Curly Flow Matching for Learning Non-gradient Field Dynamics

We thank the reviewer for their time and effort in reviewing our work. We are pleased that the reviewer found our method to “perform well” and that the reviewer agrees that CurlyFM can handle non-gradient field dynamics by leveraging velocity information. We now address the main questions posed in the review.

Compatibility of the model with Schrödinger Bridge

We acknowledge the reviewer's questions regarding the compatibility of our diffusion terms $g_t$ in various places of our Schrödinger Bridge formulation. We now answer each question in turn.

In eq (9), with general diffusion term , should the KL divergence being $E_P \int_t g_t^{-1} \|v\|^2_2$? Similar question holds for the unnumbered equation below eq. (10).

The reviewer is indeed correct in their observation. However, for our problem, we can drop the scaling factor as it doesn’t affect the optimization problem, so we drop this term for simplicity. We will add this as a note to clarify this point..

Compatibility with different diffusion terms

The reviewer is again correct in their statement that SDEs in the SB problem that do not share diffusion terms could lead to problems. However, in this work, we consider the KL divergence between SDEs with matching diffusion terms, and always construct $\mathbb{Q}$ as a mixture of Brownian bridges with the same diffusion term as $\mathbb{P}$. We hope that this clarifies this technical point.

Velocity data in transcriptomics and oceans datasets

We thank the reviewer for their helpful suggestions here. We are happy to provide further discussion on velocity measurements and the assumptions on them in each application. The reviewer is correct to point out that the assumptions on velocity measurements may be quite different between applications, and the quality of the velocities may vary greatly as well.

Relating RNA velocity to velocity in the SDE model

We agree with the reviewer that RNA velocity attempts to compute the rate of transcription of each gene, which is not directly analogous to the velocity used in the SDE model. We follow [1] in assuming that the RNA velocity estimate is a reasonable estimate (up to a scaling factor) of the SDE velocity. There are a number of discrepancies between these two types of velocities, but they nevertheless seem correlated enough in practice to provide improved estimates of the SDE velocity field as demonstrated in a number of subsequent works [2, 3, 4].

Intuitively, this interpretation makes sense, as assuming we have a perfect estimate of transcription and degradation rates and at an infinite number of locations, then following the continuous vector field in RNA state space defined by these rates should be a reasonable model of cell dynamics (at least at the RNA level). These, of course, are unrealistic assumptions, but they give intuition on how RNA velocity estimates may be useful in SDE models of cell dynamics.

Sparse velocity interpolation

Yes, we thank the reviewer for their insightful comment on velocity interpolation. This is indeed a necessary step in both the oceans and single-cell experimental setups—the most realistic setting—we consider in CurlyFM, as most velocities will be measured at sparse locations both in time and space.

In this work, we primarily explore how to use a continuous velocity prior in flow matching models and leave detailed exploration of this interpolation to future work. More precisely, we rely on using simple kernel smoothing following [5] for all applications with sparse velocities.

How good do the velocity measurements need to be?

We thank the reviewer for their insightful remark on how the quality of velocity measurements might affect the performance of our model. To investigate this, we ablate the velocity quality by adding various levels of noise. In particular, we construct a noisy drift $f_{t}^{\text{noise}}$ as a linear combination of the ground truth reference drift $f_{t}$ and noise from a standard gaussian distribution ($f_{t}^{\text{noise}} = (1 - \beta) * f_{t} + \beta * \text{noise}$). Note that $\beta = 0$ (no noise) corresponds to the setting of CurlyFM used in the main paper. The results of this ablation are presented in the Table R1.

We find that the performance between $\beta = 0$ (no noise, i.e. regular CurlyFM) and $\beta = 0.25$ (25% noise) are similar whilst the performance for $\beta = 0.5$ and higher gradually becomes worse, as the noise dominates over the ground truth reference drift in $f_{t}^{\text{noise}}$. This shows that CurlyFM is robust to moderate amounts of noise added to the ground truth reference drift.

Table R1: Ablation with noisy reference drifts

Concluding remarks

We would like to thank the reviewer again for their time reviewing our paper. We hope that our rebuttal here was successful in answering the reviewer's great points and allows the reviewer to continue positively endorsing our paper. We are also more than happy to answer any further questions that arise in the discussion phase.

