A critical limitation in both physics and machine learning is that the exact symmetry group must be known a priori. In practice, the underlying symmetries are often unknown, approximate, or domain-specific. In physics, many systems exhibit hidden symmetries, from a simple 1D harmonic oscillator to black holes (Gross, 1996;Liu & Tegmark, 2022). In material chemistry, non-obvious symmetries such as valency and electronegativity can reduce the combinatorial search space (Davies et al., 2016). In computer vision, objects often contain local symmetries, and real-world scans are often partial or noisy (Mitra et al., 2006;2013). These applications share a common need: the ability to automatically discover and characterize symmetries from data, without relying on domain expertise or manual specification.

Several prior works have focused on symmetry discovery in restricted settings, such as rototranslations in images (Rao & Ruderman, 1998;Miao & Rao, 2007), commutative Lie groups (Cohen & Welling, 2014), or finite groups (Zhou et al., 2020;Karjol et al., 2024). Recent studies on learning Lie groups also have some limitations: Benton et al. (2020) assume a fixed Lie algebra basis and a uniform distribution over the coefficients; Dehmamy et al. (2021) produce non-interpretable symmetries; Yang et al. (2023) assume a Gaussian distribution over the Lie algebra coefficients and use potentially unstable adversarial training; and Allingham et al. (2024) use a fixed Lie algebra basis and evaluate only over images.

In this work, we propose LieFlow , which discovers symmetries via flow matching directly on Lie groups. We formulate symmetry discovery as learning a distribution over a larger symmetry group, which serves as a hypothesis space of symmetries, and the support of this distribution corresponds to the actual transformations observed in the data. Using flow matching allows us to capture both continuous and discrete symmetries within a unified framework and accurately model the highly multi-modal nature of symmetries. Unlike standard flow matching (Lipman et al., 2023), which operates in data space, LieFlow learns flows directly over Lie group manifolds, conditioned on the data samples, enabling the generation of new plausible data that respect the underlying symmetry structure.

Our key contributions are to:

• formulate symmetry discovery as a flow matching problem on Lie groups to match the true data symmetries, • provide a unified framework to discover continuous and discrete symmetries, including reflections via flow matching over the complex domain, • identify the “last-minute mode convergence” phenomenon, causing near-zero vector fields for most timesteps, and introducing a novel time schedule for symmetry discovery.

Lie Group. A Lie group is a group that is also a smooth manifold, such that group multiplication and taking inverses are also differentiable. This makes them particularly suitable for describing continuous symmetries. For example, the group of planar rotations SO(2) is a Lie group consisting of one parameter θ, where the rotation by angle θ can be represented as R θ = cos θ -sin θ sin θ cos θ . Each Lie group G is associated with a Lie algebra, denoted as g, which is the vector space tangent at the identity. The Lie algebra captures the local (infinitesimal) group structure and describes how the group behaves near the identity.

The exponential map exp : g → G defines the relationship between the Lie algebra and the Lie group. For matrix Lie groups, which we consider exclusively in this work, the exponential map is the matrix exponential exp

If the exponential map is surjective, we can restrict its domain and define the logarithm map log : G ⊃ U → g for the neighborhood U around the identity such that exp(log(g)) = g, ∀g ∈ U . Note that any Lie algebra element A can be written as a linear combination of the Lie algebra basis, and the group element exp(A) is generated by exponentiating the linear combination. This allows us to generate the entire connected component of the Lie group using the Lie algebra, and thus the Lie algebra basis L i ∈ g is also called the (infinitesimal) generators of the Lie group.

Flow Matching. Flow Matching (FM) (Lipman et al., 2023;Liu et al., 2022;Albergo et al., 2023) is a scalable method for training continuous normalizing flows. The goal is to transport samples x 0 ∼ p 0 drawn from a prior (e.g., Gaussian noise) to data samples x 1 ∼ p 1 . This transport is described by a time-dependent flow1 , a family of maps ψ t : R D → R D for t ∈ [0, 1], where ψ t := ψ(t, x) and ψ 0 (x 0 ) = x 0 and ψ 1 (x 0 ) ≈ x 1 . This flow is defined through the ODE d dt ψ t (x) = u t (ψ t (x)), where u t is a time-dependent velocity field governing the dynamics. To construct a training signal, FM interpolates between x 0 and x 1 over time, e.g., through the straight-line interpolation

defining a stochastic process {x t }, also referred to as a probability path.

