces such as hyperbolic manifolds, where one can define the distribution in the tangent space at point μ and then use the exponential map project onto the manifold.
Comparison to vMF. A closely related distribution is the von which is traditionally defined on the sphere 5 by
(vMF) distribution,
(= μ4, k))C (xp(x<2,
with j 5 and (:.)) denoting the standard dot product. The vMF distribution is based the notion of directional data and an inner product structure that measures alignment. In contrast, the RG distribution is inherently tied to the Riemannian metric, making it applicable class of manifolds. Generalizing the idea behind the vMF distribution to other geometries often requires embedding the manifold into larger ambient space and defining suitable bilinear form (such as the Minkowski inner product in hyperbolic geometry). In this sense, the RG approach offers a more natural and geometrically intrinsic formulation.
much wider
In summary, the Riemannian Gaussian distribution is defined in terms of the geodesic distance and the corresponding volume element, and it adapts the underlying geometry of any Riemannian manifold.
D VF AND LINK WITH RFM
D.1 DETAILED DERIVATION OF RG-VFM OBJECTIVE
Proposition D.1. If the manifold (M,g) g) homogeneous, the normalization constant
C = J exp (- distg(z, d.M
(24)
202
independent of the mean Jr.
Proof. We initially rename the normalization constant by making the dependency the mean explicit, referring to it as C(r). In this setting, we want to prove that for two arbitrary mean values ç and jz, we have C(ñ) C(ī).
By definition, a Riemannian manifold M is homogeneous if Vx, M, M - M such that f(z) y and with being an isometry, meaning that distg (1,tg(11))) dist. f(y))
We can then assume that satisfies ji f(ā), getting the following:
dist, (2)
C(ū)
exp
dM2.
20
JM
distg (J, 12)
C(ū) exp
d.yy,
20
.
disty (y,, f(ā))2
C((())) JM
exp
dMy.
2a²
with C(τ) C(f(ā)).
Let's suppose that y = f(s), for some E M. By the definition of isometry, we have (y. distg(f(s), o ..M- R and isometry f:
dist p). Furthermore, for any integrable scalar function
S plv)
dl dM, = [ [f(s)) aM..
17

---

Published as a conference paper at ICLR 2026
By applying these two facts to case, obtain the following series of equalities:
[M d.y
((
exp
20
M
disty (f(s), f(1))
exp
dM.
20²
M
distg(s (1)
exp
d.M, = C(.).
20²
Proposition 3.1. Let M be a homogeneous manifold with closed-form geodesics. Then, the RG- VFM objective reduces to
LnG (f) Ba [II loga, (44 [1) Ea, [dist ( (
(9)
where log denotes the logarithmic map on the manifold and dist the geodesic distance.
Proof. The objective of VFM is defined as (VF -E [log de (z) [x)]
We define the objective function of RG-VFM by setting the posterior probability as the Ricmannian Gaussian, i.c.
(x1 [z)
NRsim 14 (x). oz (x)),
so that
CRG-VFM(O) -,, logoNiiom
[
More explicitly, have
CRG-VEM(0)
Br [log de (x11
Eg E,,01
log NogeNiiem
(((())
dist
-E,,,
log
((1((x)) exp
(x,(2))
distg (21,, ui (x)))
log
C(i(()))
20, (x)2
disty (11 (x))) 20, (x))²
EE,
log
Et,21
(()((x))
where distg() denotes the geodesic distance induced by the Riemannian metric g.
Without any regularity assumptions on M, no further simplification is possible. However, under the following assumptions the objective becomes more tractable:
1. Homogeneity: If the manifold (M, g) is homogeneous, the normalization constant
(i,tg(z,
C
exp
202 dM-
M
is independent of the mean (see proposition D.I). Hence, defining
(C(n)(a)))
(((u(())) )
K - Er
log
which is constant with respect to 0, we obtain
distg( 2 20,(2))
LRG-VFM (f) K + E
Since K a constant that independent of the model's parameters 0, the minimization objective becomes
distg (I 1 (x))))²
LRG-VEM(0) E
20, (x:)2
18

---

Published as a conference paper at ICLR 2026
2. Closed-form Geodesics: If the geometry allows closed-form expressions for geodesics, namely
(t) expx log, (v))
then the geodesic distance between points is given by:
distg(z, log, (N)llx
In this setting, we can write
distg ( M4 (x))²
loga (1) (x))
so that the objective becomes
]
LRG (RG)V B log (((((=)))
+ E, Er,.,
log (x)())))
(x)
3. Combined Assumptions: If both conditions hold, the objective simplifies to
LRG-VEM Er,
(2)(2)2 log,, (14 (z))
If we further assume that (x) is constant, this reduces to
vRa Ea [1] log,, (1((
Remark the definition of (x)). In the previous proof, the result is obtained by assuming 0z(2)) to constant. More general, we could maintain the presence of (x(2)) explicit in the loss, obtaining (RG-VFM(0) E, [0) logz, (4g (x)) By being time dependent, o.(x) can for example be defined as the normalization constant 1--- Despite the generality that it allows, for the sake of simplicity we make the choice to assume (x))) being constant, or implicit in the loss
definition.
Examples of simple geometries. A homogeneous manifold does not necessarily imply that geodesics admit closed-form expressions. Conversely, the simple geometries with closed-form geodesics considered in the RFM setting-such as hyperspheres S", hyperbolic spaces H flat tori T = [0, 2TT and the space of SPD matrices S with the infine-invariant metric- are ho- mogencous. Thus, when restricting these geometries for comparison with RFM, we are in the combined case.
Special euclidean space. In the Euclidean case (which also falls into the combined case), the objective simplifies further to
(RG-VTM(0) Er
(ala__[llll [1])((x) - x 113]
D.2 How DOES RG-VFM FIT IN THE EXISTING FLOW MATCHING FRAMEWORK?
Figure (left) illustrates how RG-VFM fits within the framework of related FM models. In VFM, probabilistic generalization of CFM is obtained by making the posterior distribution explicit and customizable, obtaining standard CFM under the choice of specific Gaussian (see Dijkelboom et al. (2024)). In contrast, RFM serves geometric generalization of CFM, where the model and its ob- jective take into account the intrinsic properties and metric of the underlying Riemannian manifold. The same happens for the variational models: VFM with a Gaussian posterior particular in- stance of RG-VFM when the geometry is Euclidean. In Fuclidean space, log,, (14f (x)) reduces to ||, x ll2, thereby recovering the VFM objective.
further comparison can be made between the simplified version RFM and RG-VRN KG-VFM-M. where M is homogeneous manifold with closed-form geodesics. The variational model (RG-VFM) not
19