References

[1] Tong, Alexander, et al. "Trajectorynet: A dynamic optimal transport network for modeling cellular dynamics." International conference on machine learning. PMLR, 2020.

[2] Koshizuka, Takeshi, and Issei Sato. "Neural lagrangian schr" odinger bridge: Diffusion modeling for population dynamics." arXiv preprint arXiv:2204.04853 (2022).

[3] Neklyudov, Kirill, et al. "A computational framework for solving wasserstein lagrangian flows." arXiv preprint arXiv:2310.10649 (2023).

[4] Qiu, Xiaojie, et al. "Mapping transcriptomic vector fields of single cells." Cell 185.4 (2022): 690-711.

[5] Shen, Yunyi, Renato Berlinghieri, and Tamara Broderick. "Multi-marginal schr" odinger bridges with iterative reference refinement." AISTATS (2025).

We thank the reviewer for their time and effort in reviewing our work. We are glad the reviewer found our work “well explained” and found the range of experiments interesting. We now address the main concerns raised in the review, grouped by theme.

Motivation for CurlyFM

Our work is motivated by practical applications, which are similar to Schrödinger bridge problems and motivated by them, but have different criteria for success. Namely, we aim to model dynamical systems well empirically in a variety of diverse settings, where the reference drift may only be observed at measurements rather than everywhere in space and in time. This necessitates the need to devise a learning framework that is efficient to scale to these problems and to iterate quickly, two points that we have sought to demonstrate empirically through our selection of benchmarks and results.

In more detail, our work attempts to find a computationally efficient solution to model our high-impact application settings, such as trajectory inference in single-cell RNA with RNA velocity, rather than computing the best numerical approximation to the Schrödinger bridge problem in a general setting. Given the vast and storied literature on SBs with many numerical approximations that require simulations to solve, we position CurlyFM as a simulation-free method that takes key ideas from the SB problem and allows modeling of complex non-gradient field dynamics found in real-world systems, which may have relatively low noise relative to the scale of the drift.

Also of note, in these applications, previous work [1] has found low or no stochasticity to work the best empirically. Consequently, most of our approximations are optimized assuming that we will likely be in the low or no-stochasticity settings, as motivated by our choice of experimental benchmarks. We understand that this motivation may not have been clear in our original draft, and we will update the paper accordingly.

Approximation of SB problem in the stochastic case

We thank the reviewer for their insightful question and opportunity to clarify our work. Despite our work being motivated in the little to no stochasticity regime, we now consider an ablation below where $\sigma > 0$ to demonstrate that this is a simple modification to CurlyFM that does not lose any of its efficiency.

Behavior with respect to $\sigma_t$ and $g_t$

We find, similar to previous work [1], that low values of $\sigma$ perform the best on all metrics. As a result, we recommend setting $\sigma_t$ to zero unless some reference $\sigma$ value is known. Therefore, all of our experiments in the current manuscript are with $\sigma_t=g_t=0$. To show this in our setting, we perform a new ablation of $\sigma_t$ and $g_t$ on the oceans application. We find that larger sigma monotonically decreases performance on our tasks, justifying our choice of $\sigma_t=0$ for our empirical work.

Table R1: Ablation on stochasticity for CurlyFM

We further note that bridge matching approaches based on iterative refinement do not converge well with small values of $\sigma$ and therefore are only really practical in the large $\sigma$ setting [1]. In settings where a large $\sigma_t$ is required, we believe that these methods may be more well-suited to the task, but present a different set of tradeoffs with better theoretical approximation of the bridge, but in contrast, may incur larger computational complexity due to simulation.

Comparison with GSBM

We appreciate the reviewer’s suggestion on the inclusion of GSBM as a baseline. We now include this baseline adapted to the Ocean currents dataset below using spline neural interpolant $x_t$ (as in GSBM) as well as a comparison of the training times between GSBM and CurlyFM.

Table R2: Comparison with GSBM

We see that CurlyFM performs significantly better than GSBM, albeit at a higher cost which represents the expected tradeoff between expressivity and performance. We will include these baselines in our update to the paper.