In Flow Matching, we define the velocity field u t that generates the desired probability path, and train a neural velocity field v θ t to match it:

Since u t can be written as the conditional expectation of these per-sample derivatives,

we obtain the practical Conditional Flow Matching (CFM) loss

where x t and ẋt come directly from the chosen interpolation.

This highlights the key advantage of FM: the transport dynamics can be learned entirely in a selfsupervised manner, since interpolations between noise and data provide training targets without requiring access to likelihoods or score functions.

Symmetry Discovery. Many works on symmetry discovery differ on the types of symmetries that they can learn. Early works (Rao & Ruderman, 1998;Miao & Rao, 2007) used sequences of transformed images to learn Lie groups in an unsupervised way, but were limited to 2D roto-translations. Some recent works such as (Zhou et al., 2020;Karjol et al., 2024) can only discover finite groups, while others learn subsets of known groups (Benton et al., 2020;Romero & Lohit, 2022;Chatzipantazis & Pertigkiozoglou, 2023). LieFlow is flexible enough to learn both continuous and discrete symmetries and highly multi-modal distributions. Other related works focus on continuous Lie groups, as in our work, but suffer from other limitations. Dehmamy et al. (2021) uses Lie algebras in CNNs to discover continuous symmetries, but learn the symmetries end to end along with the task function, leading to non-interpretable symmetries. Moskalev et al. (2022) extracts Lie group generators from a neural network to analyze the learned equivariance, which is different to our objective of learning which symmetries exist in the dataset. Yang et al. (2023) use GANs to learn the Lie algebra basis, but assume a known distribution, often Gaussian, over these elements. Otto et al. ( 2023) provide a unified framework based on the Lie derivative to discover symmetries of learned models via linear algebraic operators, whereas LieFlow focuses on learning distributions over group transformations from data. Shaw et al. (2024) propose a method based on vector fields that discovers continuous and discrete symmetries of machine learning functions beyond affine transformations, focusing on Killing type generators rather than probability flows on a Lie group as in LieFlow. Ko et al. (2024) learn infinitesimal generators of continuous symmetries from data via Neural ODEs and a task-specific validity score, capturing non-affine and approximate symmetries, whereas LieFlow models a full distribution over transformations within a prescribed matrix Lie group.

The closest to our method is Allingham et al. (2024), where they first learn a canonicalization function in a self-supervised fashion and then learn a generative model that outputs the transformations for each prototype, given a large prior group. While the objective of learning a distribution over symmetries is similar, Allingham et al. (2024) is a two stage optimization process and uses costly maximum likelihood training. They also only evaluate on images, while we consider point clouds.

Generative Models on Manifolds. Several recent works have extended CNFs to manifolds (Gemici et al., 2016;Mathieu & Nickel, 2020;Falorsi, 2021) but rely on computationally expensive likelihood-based training. Some works (Rozen et al., 2021;Ben-Hamu et al., 2022;Chen & Lipman, 2024) proposed simulation-free training methods, in particular, Chen & Lipman (2024) consider flow matching on general Riemannian manifolds. However, none of these works specifically consider Lie groups or the task of discovering symmetries from data. Other flow matching works directly incorporate symmetries (Klein et al., 2023;Song et al., 2023;Bose et al., 2023), but do not discover symmetry from data. Other works consider score matching on Lie groups (Zhu et al., 2025;Bertolini et al., 2025), but again assume a priori knowledge of the group. Closely related to our work is Sherry & Smets (2025), who define flow matching specifically for Lie groups. While methodologically similar to our work, they assume prior knowledge of the symmetry group and use specialized implementations for different Lie groups.

4 LIEFLOW: SYMMETRY DISCOVERY VIA FLOW MATCHING

A central challenge in symmetry discovery is to identify the subset of transformations actually expressed in data without assuming explicit prototypes or fixed augmentation rules. Existing methods often impose such structures externally, limiting scalability and generality. We address this by casting symmetry discovery as flow matching on Lie groups, which allows distributions over transformations to be learned directly from data while respecting the geometry of the underlying group.

Problem Formulation. Our goal is to recover the unknown group of transformations H which preserve the data distribution q on data space X . To discover the hidden group H, we first assume a large hypothesis group G which acts on X that contains the most common symmetries. We wish to find the stabilizer subgroup H ⊂ G such that q(hx) = q(x) for all x ∈ X and h ∈ H.