---

Published as a conference paper at ICLR 2026
direct generalization of vanilla RFM because, unlike in Euclidean space, tangent spaces at different points on a manifold do not coincide. This difference reflected in the models' outputs (fig. 2): vanilla models predict velocity fields, which integrated ODEs construct flows, whereas variational models predict endpoint distributions, ideally aligning with the target distribution p1-
In Euclidean space, the difference between vectors starting x: and pointing different end- points is simply the vector between those endpoints, leading to identical L2 terms in the objectives, ie. (n(()) x ll for VFM and ||u x1) of (==]|l for CFM. However, since T T Ta, M in general, in their geometric counterparts this equivalence no longer holds: indeed, the difference vector in the RFM objective, ef (x) (1-t)
(m1) is in T,,, while log,, (vf is in M. This
fundamental distinction separates RG-VFM from RFM. More details are in the following Section.
Algorithm CFM
Algorithm 4 VFM
Require: base p e IRd, target q R4. # Training Phase
Require: base p Rd. target e Rd. # Training Phase
Initialize parameters of Ur
Initialize parameters 0 of j4x
while not converged do sample 2(0 1). 20 21
while not converged do sample t (0, 1), 20 p, ~
compute linear interpolation:
compute linear interpolation:
St 21 + (1 t) æo
St t. 21 (1 8) 20
compute corresponding velocity:
{(0) E [1 11
á; = - 2,)/
0 = opt imi wer.zeer.ssp(())
(()) E, (zz; 2ell
end while
0 opt im zer step(((()) end while
# Generation Phase sample noise zo p
# Generation Phase sample noise Ta
compute corresponding velocity:
21 = solve ODE([0, 11, ro, v)(=);))
=sooool ru,
Algorithm RFM
Algorithm 6 RG-VFM (general)
Require: base E M, target g M. # Training Phase
Require:
base p. target E M.. Training Phase
Initialize parameters of 0z
Initialize parameters 9 of μ4
while not converged do
while converged do
sample ~ (0, 1), xo ~ p.
sample / ~ (0, 1), 2i ~ p, ~
compute geodesic interpolation:
compute interpolation:
exp... (t- 10 (z1))
21 = int (t, xo. I1)
compute corresponding velocity:
{(0) = E,
2; PTTo--aa (1og (x1))
0 = opt imi zer (em.zeestee(((())
(()) = B an
[Ive( #e]l?]
end while
opt imm zer zer_step(((0)) end while
Generation Phase
sample noise xop ~
compute corresponding velocity ă; @esolve_ODE(() 1,,,1,,,1.. vo, i.)
# Generation Phase sample noise To
æ = so solove www.oleee.e((((( (DE([O, 1], xo, vx(x))))
20

---

Published as a conference paper at ICLR 2026
Algorithm 7 RG-VEM intrinsic
Algorithm 8 RG- VFM extrinsic
Require:
base E M, target e M. # Training Phase
Require: base p TRd target g E M.
# Training Phase
Initialize parameters 0 of μ4
Initialize parameters 0 of ftr while not converged do sample t 24 1). 2g ~ p. ~ compute linear interpolation:
while not converged do sample (/ 1), 20 p1 ~ q
compute geodesic interpolation:
Zt exPx, (t- 10g20 (z1))
3t = L æ +
20
((0) E, ,, [dist] (2), (z 0))
(()) - E,,,, dist (1r, (e(
opt imi zer zer.stte(((())
0 = opt imi zer zersstee(((())
end while
end while
# Generation Phase sample noise To p
Generation Phase
sample noise xo ~ p
compute corresponding velocity:
compute corresponding velocity:
d 109., (14 (x))
i [,
1-t
21 =solee [S0 10,
=000v 1], ro,
D.3 RG-VFM vs RFM ON HOMOGENEOUS SPACES WITH CLOSED-FORM GEODESICS
The objective of RG-VFM defined as
LRG-VFM(0) = Er, 01 log,r (nl [2) while the objective of RFM, in the case of closed-form geodesics, is given by
(RF E, [10](()) (x))
with g being the metric tensor at a P=(x(<11)
Ignoring multiplicative constants that depend only and æ. comparing the losses reduces to comparing the quantities
log,, (p) and
| (z) - (x1))
Euclidean space. In Euclidean space R", the tangent space each point naturally identified with R". In this setting,
log,, (4f = (z) 21 and log, (x1) = 21 2.
Notice that (x(x))
14(()) - x -1,(x)) L) (x1 x) (n,(x) x) - -loga
If we define (ignoring multiplicative constants such as 1/(1 -t)) t))
(x(x) log (1((2)) = 11 (x) 2,
then follows that implying
log,, (4f log, (1g (x)) - log, (I1),
loga, (H. (z)) ||0 log, (21))
Thus, LRG-VFM ()) and (f) are equivalent up to an additive constant. This result is consistent with the known equivalence between (9) and LCrM(9).
General geometries. non Euclidean spaces, however, the quantities
log., (+)() and [lo{(x) - log, (z1)/(1 (
are not necessarily equal. This is because log,, (14 is vector in M, while
log, (44(2) (z)) log, (x1) lies in T.M. and in general Tr M * T.M. Establishing relation
between these vectors not straightforward and can be illustrated by comparing the law of cosines in Euclidean, hyperbolic spaces, and on hyperspheres.
21

---

Published as a conference paper at ICLR 2026
D.4 A COMPARISON BASED ON JACOBI FIELDS
In this section, we report the notations used for explaining the comparison based on Jacobi fields, in table | as well as the proofs of the propositions section [4
Symbol
Name
Description
T-- 21.27
z0 2a
base point
Base point obtained at Predicted and reference data points after training.
generated and target end point
v',00
generated and target vector
Predicted reference vectors after training.
CRO-VMM = Br [xist(=).=)))
RG-VEM loss
Geodesic distance used in the Variational Riemannian FM loss function.
CRrM Bea 11|0 - =01
RFM loss
Norm of vector fields used in the Vanilla Riemannian FM loss function.
Table 5: Synthetic notations used in this section for the Jacobi field and Riemannian flow matching losses.
Definition 4.1 (Jacobi field at a vanishing starting point). The family of geodesics defined as:
a(s,T) T) := :T exp2o (T(v + sw)),
with [0, 1] and + [0, v, WE Ta,M, M, is smooth family of shooting geodesics with (0) = zo. (0 V and in (0) = = w.
For each fixed [0, 1). there exists vector field, called Jacobi field,
J(T) ),
along the geodesic 7o (T( a(s,T) satisfying the ODE equation: DR.J + R(J, 9/2 0 with R the Riemannian curvature tensor of the manifold. In particular, this Jacobi field is uniquely defined by the initial conditions and at 0 one has the initial conditions: J(0) = 0, and D+J(0) w.
Proof. Lee 2018 [Lemma 10.9. and Proposition 10.2.]
Since - (s,,T) r) is geodesic for each s, we have D O. Differentiate with respect to s and the torsion-free, metric connection to get (² R(0,a, ,, O, which the Jacobi equation for ((T) = a,a (a)(ssT)ss=z s= Because ass, 0) = for all s, we get J(O) 0. Also a, 0)= u + sw, so J, Ō₂(u sw) == w|s=0 w.
The Jacobi equation is a linear second-order ODE along with smooth coefficients, there is a unique solution with any prescribed initial data (J(0), ((0) (0, w).
For = 0, we are interested the geodesic "/o : r exp_ (TV"), with the target velocity, and for s 1, the geodesic ?1 T - expx, (To") is defined with v the velocity learned by the model, = 0 v.1 corresponds to the discrepancy between the learned and the conditional initial velocity, and their norm is exactly what is minimized in the vanilla Riemannian Flow Matching (Chen & Lipman| 2024).
Proposition 4.1 (Loss functions as evaluation of Jacobi fields). Consider Jacobi field J(T) == ( T) )= as defined definition 4.1 We denote CREM the loss function of the (vanilla) Riemannian Flow Matching (Chen & Lipman 2024), and LRG-VEM the loss function for our proposed Riemannian Variational Flow Matching. Then the following equalities hold:
CRFM(()) Er, .r [lue( xz) of (x) = E,,,, .rr [D))(())]
(10)
(RG) Ey, [Ilog (n.()))) E. [11
(11)
22