Use of kernels

The reviewer is correct that if we have direct access to the reference $f_t(x)$ which is continuous in time and space, then kernels are unnecessary. However, in the practical settings that we investigate, a continuous ground truth reference field does not exist $f_t(x)$. In our applications, we only have access to the reference field at discrete points in space and time that correspond to ground truth data at the different time marginals. Consequently, we use a kernel to build a continuous reference field $f_t(x)$. We will clarify this further in the main text and thank the reviewer for their comments on how to improve the readability of our work, and we will add motivation for each step in the methods section.

Cost of OT plan

The reviewer makes an insightful observation that the optimal plan $\pi^* = \min KL( \pi | Q_{0,1})$ is indeed intractable. Consequently, we make several approximations to these couplings that enable faster throughput as offered in simulation-free training. In particular, we make three approximations to the cost $c(x_0, x_1)$ between two points, and one on the coupling given this cost. These are the following:

• 1. Algorithm 1 has learned a path or set of paths for the means of Gaussians not too far from the optimal $Q_t(x_t | x_0, x_1)$.
• 2. The diffusion term $\sigma_t$ is small, and therefore the cost is close to the distance the means travel.
• 3. The integral of the squared length of the curve is approximated empirically using a Monte Carlo estimate

After approximating $c(x_0, x_1)$, we use mini-batch OT to approximate the entropic OT problem following [1]

Interestingly, in the low stochasticity setting [1] found that mini-batch OT with no stochasticity was empirically a better approximator of $\pi^*$ in the Gaussian case. This is because mini-batch transport adds some amount of “entropy” to the plan due to its approximation. We follow this approximation in our setting. We hope that this clarifies some of the approximations made in the OT computation for our method, and will add a detailed explanation of these choices to the appendix, as well as a brief explanation in the main text.

Tradeoff between directly computing OT plan vs. iterative refinement

This is an interesting question. We have not tried turning CurlyFM into an iterative refinement-based algorithm. However, we argue that this is a fairly easy adaptation, but not suitable for our application domains, where we assume small stochasticity levels. We also note that iterative refinement-type algorithms all require extensive simulation for each refinement step, and hence are not really simulation-free methods, which is a key goal here. However, we do acknowledge that iterative refinement-based approaches can still be significantly faster than methods that require backpropagation through simulation (e.g. [2,3]). We will update this within our related work and also discuss the tradeoffs depending on stochasticity levels in the appendix.

Other suggestions around clarifications

We appreciate the reviewer's thoughtful suggestions, and we will update the manuscript accordingly, taking into account all three suggestions.

Ablation to evaluate its robustness in a toy setting

We thank the reviewer for their insightful suggestion. To assess robustness, we designed a toy experiment with an analytical SB solution and non-gradient dynamics based on [1,4,5]. We bridge two Gaussians in the presence of a spiral reference field across various dimensions $d$ and noise levels $\sigma_t​=g_t$​. We compare against the closed-form solution and OT-CFM, measuring KL divergence, Wasserstein distance, and the cosine distance to the ideal rotational angle. Results confirm CurlyFM performs best in low-dimensional, low-stochasticity settings, consistently achieving high cosine similarity.

Table R3: CurlyFM Ablation

Concluding remarks

We thank the reviewer again for their time, effort, and thoughtful suggestions, which have enabled us to clarify technical details and also include ablations and baselines that strengthen the empirical results. We further hope that this rebuttal was successful in answering all the reviewer's great points and alleviating the concerns shared by the reviewer. If the reviewer is satisfied, we invite the reviewer to consider a fresher evaluation of our work with this rebuttal in context. We are also more than happy to answer any further questions the reviewer might have. Please do let us know.

References

[1] Tong, Alexander, et al. "Simulation-free schrödinger bridges via score and flow matching." AISTATS. 2024.

[2] Tong, Alexander, et al. "Trajectorynet: A dynamic optimal transport network for modeling cellular dynamics." ICML. 2020.

[3] Koshizuka, Takeshi, and Issei Sato. "Neural lagrangian schrödinger bridge: Diffusion modeling for population dynamics." ICLR. 2023.

[4] De Bortoli, Valentin, et al. Diffusion Schrödinger bridge with applications to score-based generative modeling. NeurIPS. 2021

[5] Shi, Yuyang et al. Diffusion Schrödinger bridge matching. NeurIPS. 2023

Thank you for pursuing the discussion. I will try to answer point by point.