Discovering Symmetry as Distribution Learning. We frame the problem of finding H as learning a distribution over the hypothesis group G that concentrates on the subgroup H. To achieve this we train an FM model which flows a chosen prior distribution over G to a distribution p θ (g) supported on H. The structure of the discovered symmetries emerges from the concentration pattern of p θ (g): continuous symmetries maintain a spread distribution across the manifold of admissible transformations, while discrete subgroups exhibit sharp peaks at a finite set of modes (e.g. the four rotations of C 4 ). Thus, LieFlow effectively filters the large hypothesis group down to the subgroup H ⊆ G consistent with the data.

Hypothesis Group and Prior over Symmetries. We assume that the hypothesis group G is connected, ensuring a well-defined logarithm map for compact groups. For the prior distribution p over the group G, we use a uniform prior for compact groups and Gaussian priors over the Lie algebra coefficients for non-compact groups. When the exponential map is not surjective, we define the prior implicitly by sampling from the Lie algebra g and applying the exponential map to construct group elements, ensuring well-defined training targets. In experiments, we consider G = SO(2) (planar rotations), GL(2) (invertible matrices) for the 2D datasets and G = SO(3) (3D rotations) for the 3D datasets. Although SO(d) and GL(d) for larger d are certainly feasible in our framework, scaling our method to large or non-compact groups introduces additional numerical challenges, which we leave for future work.

Interpolant Paths Along Orbits. Unlike either traditional FM models which operate in the Euclidean data space or Riemannian flows which operate entirely on G, we propose an FM model that is conditioned on data but outputs group transformations. More precisely, consider a data sample x 1 ∼ q and a transformation g ∼ p(G) from the prior distribution. Let x 0 = gx 1 . We wish to learn to flow from x 0 to x 1 using only transformations in G. Let A = log(g -1 ) ∈ g. We define the interpolation in the data space (an exponential curve) via the group action generated by A, where:

This yields a probability path p t (x t |x 0 ) in data space that stays on the group orbit of x 1 .

The Target Velocity Field. The derivative of the path d dt x t ∈ T xt X defines a vector field along x t . This defines a flow over X which stays within the orbit of x 1 , only moving in directions dictated by the group action. To achieve symmetry discovery, we must learn both the flow from x 0 to x 1 and reconstruct the transformation g that generated it. For convenience, we configure the flow network to output the Lie algebra element A instead of d dt x t . We can then update x t by scaling and exponentiating A and then applying the group action as in Algorithm 2. This allows us to reconstruct g = exp(-A). That is, we set u t (x t |x 1 ) = A. (Note that A pushes forward to the velocity vector d dt x t under the group action. See Appendix A for a more detailed derivation.)

Objective. The CFM objective on Lie groups is similar to the Euclidean case and is given by

where x t follows the exponential curve defined previously, p t (x t | x 1 ) is the distribution of points along the exponential curves, and G is a Riemannian metric on the Lie group, which equips the manifold with an inner product on each tangent space, allowing us to measure distances along the manifold. We use a left-invariant metric as it is completely determined by the inner product on the Lie algebra g and the push-forward of the left action is an isometry, making computations simpler.

The specific form of this metric is determined by the structure of the Lie group under consideration.

Training. Using exponential curves and the flow matching objective, we can now derive the training and sampling algorithms for our method. The training algorithm (Algorithm 1) directly implements the Lie group flow matching objective by constructing training pairs (x 0 , x 1 ) where x 1 is sampled from the data distribution and x 0 = gx 1 for a randomly sampled transformation g ∈ G. x 0 is now a sample from the prior and lives in the group G. Unlike standard conditional flow matching, we construct x 0 from x 1 using g and learn the flow over the Lie algebra of group G. As G is a group, we can compute the target vector field A analytically using the sampled group element g. We sample a timestep 0 ≤ t ≤ 1 and construct a point x t = exp(tA)x 0 on the curve. Given x t , the model predicts A. This means that the target vector field for this sample pair (x 0 , x 1 ) is constant over time. Thus the network only needs to learn to predict the Lie algebra element that generates the transformation from x 0 to x 1 .