---

Published as a conference paper at ICLR 2026
Proof. In RFM, the goal is to learn velocity field of that transports the base distribution t 0 into the target distribution at = 1. The loss penalizes the discrepancy between the conditional velocity u, ( 21) and the model's prediction 2 (z), averaged over all time steps t. target samples 31 ~ Pdata and intermediate samples x ~ pr(2 21).
When introducing Jacobi fields, we first define them for fixed generative step specific target point 2 = xf= and an associated intermediate point. As an abuse of notation, we suppose here that the Jacobi fields are obtained for all i, 1 x, allowing us take expectations over these variables.
1. Let us prove that:
(RFM Ex, [1 x1) (x) E [I
By definition of the initial conditions of our Jacobi field, we have the target velocity field defined as (==)(() :- 60 u,(x 21 and the learned velocity field defined as )0=1(0)= := v2 of (ar). By definition 4.1.
[| 21) of (x)] | - o'1
will
|w|
(O)I
2. We want to prove the other equality:
(RG-VFM(0) = Er [| (4 [x)) = E. [1)(1)
We observe that log,, (4f = (1) Letp = 71()) and := 7 (1), the
respective end points of the geodesics "N and ?1- If q lies the injectivity radius of p. then there is unique minimizing geodesic a p [0, 1] exp, with u log, (4)
with (0) p ((1) q-
a) We can then consider the Taylor expansion for the exponential map exp, and consequently the geodesic defined from it (p), as in Monera et al. (2014);
a(p) o(0) + o'(0)) + 1 o00(0) d² + O(llpu))).
By substituting the values, we obtain:
(p) = p + up + (0) x² + O(llpul)).
We want now to reparametrize the geodesic with respect to the variable s, instead of P E [0, 1]. For this, we reparametrize it with a new initial velocity vector w such that (s) (p(s)) for smooth reparametrization function p p(s). In this setting,
we still have ow (0) = and o (1) = q. Hence
(s) = + + ws + (0) + ((
and for s = 1:
(1) p
+ p
o0 + O(1
b) From the perspective of the family of geodesics
(T( exp (T(V + sw))
and the corresponding Jacobi field J (T), we can instead derive:
7c(1) = 70(1) + J(1) + J(1) + O(IJ(1))11)
that for 1 gives:
(1) + J(1) J(L) + O(I(()))
We can observe that, for s 1, the two Taylor expansions in (a) and (b) coincide, and we also have for the linear terms:
log, q w (1).
From this, we derive the fundamental equality log,, (447 (z)) log, q = J(1), which exactly corresponds to what we want prove.
23

---

Published as a conference paper at ICLR 2026
Proposition 4.2. (0) is linear approximation of J(1).
Proof. Let ( (0) D (() be the k-th derivative of J with respect to then evaluated The value of the Jacobi field at timestep T can be computed through Taylor expansion of the Jacobi field centered in 0,
33 (( O(ww|>))
= ( + ( 2
Since the exponential map is smooth, the associated geodesics and Jacobi fields smooth functions depending on т. The Taylor's theorem guarantees that we can approximate J(r) around with the big-O term quantifying the size of the error. particular, the expansion centered at = 0 remains valid when evaluated at any provided that the exponential map stays well-defined and smooth.
In our setting, J(0) O, and we are interested in the timestep 1, for which we get
((1) - )0) / (2) 1 ( [[[w]]
and if stop at the linear term:
((1) ~ (0),
in the sense that J(L) is a linear approximation of J(1).
Proposition 4.3 (Difference of loss functions as curvature term). Consider a Jacobi field J(T) == ) as defined in defimition 4.1 and the equivalences shown in proposition 4.1
The difference between LRG-VEM and LRFM encodes the manifold curvature through:
(RG-VFM(O) (RFM(() LC D(), v) Enigher]
(13)
curvature-lepeedeet term
where the leading-oree oordrr curvature functional is:
C(R, = ()) -VV. (1), D,J()))
(14)
and Enigher = 0011--
(15)
with R the Riemannian curvature tensor and to the reference geodesic velocity. The higher-order term Enigher encodes curvature variation along geodesics through covariant derivatives of R. In terms of the RFM loss terms, ue(x x1) and D+J(0) = ui (w) - uel 21).
Proof. The proof consists of three steps: (1) we will detail the Taylor series of the norm of the Jacobi field, noted S = ||] (2) we will compute up to the 5th order of those terms evaluated at and (3) we will simplify the expression have the approximation.
Let's look at the Taylor expansion of the Jacobi fields. The Jacobi fields are indefinitely differentiable at a point O, and so have a Taylor series on this point. They can be expressed as
n! p)" n ( (
(T (T-
((T)
Σ k=0
(0)
k!
with ((() = D the k-th derivative of with respect to then evaluated at 0. R. S (r-p3" ((++1) is the remainder term.
n!
24

---

Published as a conference paper at ICLR 2026
We further note ST)) the linear product of two Jacobi fields, defined as:
(m)) (0), Σ T" J (0))
S(T) =- [J(T]| (J(+),
n!
n20
Note that here and in the following, <+')> 7 g. but omit g to avoid overcharging equations.
By bilinearity and using the Cauchy product, have:
S(T) = Σ
m++n
(m) (0),
(())))
= Σ
" T' r20 r!
(
m!n!
m2n00
with (0)) = Lio
2. Let us compute the Jacobi terms (())(0) :- Dス J()) and their norm S* (0).
We know that the Jacobi fields satisfy the following ODE equation
f(2) + =J=0, = 0,
with A(T)J(T) = =()( +() with 7(0) = v is the initial velocity and
(++(T) 0 since is a geodesic. We know the initial condition J(O) 0 and (0) == (0) w. Noting (#) = D A(T), we can compute with the chain rule:
D,A(T))(F = [M(
D+)) (B00::/00001100000001--------,,0 (V.B:UU))))y-+-D/JJ....)>) since D
D2A(T))(T) DAN1:8001T1,,1----.0000011111,11
For all I, we can express the derivatives of (()) as
D. [A(F)J(7)] [=[A())] + A(r)
[J(r]], with in general:
D [A(T)] = (V))) and A(T) [J(T)] = RDD
(()( (0) = 0
(0) (0(()
(()( (0) A(())(()) =0
(0) - (O)(()) (()) - -R(w, v)b
(() (0) (00 A(0)) 22( v)c
And we have the following norms:
25