1. About the deterministic variant of the GSB problem. Thank you for detailing the set up of the stochastic (and canonical) version of the bridge in the SB problem. To be clear, my concern was related to the fact that only a deterministic setting of the bridge was given (see Equation 11 in the main), and no extension to the stochastic setting was given besides this in the paper. As I understand, when the volatility of the reference dynamics $g_t$ is positive, the authors propose to model the bridge as $Q_{t|0,1}= N( x_t; tx_1 + (1 - t)x_0 + t(1 - t)φ{t, \eta}(x_0, x_1), g^2_t)$. However, I am afraid that this modeling is not correct and surely introduces numerical bias when the volatility increases. For instance, think of the case where the reference is zero and $g_t=\sigma$, where $\sigma>0$ is constant: in this case, the bridge $Q_{t|0,1}$ is tractable (this is a scaled Brownian bridge), and its marginal variance is given by $\sigma^2 t(1-t)\neq \sigma^2$. Hence, even in a simple case, the proposed modeling is not able to provide a good approximation when $g_t$ is not very low. For general bridge, the marginal variance should notably cancel out at the boundaries, which is not the case with the proposed modelling. As I explained in my first review, a well-suited formulation would be to follow the setting of GSBM, which lets flexible the variance and learns it along the marginal mean. I have the feeling that this set up explains why the proposed approach cannot work with non-zero volatility.

2. About the comparison with GSBM. Following my first point, I am afraid that the comparison you propose with GSBM is not fair, especially related to the fact that you use the loss at Line 162, instead of their variational loss (6a) which is more theoretically grounded. Indeed, when the bridge interpolant $x_{t,\eta}$ has a variance (which is the case that we treat here), then the exact drift of the corresponding marginal-preserving SDE is not simply the conditional expectation of the interpolant given by $\mathbb{E}[x_{t,\eta}|x_0,x_1]$ -> this quantity is the drift of the corresponding marginal-preserving ODE (which amounts to the deterministic case advocated by the authors). To compute the drift of the SDE, one has to take into account an additional term in the conditional expectation, which is related to the variance of $x_{t,\eta}$ -> see Equation (8) in GSBM paper. Hence, this implies a modification of the loss $\mathcal{L}(\eta)$ proposed by the authors, which , as I explained in my original review, is only valid in the deterministic case.

for _ in range(n_iter):
    x0 = sample_x0(batch_size)
    x1 = sample_x1(batch_size)
    x0, x1 = coupling(x0, x1)
    t = torch.rand(batch_size).type_as(x0)
    eps = torch.randn_like(x0)
    lambda_t = (2* torch.sqrt(t * (1-t))) / sigma
    xt, xt_dot = get_xt_xt_dot(t, x0, x1, geodesic_model) 
    xt = xt + eps * sigma
    vt = drift_model(xt, t)
    st = score_model(xt, t)
    flow_loss = torch.mean((vt - xt_dot) ** 2)
    score_loss = torch.mean((lambda_t * st + eps) ** 2)
    loss = flow_loss + score_loss

for t in torch.linspace(0, 1, 100)[:-1]:
    stochastic_drift = drift_model(x,t) + score_model(x,t)
    x = stochastic_drift * dt + sigma * torch.sqrt(dt) * torch.randn_like(x)

We thank the reviewer for their time and effort in reviewing our paper. We appreciate that the reviewer found our paper “technically sound” and that the majority of our claims “are well supported by the experimental findings,” which have the potential to be impactful to the community. We are also heartened to hear that the reviewer finds our paper to be “clearly written” and that the work is clearly motivated. Finally, we thank the reviewer for acknowledging that CurlyFM “presents a new method that clearly differs from most of the prior works”. We now address the key points of clarification raised in the review, grouped by theme.

Questions regarding the clarity of presentation

We value the reviewers' feedback on improving the presentation clarity of our paper. We now answer the specific pain points raised in the questions.

When I first encounter $\eta$, it is not clear what this represents …

$\eta$ represents a general mixture of conditional bridges $\mathbb{Q}_{\eta, t}$. As a modelling choice, we then set $\eta$ as the parametrized neural path interpolant, which we use to parametrize a mixture of Brownian bridges as stated in line 154. We understand that this point may not have been clear, and we will update the notation to improve clarity.