Algorithm 1 Training 1: repeat 2:

t ∼ U(0, 1)

5:

x 0 = gx 1 6:

A = log(g -1 )

7:

x t = exp(tA)x 0 8:

Take gradient descent step on 9:

Sampling and Generation. During sampling (Algorithm 2), we integrate over the learned vector field to produce a new sample x ′ 1 . The learned vector field predicts Lie algebra elements that are integrated using Euler’s method with the exponential map x t+∆t = x t exp(∆tA t ), ensuring that the trajectory remains on the Lie group manifold throughout generation. We also output the composed transform M , where x ′ 1 = M x 0 as it is required for generating new group elements. Algorithm 3 Generating Group Elements 1:

2 Lines 3-11 5: return M g Algorithm 3 generates group elements h consistent with the target group H. Given a data sample x 1 , we transform it by g ∼ p(G) to obtain x 0 = gx 1 . We then run Alg. 2 partially (lines 3-11) on x 0 , obtaining outputs x ′ 1 and M . The composed transform M g forms a group element h ∈ H that maps x 1 to x1 . Note that the new sample x ′ 1 does not always match x 1 and in fact converges to the closest mode in the orbit (see Section 5.4 for more details), and thus this procedure does not always produce the identity element.

Comparison with Standard Flow Matching. While standard flow matching operates in Euclidean space, our approach operates on Lie group manifolds, learning vector fields in the tangent space (Lie algebra) that generate curved trajectories respecting the group structure. The exponential and logarithm maps replace Euclidean interpolation with geodesic-like paths on the group manifold. More importantly, we learn a generative model of transformations rather than data points, yielding a lower-dimensional problem on the group manifold itself, not over the entire data space. We formulate symmetry discovery as flowing from a broad hypothesis group to the specific symmetries in the dataset, with challenges analyzed in Section 5.4.

To evaluate LieFlow , we generate several datasets with known symmetries of simple 2D and 3D point clouds of canonical objects (Figure 9 in Appendix B). For the 2D datasets, we consider the target groups H = C 4 , D 4 and perform flow matching from SO(2) → C 4 , GL(2, R) + → C 4 , and GL(2, C) → D 4 . To learn D 4 symmetries, we perform flow matching over the complex domain, where the data samples are first converted into complex numbers and passed through the complexvalued network, which uses twice as many neurons. We find that 20 steps for inference works well.

For the 3D datasets, we use an object with no self-symmetries and consider the tetrahedral Tet, octahedral Oct and rotation over z axis SO(2) as target groups. We consider flow matching from SO(3) → Tet, SO(3) → Oct, SO(3) → SO(2) and use 100 steps at inference. To quantify and comapre the quality of the learned symmetries, we compute the Wasserstein-1 distance between the generated group elements and the ground truth group elements. For all experiments, we use a simple 3-layer MLP with GELU (Hendrycks & Gimpel, 2016) activations. See Appendix B for more details.

Figure 2 shows generated samples from random x 1 in the test set. We see that most samples correctly align with the target distributions C 4 or D 4 . Surprisingly, we find that flow matching in the complex domain, from GL(2, C) to D 4 , correctly recovers all 8 group elements. Considering that we are flowing from continuous groups to discrete groups, our method works quite well, with little variance at the target modes.

Figure 3 visualizes the centroids of a 100 generated samples over t ∈ [0, 1]. The x coordinate of the centroids are offset as t increases. As expected, we can see the centroids converge to 4 modes for the C 4 target groups and 8 for the D 4 group as t → 1. These figures show that the learned distribution over the prior group becomes quite peaky, suggesting that it does indeed learn a discrete group quite well and learns the correct modes. Specifically for SO(2), we can decompose the learned transformation matrices (which should be very close to elements of C 4 ) and plot a histogram of the angles. As we have the ground truth transforms of the dataset, we first canonicalize M in Alg. 2 and perform polar decomposition over it to obtain the rotation matrix and finally convert them to angles. Figure 3d shows that the learned distribution gives the correct peaks at the C 4 elements and is close to the correct distribution (mixture of 4 delta distributions). ). Interestingly, we see that depending on the random transformation to produce x 0 (line 3 in Alg. 2), the final generated sample does not match the original x 1 and seems to flow to the closest group element in the orbit of x 1 of the target group, i.e., arg min gx1 {∥gx 1 -x 0 ∥ : g ∈ G}. As the dataset contains all C 4 or D 4 -transformed samples of the object, the flow still produces samples that are close to the data distribution. Furthermore, the progression shows that the vector field v t is close to 0 for the earlier timesteps (the determinants are close to 1) and only induces a flow towards the closest gx 1 , g ∈ T when t is close to 1. We analyze this “last-minute mode convergence” phenomenon in Section 5.4. We also generate transformed point clouds given by our FM model for Tet and SO(2), as shown in Figure 13 (Appendix C.2). We can see several repeated point clouds, and see that some samples are rotations or reflections of other samples. We also show the intermediate x t during generation in Figure 16 and Figure 17 (Appendix C.2). For Tet group, as in the 2D case, we see that the transformed point clouds converge to the closest group element in the orbit. For the SO(2) group, we can see that the model produces rotations to turn one node of the tetrahedron to be aligned with the z-axis and one triangle face to be in the xy-plane, which corresponding to SO(2) symmetries around the z-axis. For the Oct group, we show its generated point clouds and the intermediate x t in Appendix C.2, where we can see that the FM model produces outputs close to the identity at most steps and fails to properly discover Oct. Motivated by our analysis on “last-minute mode convergence” in the next subsection, where we observe that the entropy of the posterior remains stationary until large times, we introduce a novel time scheduling scheme to sample more timesteps near t = 1. Instead of sampling time from a uniform distribution U (0, 1), we propose using a power distribution where the density is computed as p(x) = nx n-1 , x ∼ U (0, 1) and n is the skewness parameter. The effect of different skewness values on the density is shown in Figure 19 (Appendix C.3). In our experiments, we chose n = 5.