---

Published as a conference paper at ICLR 2026
(0) (J0))
(t) (0) - (0)) -0
80) (0)
+ ((000))))))00))00000000)000) 2/
8(a) (0) = (((00).000))+(((00)))0000)0)))), + (0),
=
((4) (0) =
+
(0)) (((w,)) (0)) +
s (0)
+ (,10,,,0000000,,,0000000,,,,,,,0
(0) (0)
(0) - w)
3. We can finally express || in terms of Taylor series
|++T||1
Σ Str) (
(0)
r!
(0) + (0) + T2
73 (0) + ( + 5(5) (0)
75
(0) (0)
remainder
6
24
120
(R(w, v)v, w) - 7" - (V w) + O(7 ||/ (1/11))
= => T 3
with (VV [t) e)., w) < | [[]] and we assume the curvature of our manifold being bounded, VR < M, with M e R. Setting now J 1 and w (O), we get
|+(T]| = |D,[())] + C(R, D,J(0), v) + Enigher
with (.J(O, (RDD, (, (,)()) (V)))
and Etigher 0(11)_((0))111))
where the higher-order term Enigher encodes curvature variation along geodesics through covariant derivatives of R.
Taking the expectation with respect to the variables i, 11,1, x and considering the result of proposition 4.1 we obtain the desired result
(RG-VPM()) (RFM(() + E [C(R, (,J(O,, + Enigher] ]-
(25)
Corollary D.1. For 00 u1(x 21) and v¹ elf (z). the following holds:
C(R, D,J(O),)) v) C(R, v. u. vt) K(o', " a 00 - R) "ooo,,o)) "000 vt>. (26)
(V
with K the sectional curvature, ^ the wedge-product R the Riemannian curvature and V the covariant derivative. For constant sectional curvature K, we further have V,o 0:
(RG-VFM(() (R) - - K| A =
(27)
Proof. The sectional curvature is way measure locally the normalized deviation between two geodesics. It defined as:
K(w, v) (R(w, vyu, w) ||w A v|l
26

---

Published as a conference paper at ICLR 2026
with || A el| the area spanned by the vector w and Furthermore, for the sphere and the hyperboloid, the sectional curvature constant, being K and K = -1 respectively.
With this definition, can first of all re-express, in point (2) of proposition 4.3
(0)) -8 -8(R(w; R(w, u)u, w> (8K(w, w)|lw vll?
Consequently, we get the following expression in part (3):
|J(T)| Σ - (0) r!
r30
72
7 5t5) (0) + remainder
(0) +5() +
(1)
5th) (0)
( +
2
24
120
1 (w, v) w)|lw
w|| - ( (V R) (w, o)e, w) O(- || |1])(²)
K
With the initial velocity vectors r. and w vt v. we can express the Riemannian curvature tensor (R(w, w)e, w) = (R(ot, vt, v0)e0 r²)
[[((T)]] = - - o1 - K(v' oll a 1 - 6
R) (V wn) 10
Hence, we have, evaluating the Taylor expansion r 1 and considering it in expectation:
CRG.VFM CRMM E,
3 K
10 1 (V R)( oo)o", at> 01110"))
with K the sectional curvature, the covariant derivative.
R the Riemannian curvature and V the
For constant sectional curvature K, further have V,a R 0:
LRG-VEM = CREM 3 K a r oll + o 11+10111911111 |13)
Let us consider the angle between the vectors defined as cos(I) = <10000 |1-- In that case:
||od A 001 = 100 ([l==1) | cos 0)² = l/u- 22 cos? 2 0) = || 100 sin²
For a sphere 8² (K=
= +1), we have:
dist())(1,, =1|u' 111
11 2 ssn" 0 0011011"10011
1
3
For a hyperboloid H2 (K = 1)), we have:
dist(ro(1), (1)) ||u² =012
1
11 sin 0 11
3
27

---

Published as a conference paper at ICLR 2026
E SYNTHETIC EXPERIMENTS ON HYPERSPHERE AND HYPERBOLOID
In this section, we present further results from the experiments described in section 5.1
E.1 DEFINITION OF THE MANIFOLDS
The hypersphere is defined as 8" (ar E R (xx, (2, 1), with the standard Euclidean inner product (ar, F) = Tolo + 211/1 + T21/2 +.... Inlln. Instead, we adopt the Lorentz model for the hyperbolic space, which defined as the upper sheet of the hyperboloid embedded in Minkowski space. The Minkowski space is the manifold g=+1 equipped with the Lorentzian inner product (x. y/C = ToAlo + 21 + T28/2 + ................................ + In.n. In this setting, the Lorentz hyperbolic model is defined as H = fx Ru+! +1 (x, 1/K, zo 0, K < 0/, where set K -1
E.2 EXPERIMENTAL SETUP
In all experiments, the target distribution P1 is the spherical checkerboard, so its support is S The noisy distribution Po varies by model: for CFM, VFM, and RG-VEM VFM-R*, Po is the standard normal distribution in RR,, while for RG- VFM-M and RFM, it is obtained by wrapping the standard normal distribution on either S2 or H 3. every case, we train a five-layer MLP with 64/128 hidden features for 3000 epochs on 10000 training samples, that we use to generate 10000 samples using a Euler ODE solver. For the intrinsic geometric models, the Euler solver is manifold-awe aware, meaning that it's defined with the log and exp maps the manifold. Additionally, for the variational models we used a clipping technique during sampling, in order to make sure that the normalization term would not be too high for values of t approaching 1.
E.3 RESULTS
Figure fig. 10 and fig. illustrate the generative flow trajectories time, from the initial distribution po to the generated distribution at 1.
Figures 6 and 13 displays the generated distributions unwrapped onto flat surface for easier visualization and comparison. These results visually confirm the observations presented in section
Finally, figs. 8 and 15 show histograms of the norm values of the generated samples. As discussed in section 5.1 this metric differentiates the Euclidean models (CFM and VFM) from the others. Ideally, points should have Euclidean norm of if lying on S H2, However, because the Euclidean models lack explicit geometric information, their points deviate slightly from the ideal norm, with CFM exhibiting a larger divergence. In contrast, the geometric models consistently generate points that lie almost exactly on the sphere.
Minkowski norm of -1 if on
E.4 LAPLACE POSTERIOR PROBABILITY
We explore the definition of the VFM training loss the absolute value of the geodesic distance, instead of the squared geodesic distance, which would be obtained by ideally defining the posterior distribution of in the VFM loss (eq. (5)) as the Riemannian version of the Laplace distribution. This would be defined as in eq. (7), by replacing the L2 norm of the geodesic distance with the _ norm, obtaining
Lap VFM
[| (1((2) Ee, [dist (1), (x)))-
(28)
We observe that using Laplace distribution as the posterior for the Riemannian VFM models yields better results, both visually and with respect to the considered metrics. This effect is particularly evident the hyperbolic case, and we hypothesize that it arises from the different impacts of using _ versus L norms in this space. The numerical results reported in table 6 the probability paths in figs. 5 and 12, the sampled densities figs. and [4 and the norm histograms in figs. 9 and 16
28