After line 159, $f^\star_\eta$ seems to correspond to the minimum value of the integral. Is that correct? …

Indeed, this is the correct interpretation. We use the star notation to denote the solution to an optimization problem, such as $f^\star$ or the optimal coupling $\pi^*$, which leads to the Schrödinger Bridge.

What is the role of $\kappa$

The kernel allows us to construct the reference process everywhere in space and time by smoothening out ground truth instances of the reference drift found in the data samples from each time marginal. This is part of the problem setup and is considered given.

what is the precise function of the neural path interpolant?

The neural path interpolant is used to match the reference drift $f_t$, which in general is no longer a straight path between $x_0$ and $x_1$. Intuitively, this makes our paths bend and can be thought of as the mean of the conditional mixture of Brownian bridges. To select which $x_0$ is paired with $x_1$ we use the neural path interpolant as part of Algorithm 2 to compute the OT cost and consequently solve the SB problem.

Differences with related work (e.g. Shen et. al 2024 and Bartosh et. al (2024 and 2025))

Shen et. al 2024

Shen et. al 2024 use an iterative bi-level algorithm wherein step 1) they solve a Schrödinger bridge and estimate the forward-backward drift given a current guess of the reference drift, 2) Given the solution to step 1, they simulate trajectories to estimate a new best reference guess. This bi-level approach is considered since the reference drift belongs to a family of possible reference drifts rather than a single prescribed one. In contrast, CurlyFM employs a 2-stage algorithm that is not iterative and also does not search over a family of reference drifts. More critically, CurlyFM is also simulation-free, which is achieved by making the modeling assumption that the conditional mixture of bridges is modelled as Brownian bridges. In practice, this means that CurlyFM is significantly more efficient to train. We report in Table R1 the computational cost in wall clock time for each approach and notice that CurlyFM completes the Ocean currents problem in a matter of minutes with better performance (see Table 2 in main paper), while the method proposed in Shen et. al 2024 is in the order of multiple hours— thus clearly demonstrating the computational benefit and empirical caliber of CurlyFM. Finally, note, both Shen et. al 2024 and CurlyFM can model non-gradient field dynamics, but only the latter is simulation-free.

Table R1: Compute cost in hours

Bartosh et. al (2024 and 2025)

Both papers from Bartosh et. al consider Latent SDEs with simulation-free training. The key difference between these papers and CurlyFM is that they do not solve a Schrödinger Bridge problem as considered in our work. Furthermore, SDEs in CurlyFM are not Latent SDEs. More precisely, Latent SDEs learned in Bartosh et. al 2025, have no theoretical basis to converge to a Schrödinger Bridge—i.e. the learning process does not minimize the KL between a reference process, and they do not learn a bridge in their current presentation of the paper. Consequently, we argue that the approaches and goals of these papers are different from CurlyFM and cannot be meaningfully compared experimentally, despite both approaches learning an SDE in a simulation-free manner. Whilst it may be possible to build a computational approach to solving the SB problem with latent SDEs, this is an orthogonal research direction to this paper. We will revise our related work section to include a more detailed commentary.

Handling more than two time points?

We acknowledge the reviewer's question. Please note we handle the multi-marginal case as done in prior work (e.g., MFM [1]), where by we construct CurlyFM between two time marginals by skipping an intermediate time marginal, for example, $t=1$ is skipped when $t=0$ and $t=2$ data is used. After which we proceed sequentially by constructing CurlyFM (instantiated from the previous iteration) on the next two time marginals and skipping the next time marginal, i.e., $t=2$ is skipped and $t=1$ and $t=3$ are used.

Access and role of the reference velocity field

In our considered setting, we assume access to the ground truth reference drift at samples $x_0 \sim \mu_0$ and $x_1 \sim x_1$ as part of the problem setup. Note that in the high-impact application domains we consider, such as trajectory inference in single-cell data, we have access to the RNA velocity [2,3]. As we need to estimate CurlyFM everywhere in space and time during training, we construct a smoothed version of the reference drift by using a kernel $\kappa_t$. This allows us to construct the reference drift $f_t (\mathbf{X}t)$---using knowledge from the ground truth reference drift in the existing dataset— in places where there are no ground truth samples. Consequently, we take $f_t$ in these intermediate points as our ground truth reference drift.