In Figure 5(c), our modified time schedule now enables our model, whose network is unchanged from before, to correctly learn the octahedral group. Figure 21a (Appendix C.3) shows that our model correctly distributes probability mass towards the 24 Oct group elements. To test the limits of our time schedule, we also consider the icosahedral group, with 60 elements. However, our method fails in this case, where the model seems to output the identity for all inputs (see Figure 21b in Appendix C.3). It is clear that while our time schedule can alleviate the last-minute mode convergence issue and allow our method to discover more complex groups, it still cannot handle more “symmetric groups”, i.e., higher-order subgroups.

To quantify the quality of the discovered symmetries, we compute the Wasserstein-1 distance between the empirical distributions of the generated group elements and the ground truth group elements in the test set consist of 5,000 samples. We choose LieGAN (Yang et al., 2023) as our baseline since it is a recent generative approach that explicitly learns Lie algebra parameters for group-like transformations. For 2D datasets, we set the number of Lie generators to be 1, 2, and 4. For 3D datasets, we set the number of Lie generators to be 1, 3, and 6. The results are shown in Table 1 and Table 2. We can see that our method outperforms LieGAN in all experiments, demonstrating the effectiveness of our flow matching approach for symmetry discovery.

Given the challenges in learning the octahedral group, we hypothesize that using flow matching for symmetry discovery is in fact quite a difficult task. We illustrate with an even simpler scenario of flow matching directly on group elements (matrices), from SO(2) to C 4 . As the target vector field u t and the probability path p t (x t | x 1 ) are important quantities in our FM objective (equation 5), we visualize them for a single sample in Figure 6.

Each target mode x 1 is colored differently and the velocities (group difference) to each C 4 group element are shown with the colored arrows. Near t = 0, the probability path p t is near uniform as x 0 is essentially noise and can be far away from x 1 , i.e. there is still a lot of time t for it to move to any mode. As p t is nearly uniform, the average of the velocities dominate and pulls x t close to the middle of the red and blue modes. Here, the vector field produces nearly 0 mean velocity, as it is equidistant to the red and blue modes, and equidistant to the green and yellow modes. The distribution p t becomes peakier towards the closer modes (red and blue), but still remains somewhat uniform. As such, the training target averages out close to 0. Near the end of training, when t is close to 1, here the velocities become larger (due to the scaling factor t inside the exponential) and p t now becomes more uni-modal, picking the closest mode (as there is little time left to go to the other modes, making them more unlikely). Thus, our task of finding a subgroup within a larger group is challenging precisely because the modes are “symmetric” (by definition of the subgroup). Flow Matching Directly on Group Elements. To verify our hypothesis, we perform flow matching directly on the group elements from SO(2) to C 4 . The group elements can be parameterized by a scalar θ, allowing us to analyze the behavior of the learned vector field more easily. Each data sample x 1 is a scalar from the set {-π, -π/2, 0, π/2} and the source distribution is q = U [-π, π].

We use the same network architecture as in the 2D experiments. We can further compute the reverse conditional or the posterior p t (x 1 | x t ) that can help verify two things: 1) that x t remain nearly stationary until t approaches 1, i.e. p(x l | x t ) remains nearly uniform, and 2) x t chooses the closest group element x 1 . The exact details of how p t (x 1 | x t ) is computed is given in Appendix D.