---

Published as a conference paper at ICLR 2026
Table 6: Results of synthetic experiments with Laplace posterior. Abbreviations: Eucl. = Eu-
clidean, Riem. = Riemannian, Ext. extrinsic, Int. intrinsic, Van. vanilla, Var. variational.
Sphere
Hyperboloid
Coverage
C2ST
Distance
Coverage
C2STI
Distance
Eucl/Ext/Var (VFM)
89.92
59.98 + 0.034 0.042
87.63
57.26
0.001 0.133
Riem/Ext/Var (Ours)
95.04
61.33 + 0.23
0.008 # 0.034
91.98
62.55 # 0.30
0.041 # 0.113
Riem/Int/Var (Ours)
90.56
57.39 + 0.70
86.23
56.04 0.41
29

---

Published as a conference paper at ICLR 2026
(a) Model: CFM; supp(po) == R$, po: standard normal distribution in R
(b) Model: VFM; supp(po) = R po: standard normal distribution in R
(c) Model: RG-VEM; supp(po) == R po: standard normal distribution in R
(d) Model: RFM; supp(pu) : S² pu: standard normal distribution on S?
(e) Model: RG-VFM; supp(ro) := S po: standard normal distribution on S².
Figure 4: Flow trajectories of 10,000 samples, initially drawn from the noisy distribution po at t = 0, evolving reach their final configuration by 1. In all variational cases, the posterior distribution Normal, and pr is the checkerboard distribution on S
30

---

Published as a conference paper at ICLR 2026
(a) Model: VFM; supp(po) R po: standard normal distribution in R
(b) Model: RG- VFM; supp(po) == R po: standard normal distribution in IR
(c) Model: RG- supp(po) S²; pot standard normal distribution S?
Figure 5: Flow trajectories of 10,000 samples, initially drawn from the noisy distribution pa at t = 0, evolving reach their final configuration by In all variational cases, the posterior distribution is Laplace, and is the checkerboard distribution on $2.
31

---

Published as a conference paper at ICLR 2026
03
455
-1.0
-10-
(a) Model: CFM; supp(po) = R Pui standard nor- mal distribution R.
(b) Model: VFM; supp(po) normal distribution in R
R²; Pui standard
0.0
0.0
-05
-05
-10
-10-
(c) Model: RG-VFM; (pp)(po) supppr) normal distribution IR.
R2 Pot standard
(d) Model: RFM; supp(po) S?: standard nor- mal distribution S
(e) Model: RG-VFN supp(po) = S² Po: standard normal distribution on S?
Figure 6: Sample distributions generated by different models (representing the flow configuration at t = 1) unwrapped from 52 to E²2 for improved visualization. The true checkerboard distribution is shown in gray in the background. In all variational cases, the posterior distribution Normal, and Pi is the checkerboard distribution on ?
32

---

Published as a conference paper at ICLR 2026
0.-
0.5
-1.
(a) Model: VFM; supp(p))
R pot standard
normal distribution in R
0.0
-05
-10
(b) Model: RG-VFM: supp(po) R²: pot standard normal distribution RR
(c) Model: RG-VFM supp(po) = S² Pot standard normal distribution on S?
Figure 7: Sample distributions generated by different models (representing the flow configuration at ? = 1) unwrapped from 32 to E for improved visualization. The true checkerboard distribution is shown in gray the background. In all variational cases, the posterior distribution is Laplace, and Pi is the checkerboard distribution on ?
33

---

Published as a conference paper at ICLR 2026
lontin of Point Norms (Should be -1.0 for Sphere)
Distribution of Point Norms (Should be +1.0 for Sphere!
Peddect spteee (nommoll
200
Perfect sphere (nomm-1l
150
125
60-
100
20
Indaa
0.500
0,975
1.000
1,050 2,075
3.300
0.605
1.525
1.050 1.575 3.300
(a) Model: CFM; supp(po) = R Put standard nor- mal distribution R.
(b) Model: VFM; supp(po) == R² Pot standard normal distribution in R
Distributtin of Point Norms (Should be -1.0 for Sphare)
Distibution of Point Norms Ishould be -1.0 for Sphere)
Perfect sprane (narm1lll
10000
Parfes sphare (nomm1))
330
0020
4000
50
2020
0.500
0,525
0:550
0,915
1.000
1,585
1,550
1075
1.300
0.500
3,025
0.350
0.975
1.000
1.025
3,050
1.005
1.000
(K]]
[[]]
(c) Model: RG- supp(p)) R² Po: standard normal distribution IR.
(d) Model: RFM; supp(po) mal distribution S
S²: po: standard nor-
Distr on Norms (Should =1.0 Sphori)
sphere (nommell)
6002
4000
2003
9.050
Nomm
(e) Model: RG- supp(po) = S² Po: standard normal distribution on S?
Figure 8: Histogram of the norm values of the 10,000 samples describing the generated distribution. In all variational cases, the posterior distribution is Normal, and p1 is the checkerboard distribution on S
34

---

Published as a conference paper at ICLR 2026
Distribution of Point Norms (Should be -1.0 for Sphere)
Pedfect sphere (nomm-1l)
125
90
0.500
1.000
1,025
1.575
3.500
(a) Model: VFM; (pp) == R Pot standard normal distribution in R
Distribution of Point Norns (should be 3.00 for Sphera)
Perfoc, spfarre (nseswll)
203
30
0.500
0.505
0.550
0.515
1.000
1,505
1,050
1.75
1.300
(Ic]
(b) Model: RG-VFM: spp)(po) R pot standard
normal distribution RR
Distribution Point (Should Sphreed
10030
(oommell)
8030
6020
4030
2020
(c) Model: VFM, supp(po) = S² Po: standard normal distribution on S?
Figure 9: Histogram of the norm values of the 10,000 samples describing the generated distribution. In all variational cases, the posterior distribution is Laplace, and p1 is the checkerboard distribution on S
35

---

Published as a conference paper at ICLR 2026
(a) Model: CFM: supp(po) :- R², Po: standard normal distribution in R
(b) Model: VFM; supp(po) = R po: standard normal distribution in R
(c) Model: RG- VF-: supp(po) :- RR po: standard normal distribution R.
Figure 10: Flow trajectories of 10,000 samples, initially drawn from the noisy distribution Po at t= O, evolving to reach their final configuration by t = 1. In all variational cases, the posterior distribution is Normal, and p1 is the checkerboard distribution on H
36

---

Published as a conference paper at ICLR 2026
(a) Model: RFM; supp(po) :- H 1: standard normal distribution on H
(b) Model: RG-VFM; == m22; : po: standard normal distribution on H²
Figure II: Flow trajectories of 10,000 samples, initially drawn from the noisy distribution po at t == O, evolving to reach their final configuration by & = 1. In all variational cases, the posterior distribution is Normal, and pr the checkerboard distribution on H2,
37