Figure 29 in Appendix C.6 visualizes the progression of x t over time for 100 samples, showing that C 4 symmetry is clearly identified. Figure 7a plots the entropy of the posterior, and shows that even in this simple scenario, the entropy remains near the maximum (uniform distribution) until t approaches 1. Figure 7b visualizes the posterior p(x 1 | x t ) averaged over 1,000 samples and separated into 4 classes, depending on the original x 0 and its nearest mode in {-π, -π/2, 0, π/2} (only two shown, other classes are similar, see Figure 30b in Appendix C.6). For all 4 classes, we observe that p(x 1 | x t ) remains nearly uniform until around t = 0.9, when it becomes uni-modal to the closest mode/component, e.g., for all x 0 that were closest to the mode µ = -π, then the posterior p(x 1 | x t ) converges to µ. This shows both that the learned vector field remains nearly 0 for most timesteps, only converges when little time is left, and converges to the closest group element.

Note that this differs from the standard FM setting, where the flow is defined on a high-dimensional space. Bertrand et al. (2025) find that for high-dimensional flow matching with images, the entropy of the posterior drops at small times, transitioning from a stochastic phase that enables generalization to a non-stochastic phase that matches the target modes. In our low-dimensional setting of symmetry discovery, we find the opposite occurs: the entropy stays near uniform until large t, where it suddenly converges. We term this phenomenon “last-minute mode convergence”.

Figure 8 shows the evolution of a uniform grid over SO(2) over time. As we hypothesized, we can see from the velocity arrows that particles move slowly closer to the midpoint between two modes (t = 0 to t = 0.60) but the velocities are relatively small. From t = 0.70 onwards, we can see that the shift towards the closest modes and the velocities become increasingly large until t = 1.

Figure 31 (Appendix C.6) shows a clearer picture of how the velocity evolves over time. From t = 0 to t = 0.6, we see that the x t ’s are nudged towards the midpoints between modes. Interestingly, we see some oscillating behavior where the vector field flips sign, suggesting that the net velocity hovers close to 0 with no defined pattern. From t = 0.8 onwards, we see that points x t are pushed to the closest x 1 mode, with increasing velocities. With near zero net velocity until t = 0.6, these plots also support the observation of “last-minute mode convergence”.

Despite the challenges revealed in our experiments, this work demonstrates that generative models, particularly flow matching, offer a promising avenue for symmetry discovery. We introduced a novel approach to symmetry discovery using flow matching over Lie groups, where we learn a map from a larger prior group to the data symmetries. Our experiments revealed both successes and fundamental challenges. In the 2D case, we successfully discovered discrete symmetries from several different continuous prior groups, including the dihedral group by learning a flow over the complex domain. The method naturally discovers the closest group element rather than exact transformations, which may be advantageous for handling approximate symmetries in real data. However, our 3D experiments exposed some limitations: while our method can discover smaller subgroups such as the tetrahedral group, it fails for higher order groups such as the octahedral group. Our analysis uncovered a “last-minute convergence” phenomenon, that the symmetric arrangement of the target modes causes the learned vector field to remain near zero for most of the trajectory, only converging to the nearest mode as t approaches 1. Our simplified flow matching experiment supported this claim and we showed that the posterior entropy remains near uniform for most timesteps. We introduced a power time schedule to skew the sampling towards timesteps near 1. Our heuristic allows use to discover the more complex octahedral group, but still fails for higher-order groups.

There are several limitations to our work. While we demonstrate feasibility on simple and noise-free point clouds, scaling to noisy real-world datasets remains an open question. Our method is currently limited to discovering symmetries within the parameterization of the hypothesis group, and it is still an open question as to whether LieFlow will scale to real-world datasets. Our method requires fitting a distribution over the prior group, which may impact the range of symmetries it can discover depending on the parameterization and boundaries. Although we have analyzed the cause behind last-minute mode convergence and proposed a time schedule according to the power distribution, it is still unclear how to discover complex symmetries. An important future direction is to investigate Riemannian diffusion-based formulations, which can sidestep the issue of discontinuities of the target vector field at the cut locus (Huang et al., 2022;Zhu et al., 2025;Mangoubi et al., 2025), and may alleviate the last-minute mode convergence behavior observed in LieFlow. Another important future direction is to improve numerical stability and scalability on high-dimensional or non-compact groups, enabling LieFlow to fully leverage its group-agnostic framework beyond the lower-dimensional settings demonstrated here.