---

Published as a conference paper at ICLR 2026
(a) Model: VFM: supp(po) :- R² pot standard normal distribution in R
(b) Model: RG-VFM; supp(po) = R²: po: standard normal distribution in IR
(c) Model: RG-VEM; supp(ro) : H2:: po: standard normal distribution HH2,
Figure 12: Flow trajectories of 10,000 samples, initially drawn from the noisy distribution po at t = O, evolving to reach their final configuration by t 1. In all variational cases, the posterior distribution is Laplace, and pr is the checkerboard distribution on HH
38

---

Published as a conference paper at ICLR 2026
0.0
0.5
055
-1.2
-10-
(a) Model: CFM; supp(po) == R Pui standard nor- mal distribution R.
(b) Model: VFM; supp(po) normal distribution in R
R²; Pui standard
0.0
0.0
-05
-05
-10
-10-
(c) Model: RG-VFM; (ppp(po) suppp)) normal distribution IR.
R. Pot standard
(d) Model: RFM: supp(po) H2 Pet standard
normal distribution on H
05-
0.
-0.-
-15
(e) Model: RG-VFM; supp(pe) 112 15 Pot
dard normal distribution on H2,
Figure 13: Sample distributions generated by different models (representing the flow configuration at t = 1) unwrapped from H2, to R2 for improved visualization. The true checkerboard distribution is shown in gray the background. In all variational cases, the posterior distribution is Normal, and pn is the checkerboard distribution on H2,-
39

---

Published as a conference paper at ICLR 2026
0.-
0.5
-15
(a) Model: VFM; supp(p)) = R Pot standard
normal distribution in R.
0.-
-05
-10
(b) Model: RG-VFM: supp(po) R²: pot standard normal distribution RR
(c) Model: RG-V CFM; supp(po) = H 1: stan-
dard normal distribution on HÉ
Sample distributions generated by different models (representing the flow configuration at t 1) unwrapped from IH,2, to I for improved visualization. The checkerboard distribution is shown in gray in the background. In all variational cases, the posterior distribution Laplace, and p1 is the checkerboard distribution H2,,
Figure 14:
40

---

Published as a conference paper at ICLR 2026
Distribution of <a,>> (Should be --1.0)
Distribution of -uux- (Should be --1.0)
40
Perfect hypersannt (crupo--l)
Pertect hyperbola of t-tpp- -11
25
Recueesy
40
:0-
70-
2,02
0.000
3.073
0.500
(a) Model: CFM: (po) == Rd Po: standard mal distribution R
(b) Model: VFM: supp(po) normal distribution in R
R pot standard
-1101
Distribution of <0x,t (Shoulid be --1.0
hyperasial
Porfect hypar [=spcc=-1
1000
0000
120
5000
4000
00
1000
60-
40-
20
-1 100 -1 2.075 -1 0000 -1.005 -1 1000 -3.575 -0 -3.955 -8 sco
-1. 10000 -1.075 -1.050 -1.005 -1 1ob 675 -n enn -n.405 N00
Mattooki Do. Prouut xa.0
Wirkowsel Dot Product wa.x-
(c) Model: RG- RFM, supp(po) := 3 Po: standard normal distribution in R
(d) Model: RFM; supp(po) == H2, Pot standard normal distribution on 1
Distribution hype
8000
3000
2000
1000
1.0 -3.075
-0.425
Wikkoes Product
(c) Model: RG-VPM; supp(po) H2 stan- dard normal distribution on H2, J.
Figure 15: Histogram of the norm values of the 10,000 samples describing the generated distribution. In all variational cases, the posterior distribution Normal, and pr is the checkerboard distribution on H2
41

---

Published as a conference paper at ICLR 2026
Distribution of <A,e> (Should be -=100
Perfect bypedbote cc
(a) Model: VFM: spo) normal distribution in R
1 pot standard
Distribution of oo,o (Should be .-1.0)
200
Perfect typorbblod rou..
(b) Model: RG-VFM sppp(po) R²: pot standard normal distribution in R
Distribution <x,a> (Should
Perfect hype rbolald (ocoo-11
1030
6000
5000
4000
3000
2000
3000
-1.025 -0975 -0.25
(c) Model: VFM; supp(po) H Pot stan- dard normal distribution on H.
Figure 16: Histogram of the norm values of the 10,000 samples describing the generated distribution. In variational cases, the posterior distribution Laplace, and pı is the checkerboard distribution on HH,
42

---

Published as a conference paper at ICLR 2026
F MOF GENERATION WITH MOFFLOW
F.1 EXPERIMENTAL SETUP
MOFFlow. MOFFlow (Kim al. 2024 flow-based generative model for MOF structures that operates at the level of rigid building blocks, i.e., metal nodes and organic linkers. A MOF is represented as $ = (B, 9, T, E), where B denotes the set of building blocks, and the model learns their roto- iro--rrnnlltions (g. T) together with the lattice parameter f. Instead of predicting atom-level coordinates, MOFFlow (Kim et al. 2024) treats building blocks as rigid bodies, reducing the search space. The generative model is defined as conditional normalizing flow po(g, & BB), trained with the Riemannian flow matching framework. Specifically, conditional flows are defined along geodesics in S3) and the lattice space
(x w (1 (og ))),
7(1)
(1 1))(0) (0) + + (2),
((t) (1-1) (1-1)el0) + 100) (29)
leading to the conditional vector fields
(a ()
log (e) (g
-00 -00-p(0 (0) Ue (-(()) (1))
u. (()) p00_600
(30)
1-t
1-t
Rather than directly modeling these vector fields, Kim et al. (2024) uses re-parameterized training objective that predicts the clean data TI, (1) from an intermediate structure 5(0:
LMOFFOO (0) Eg(n)-A,-2(0.) (g) +
(2)=(0)
+ XzCi(0))
(31)
loggros (0:) log, ( (gn) 500s
|
Ti 11 t)2
f
(BgW).,,--((..)) A1
A2
A
Ra
(1 f)²
(1
t)²
At generation time, samples are drawn from priors on rotations, translations, and lattice parameters, which are then mapped the full MOF structure by applying the predicted blockwise ioto-translation. to the input building blocks.
V-MOFFlow. Our contribution consists in adopting a variational perspective in the rotational component of CyMMOFFlow (0), by only substituting (0) with the following:
(0) logg, (g1) 5O(3)*
(32)
which corresponds to the squared geodesic distance between @i and q in SO(3). The definition of the vector fields is unchanged from eqs. (29) and (30), well as the sampling algorithm.
Implementation details. For reproducing the MOFFlow results (training from scratch) and evaluating our V-MOFFlow model, we follow the exact experimental procedure described in Kim et al. (2024) using their codebase and hyperparameter values, with only the following differences:
1. We use the Bat implementation introduced in Kim et al. (2024) instead of T meBat ch YYm et al. (023b), which processes multiple data instances per batch, leading to reduced computational requirements in terms of training and generation time in GPU hours.
2. In terms of computational resources, use x 24GB NVIDIA RTX A5000 GPUs instead of 8 x 24GB RTX 3090 GPUs.
Regarding dataset details and train/validation/test split information, we refer the reader to Kim et al. (2024). Furthermore, we choose not to report inference times in table 2 as we find the differences negligible compared to the reported MOFFlow values.
F.2 ADDITIONAL RESULTS
Results in property evaluation. Following Kim al. (2024). evaluate the quality of generated MOF structures beyond match rate and RMSE by analyzing eight key properties: volumetric surface area (VSA), gravimetric surface area (GSA), largest cavity diameter (LCC),
43