Future directions include developing specialized architectures with group-aware inductive biases, incorporating additional supervision signals to guide convergence, and extending the method to handle approximate symmetries common in real-world data. The integration of discovered symmetries into downstream equivariant networks also remains an important open question. Despite current limitations, this work demonstrates that flow matching offers a promising path toward automatic symmetry discovery, lending us insights into the data and offering improved sample efficiency and generalization for downstream tasks.

This section provides a detailed derivation of the target vector field u t (x t | x 1 ) used in our Lie group flow matching framework. We begin by introducing flow matching on Lie groups, followed by the derivation of the target vector field on the data space induced by the group action.

Interpolants on Lie Groups. Denote the left action of G on itself for any g ∈ G as L g : G → G, ĝ → gĝ. Let T g G be the tangent space at g ∈ G. The push-forward of the left action is (L g ) * : T ĝ G → T gĝ G, which transforms tangent vectors at T ĝ G to tangent vectors at gĝ ∈ G via the left action.

To derive FM on a Lie group, we need to define the interpolant as in the Euclidean case. An exponential curve starting at g 0 , g 1 ∈ G can be defined by (Sherry & Smets, 2025):

This curve starts at γ(0) = g 0 and ends at γ(1) = g 1 . The multiplicative difference is g -1 0 g 1 and the logarithm maps it to the Lie algebra, giving the direction from g 0 to g 1 . The scaled version is then exponentiated, recovering intermediate group elements along this curve. Using these exponential curves, we can define the target vector field as

Defining the Target Vector Field. Note that we couple source and target by an explicitly sampled group action. According to equation 4, given x 1 ∼ q and g ∼ p(G), and set x 0 = gx 1 and A := log(g -1 ) ∈ g. Consider the group curve g t = g exp(tA) so that g -1 t = exp((1 -t)A) and the induced path x t = g t x 1 = x 0 exp(tA). Plugging g 1 = e into the group interpolant equation 7, we have:

which is a time independent target field in the Lie algebra along this curve. To obtain the conditional target on the data path, we first define the orbit map ϕ x1 : G → X , ϕ x1 (g) = gx 1 . Its differential at g t is given as (dϕ x1 ) gt : T gt G -→ T xt X , where x t = ϕ x1 (g t ). Note that (L gt ) * : g → T gt G.

Compose them together, we obtain the pushforward operator J xt := (dϕ x1 ) gt • (L gt ) * : g -→ T xt X . Then, we push the group target to the data path and obtain the conditional target on data:

We define the stabilizer at x t as Stab(x t ) = {s ∈ G : s •x t = x t } and s xt is the Lie algebra of the stabilizer. According to the Orbit-stabilizer theorem, we have:

This means that if the stabilizer of the data is trivial, the pushforward operator is bijective. Consequently, every tangent vector

Using this coordinate identification, for simplicity, we can write u t (x t |x 1 ) ≡ A as the target vector field.

B TRAINING DETAILS 3D Datasets. We use an irregular tetrahedron with no self-symmetries as shown in Figure 9c and generate datasets containing Tet, Ico symmetries. We consider the prior group SO(3) and use a uniform distribution over it by using Gaussian normalization over unit quaternions and transforming them into 3 × 3 matrices.

For the 3D case, we use a similar but wider network architecture as the 2D datasets and use a sinusoidal time embedding.

This section contains additional results and visualizations for both 2D and 3D datasets. Figure 12 visualizes the trajectories of the centroids over time for 100 samples, and PCA is performed to project them onto the 2D plane. We can see that our model seems to learn Tet and SO(2) elements, creating some visible clusters, but fails to discover Oct symmetries. We also show the intermediate x t during generation in Figure 16, Figure 17 and Figure 18. For the Tet, we see that the transformed point clouds converge to the closest group element in the orbit. For the SO(2) group, we can see that the model produces rotations to turn one node of the tetrahedron to be aligned with the z-axis and one triangle face to be in the xy-plane, which corresponding to SO(2) symmetries around the z-axis. For the Oct group, we can see that the FM model produces outputs close to the identity at most steps and fails to properly discover Oct. In this section, we show additional results for the 3D datasets with time sampled from the power distribution. The power distribution density is shown in Figure 19. Figure 20 shows the generated transformations when using the power distribution time schedule. We can see that flow matching now works for octahedral group, but still fails to discover the icosahedral group. In this section, we show additional results for discovering symmetries in a multi-object dataset. We create a dataset consist of four different irregular 3D objects (Figure 22): tetrahedron, triangular prism, cube, octahedron. We use Tet symmetries as the target group and SO(3) as the prior group.