---

Published as a conference paper at ICLR 2026
pore limiting diameter (PLD), void fraction (VF), density (DST), accessible volume (AV), and unit cell volume (UCV). We use the same experimental implementation and code as MOFFlow, evaluating models with RMSE and distributional differences.
Results in table 7 compare our model against reported DiffCSP and MOFFlow results from Kim et al. (2024), as well as reproduced MOFFlow model trained with the Batch implementation. V-MOFFlow achieves improved RMSE for half the properties compared to the original MOFFlow paper. Moreover, the reproduced MOFFlow yields slightly higher property values than both the original MOFFlow and V-MOFFlow results. Overall, we believe that the magnitude of most values high for meaningful comparison across methods.
Table 7: Property evaluation. We report results for DifiCSP and MOFFlow as they are in Kim et al. (2024), and we compute from scratch the properties of the generated samples with the re- trained MOFFlow and V-MOFFlow, that make use of the Batch implementation. Average RMSE is computed between the ground-truth and generated structures.
RMSE 1
DiffCSP MOFFlow (Paper)
MOFFlow (Reproduced)
V-MOFFlow (Ours)
VSA (m2/cm²
796.9
264.5
289.9
265.0
GSA (m²/g)
1561.9
331.6
473.2
328.8
AV (A#)
3010.2
530.5
1935.1
714.2
UCV (A3)
3183.4
569.5
2108.5
785.8
VF
0.2167
0.0285
0.0379
0.0263
PLD (Å)
4.0581
1.0616
1.2434
1.0337
LCD (A)
4.5180
1.1083
1.2613
1.0888
DST (g/cm²)
0.3711
0.0442
0.0747
0.0446
Effect integration steps. Following Kim et al. (2024), investigate how the number of sampling integration steps affects both V-MOFFlow and our reproduced MOFFlow model (the one from section 5.2).. We randomly select 1000 structures from the set and evaluate match and RMSE across varying integration steps: [2, 5, 7, 10, 50, 100, 200, 500, 1000], using the same experimental procedure as Kim et al. (2024). The results in fig. 17 show that both models exhibit similar trends, with performance peaking around 10 and 50 integration steps before slightly declining higher step counts. The main difference between the models is the performance gap rather than the overall trend, reflecting the difference in accuracies obtained on the entire test set (table 2).
Integration Steps Comparison: MOFFlow
NOFF on
MORFMW
20
20
Match ole
RMA5E
3.5
2.5
200
100
802
1000
200
600
$00
820
3005
Integration steps
Integration steps
Figure 17: Comparison between MOFFlow (the reproduced from section 5.2) and V-MOFFlow in terms of match rate and RMSE over different timestep values: [2, 5, 7, 10, 50, 100, 200, 500, 1000].
44


Reviews:
[{"review_id": "8sjMPXkSnm", "reviewer": "Reviewer_CUa3", "summary": "This paper focuses on extending Variational Flow Matching (VFM) to Riemannian manifolds through the proposed Riemannian Gaussian Variational Flow Matching (RG-VFM). The authors argue that endpoint prediction provides a stronger learning signal by directly minimizing geodesic distances. The method formulates a variational flow matching objective based on Riemannian Gaussian distributions, applicable to manifolds with closed-form geodesics. They further demonstrate with experiments on spherical, hyperbolic, and real-world datasets that RG-VFM can better captures manifold geometry.", "questions": "Please refer to weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "- Very well-written paper with excellent explanations and visuals.\n- Variational extension of FM is well motivated; and this paper lifts this idea to Riemannian manifolds.\n- The derivations and design choices are well-justified and appear natural.\n- Good empirical results in two different real-world domains and on one synthetic dataset.", "weaknesses": "- The work appears somewhat incremental, as extending Riemannian flow matching to a variational formulation is relatively straightforward. Formally establishing a connection between RG-VFM and RFM through Proposition 4.1 is useful, but not very surprising.\n- The discussion of intrinsic versus extrinsic viewpoints is interesting, yet it remains unclear how the extrinsic perspective offers a clear advantage. While it provides additional flexibility, it essentially represents a trade‑off rather than an optimal method. Similar to the difference between VFM and FM, the primary benefit seems to arise mainly from supervision at the endpoints.\n- Although the ability to choose an arbitrary variational distribution is often presented as a main strength of variational flow matching, this flexibility is rarely reflected in practice, since most experimental setups still rely on simple distributional choices. It is not easy for the reader to establish a direct connection between this generic motivation and the experimental setups (esp. for readers less familiar with these benchmarks). It would be much nicer if the authors could elaborate how this type of generality is used in practical setup.", "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": "NWTfr2AQfe", "reviewer": "Reviewer_fzbJ", "summary": "This paper introduces **Riemannian Gaussian Variational Flow Matching (RG-VFM)**, a geometric extension of Variational Flow Matching (VFM) for generative modeling on manifolds.  \nThe authors argue that on curved manifolds, endpoint prediction (as in VFM) provides a stronger learning signal than velocity prediction (as in Flow Matching, FM, or Riemannian FM, RFM) because it directly minimizes **geodesic distances**.  \n\nKey contributions:\n1. **A variational flow matching objective on general Riemannian manifolds**, employing Riemannian Gaussian distributions with closed-form geodesics.  \n2. **A formal analysis linking RG-VFM to RFM** via Jacobi fields, showing that curvature-dependent terms naturally emerge in the RG-VFM objective but are absent in RFM.  \n3. **Experiments** on synthetic (spherical/hyperbolic checkerboards), **materials (MOF)**, and **protein backbone generation** tasks. Results indicate improved geometric consistency and generation quality over Euclidean and velocity-based baselines.", "questions": "1. How would the approach generalize to manifolds without analytic exponential/log maps (e.g., learned manifolds from data)?  \n2. Could the curvature-dependent term be approximated or regularized for arbitrary manifolds, enabling broader applicability?  \n3. In Table 3, could the authors provide variance/error bars for the protein design metrics to quantify statistical significance?  \n4. Is there any insight on the trade-off between extrinsic (RG-VFM-R³) and intrinsic (RG-VFM-M) variants beyond the empirical performance?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper provides a clear and well-motivated theoretical bridge between **variational** and **geometric** generative modeling. The idea of introducing curvature-dependent terms via Jacobi fields to analyze flow-matching losses is novel and elegant.  \n2. The derivation is rigorous, with propositions and proofs connecting RG-VFM and RFM. The mathematical treatment of curvature effects is insightful, especially Proposition 4.3, showing curvature-dependent correction terms.  \n3. The paper is well written and technically organized, with clear mathematical notation and well-illustrated figures (e.g., Fig. 1–3 showing geometric intuition).", "weaknesses": "1. The method is limited to manifolds with **closed-form geodesics**. This assumption restricts applicability to simple spaces (e.g., \\(S^n\\), \\(H^n\\)), leaving open how to handle more general manifolds.  \n2. While results are positive, the experiments lack deeper ablations (e.g., sensitivity to curvature magnitude, loss variants, or effect of variance parameter σ in the Riemannian Gaussian).  \n3. The paper briefly mentions that RG-VFM maintains simplicity of linear flows but does not analyze training/sampling cost relative to RFM or diffusion models.", "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": "yt5qowUbYv", "reviewer": "Reviewer_Kr3K", "summary": "This paper shows theoretically that training (variational) flow matching on end-point prediction in Riemannian geometry yields a more precise loss, that better accounts for the structure of the manifold. They validate empirically their losses on a variety of datasets, including large-scale ones with applications for science.", "questions": "- Did I misunderstand your usage of the words “intrinsic” and “extrinsic”?\n- What is the novelty in your method [3], beyond the theoretical guarantees: isn’t end-point prediction a rather well-known “trick”?\n- Does homogeneity alongside existence of closed form geodesics not trivialise the structure of the manifolds considered? (I am not certain at all, I would like to discuss.)\n- Have you had the opportunity to try out your method on manifolds where the higher order terms you mention appear (non-constant curvature)? I understand that you might not have, as the examples could be rare, but in which case the contribution of the paper is less significant, practically speaking. This question can also be understood as “Why does Riemannian Flow Matching work so well if it misses those important terms?” (or are they not so important therefore?)\n\nOverall, I think the paper is rather sound and a good read, but the contribution feels relatively small.\n\n### References\n\n[1] Jonathan Ho, Ajay Jain, Pieter Abbeel. “Denoising Diffusion Probabilistic Models”\n\n[2] Ricky T.Q. Chen, Yaron Lipman. \"Flow Matching on General Geometries\"\n\n[3] Bowen Jing, Bonnie Berger, Tommi Jaakkola. \"AlphaFold Meets Flow Matching for Generating Protein Ensembles\"", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is well-presented: the text is clear and simple.\n- The proofs provided in the appendix are well-detailed.\n- Proposition 4.1 (and 4.3) is an interesting contribution, showing a limitation in Riemannian Flow Matching [2] – the baseline method for Riemannian generative modelling – and demonstrating the method’s superiority in accounting for curvature terms, theoretically.\n- The proposed fix is very simple.\n- The empirical validation is rather extensive and convincing for the given examples, including large-scale ones. Overall, the method improves (arguably slightly) over the provided baselines.", "weaknesses": "- I am not certain about the initial trichotomy (or the impression that is given when stating is as follows) of “endpoint (VFM), a velocity (FM/RFM), noise (diffusion)”. Diffusion can be trained all three ways, and noise prediction was introduced in DDPM [1], as it empirically produced better performance. Same goes for flow matching. (Indeed, though, because of the linearity in the path, all reparameterisations are *theoretically* equivalent.)\n- It seems to me that the words “intrinsic” and “extrinsic” are not used properly. If I am not mistaken, intrinsic (informally) “live directly on the considered space”; extrinsic coordinates depend on the ambient space. For instance, intrinsic on the 2-sphere would be the angles $\\theta$ and $\\phi$; extrinsic would be the usual 3D coordinates. Here, the authors seem to use it for projected on the manifold or not. (See Figure 2.) Modelling intrinsically is known to be difficult.\n- I would argue that the paper is not particularly novel on the methods front (not that it is without contributions), mostly (variational) flow matching on Riemannian geometries.\n- Sampling from the Riemannian Gaussian is not trivial on certain manifolds. It seems that the authors have omitted this discussion in the paper.\n- Perhaps I have missed out on this, but it seems that there is no discussion on the hypotheses made about the manifold (homogeneity and existence of closed form geodesics).\n- Training for end-point prediction is not new [3].", "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": "XGnaTq3UO6", "reviewer": "Reviewer_423B", "summary": "This paper introduces Riemannian Variational Flow matching (RG-VFM), a method that is designed for conducting generative modeling on manifolds. RG-VFM extend variational flow matching to general geometric objects. RG-VFM replaces the velocity matching objective in RFM with a distribution matching objective that is analogue to VFM, but with the distribution defined on a Riemannian manifold. The authors then compare the Jacobi fields of RG-VFM and RFM and show theoretically that such a reparameterization of regression objective is capable of capturing the high-order curvature of the manifold. Empirical results suggest the success of RG-VFM in capturing curvature information and modelling geometries.", "questions": "- Motivations: Could the authors elaborate on how RG-VFM addresses the breakdown of equivalence between endpoint, velocity, and noise prediction on curved manifolds? Furthermore, how does this challenge directly motivate the design of RG-VFM over other approaches?\n\n- More clarification on the incorporation of high-order curvature. Can I get more clarification on what is the practical impact of the higher-order curvature term on the learning problem? For example, does its inclusion confer specific advantages when learning on heterogeneous manifolds against homogeneous ones?\n\n- Computational Complexity: Could the authors comment on the change of computational complexity when learning with RG-VFM? If learning the higher-order curvature information using RG-VFM induces more computations, relative to methods like RFM?\n\n- Empirical Reults (Table 1): Why does the external view of RG-VFM exhibit significantly better performance when learning on the hyperboloid manifold? Is this phenomenon specific to learning on Riemannian manifolds, or could it also occur in Euclidean settings under certain conditions?\n\n- This question is for my own curiosity: The relationship between RG-VFM and RFM appears analogous to that between Transition Matching [1] and Flow Matching in Euclidean space. Could the authors comment on this analogy? Specifically, is the model effectively learning a conditioned transition kernel, approximating a jump from $x_t$ to $x_1$?\n\n\n[1] Transition Matching: Scalable and Flexible Generative Modeling. N. Shaul et al. NeurIPS 2025", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The formalization and mathematical derivations are lucid and well-presented.\n\n- The manuscript is well-structured with a clear logical progression, making the core arguments easy to follow.\n\n- The novelty of the proposed method and its key distinctions from prior literature are clearly articulated.", "weaknesses": "My primary concerns relate to the empirical evaluation and the clarity of the motivation.\n\n- Empirical Results: The significance of the empirical results is not entirely convincing. In the real-world protein backbone generation benchmarks, the high standard deviations suggest that the performance improvement over existing models may not be statistically significant. Furthermore, for the structure prediction results, standard deviations are not reported, which makes it difficult to ascertain the significance of the reported performance gains.\n\n- Motivation: In the abstract, the authors state that the equivalence of endpoint, velocity, and noise prediction breaks down on curved manifolds. It is not immediately clear how RG-VFM alleviates this issue, or how this observation directly motivates the proposed method. I would appreciate clarification if this point was addressed in the manuscript and I overlooked it.", "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)