The training details are the same as in the single-object 3D experiments. Lie generators), which shows that our method significantly outperforms the baseline in discovering the correct symmetries.

To demonstrate that our method can discover structure-preserving transformations, we create a dataset by applying z axis rotations from SO(2) with angles sampled from a Gaussian distribution N (0, π/4) to an irregular tetrahedron. Thus, the data distribution is not uniform along the SO(2) orbits, but the underlying symmetry structure is still SO(2). We use SO(3) as the prior group and train our LieFlow model as before.

The results are shown in Figure 26, Figure 27, and Figure 28. Figure 26 visualizes the centroid trajectories after performing PCA. We can see that the model learns to create an incomplete circular pattern, corresponding to leanring a distribution over SO(2) group elements. Figure 27 shows the generated transformations, and we can see that the elements are concentrated along a onedimensional manifold, and the samples in the center are more dense than those in the periphery, corresponding to learning a distribution over SO(2) group elements. Figure 28 shows the histogram of the angles rotated around the z-axis extracted from the generated transformations. We can see that the distribution of angles closely matches Gaussian distribution. The Wasserstein-1 distance between the discovered distribution and the ground-truth Gaussian distribution is 0.282, showing that our method can accurately recover the underlying structure-preserving transformations. Figure 29 shows the time progression of x t when generating 100 samples from t = 0 to t = 1. We can see that the samples converge to the closest group element in the orbit.

Figure 30b in Figure 30 shows the posterior p(x 1 | x t ) for all 4 classes, depending on the original x 0 and its nearest mode in {-π, -π/2, 0, π/2}. We can see that for all 4 classes, the posterior p(x 1 | x t ) remains nearly uniform when t closes to 1. Figure 30 visualizes the posterior p(x 1 | x t ), divided into 4 buckets depending on the initial position x 0 . We see that the posterior remains nearly uniform until t = 0.95, and then x t converges to the mode that was the closest when t = 0.

Figure 31 visualizes how the velocities change over time. Until t = 0.6, the velocities are close to 0 but slightly push the x t ’s towards the midpoints between modes. From t = 0.8 onwards, we see that points x t are pushed to the closest x 1 mode, with increasing velocities.

where p(x t | x 1 ) is the likelihood and p(x 1 ) is the data distribution. Since flow matching learns a velocity field rather than explicit conditional distributions, we cannot directly compute the likelihood and need to invert the flow and integrate the continuity equation.

We first sample initial points x 0 from a uniform prior over [-π, pi) and forward simulate using the learned velocity field v θ to obtain the trajectories x t and their log probabilities through the continuity equation. To evaluate the likelihood p(x t | x 1 ) for each candidate x 1 , we invert the linear interpolation x t = (1 -t)x 0 + tx 1 to find x 0 = (x t -tx 1 )/(1 -t), then integrate the divergence along the trajectory from this inverted x 0 . Special handling is required at the boundaries: at t = 0, the posterior equals the prior p(x 1 ) (which we know in this simple scenario) due to independence, while near t = 1, we use a Gaussian approximation to avoid numerical instability from division by (1 -t).

Algorithm for k = 1 to K do for k = 1 to K do

4: for t ∈ {0, 1 T , . . . , 1} do 5:x t ← ODESolve(v θ , x 0 , [0, t]) ▷ Integrate velocity field 6: log p t (x t ) ← log p 0 (x 0 ) -t 0 ∇ • v θ (x s , s)ds p(x 1 | x 0 ) ← p 1 (x 1 )▷ Independent at t = 0 13: else if t ≈ 1 then 14:

4: for t ∈ {0, 1 T , . . . , 1} do 5:x t ← ODESolve(v θ , x 0 , [0, t]) ▷ Integrate velocity field 6: log p t (x t ) ← log p 0 (x 0 ) -t 0 ∇ • v θ (x s , s)ds p(x 1 | x 0 ) ← p 1 (x 1 )

4: for t ∈ {0, 1 T , . . . , 1} do 5:

ψt is technically an evolution operator arising from a time-dependent vector field; we follow standard terminology in the flow matching literature and use the term flow.