Could the method be adapted to scenarios where the true velocity field is known?

Yes! If the reference is known everywhere, we would simply plug it in for $f_t$ rather than use our smoothed version via $\kappa_t$.

How does the accuracy of this approximation affect the performance

This is a great question. To ablate this, we now construct a noisy reference drift for various noise levels in the tables below (for the Ocean Currents dataset). The noisy reference drift $f_{t}^{\text{noise}}$ is obtained as a linear combination of the ground truth reference drift $f_{t}$ and noise from a standard gaussian distribution ($f_{t}^{\text{noise}} = (1 - \beta) * f_{t} + \beta * \text{noise}$). We find that the performance between $\beta = 0$ (no noise, i.e. regular CurlyFM) and $\beta = 0.25$ (25% noise) are similar whilst the performance for $\beta = 0.5$ and higher gradually becomes worse, as the noise dominates over the ground truth reference drift in $f_{t}^{\text{noise}}$. This shows that CurlyFM is robust to moderate amounts of noise added to the ground truth reference drift.

We hope that our answers and ablation in Table R2 help alleviate the reviewers' concerns. We will also include this ablation in the updated draft of the manuscript.

Table R2: Ablation with noisy reference drifts

Concluding remarks

We thank the reviewer for their time and effort in reviewing our work, and we hope our efforts to clarify the main points, along with the additional ablations, allow the reviewer to consider improving their score, as they initially said they would be open to. We are also more than happy to answer any other questions that arise.

References

[1] Kapusniak, Kacper, et al. "Metric flow matching for smooth interpolations on the data manifold." Advances in Neural Information Processing Systems 37 (2024): 135011-135042.

[2] A. Riba, A. Oravecz, M. Durik, S. Jiménez, V. Alunni, M. Cerciat, M. Jung, C. Keime, W. M. Keyes, and N. Molina. Cell cycle gene regulation dynamics revealed by rna velocity and deep-learning. Nature communications, 13(1):2865, 2022

[3] V. Bergen, M. Lange, S. Peidli, F. A. Wolf, and F. J. Theis. Generalizing rna velocity to transient cell states through dynamical modeling. Nature biotechnology, 38(12):1408–1414, 2020.

We are grateful to the reviewer for their time and effort reviewing our paper, as well as their positive appraisal of our work. We appreciate that the reviewer agrees with us that learning non-gradient field dynamics “is indeed an important problem and little work in this direction has been done”. We are also heartened to hear that the reviewer finds our two-stage approach to be “reasonable and sound” and that our approach is “well described.” Finally, we thank the reviewer for finding our set of numerical experiments “extensive” and that they demonstrate settings where CurlyFM captures drift dynamics while “standard conditional flow matching fails”. We next answer the main questions raised in the review.

Tradeoff between matching the drift and marginals

We acknowledge the reviewer's question regarding the possibility of an inherent trade-off in matching the reference drift versus matching the marginals. We believe there may be a potential cause for confusion, as there is no single hyperparameter that negotiates a trade-off between Algo 1 and Algo 2. Indeed, CurlyFM is a two-stage method which first (1) matches the reference drift as closely as possible, and then (2) uses the learned reference drift to learn the optimal marginals that solve the mass transport problem. The hyperparameter and simulation-free nature of CurlyFM is a significant advantage over other methods, e.g., TrajectoryNet [1].

Our comments on the top of page 3 and lines 186-187 were simply to illustrate the fact that learning the reference drift perfectly does not necessarily solve the mass transport problem as a Schrödinger Bridge, and this is exactly the reason we need Algo 2. As a result, the perfect final model should be a valid Schrödinger Bridge, which is close to the reference as much as is required to actually solve the mass transport problem and no more. We understand that this point may not have been sufficiently clear in the main paper, and we will update the draft to include more commentary on this point.

Other mechanisms for connecting marginals

We thank the reviewer for their interesting question regarding alternative approaches to connecting the marginals without solving an OT problem, such as Action Matching. Unfortunately, to the best of our knowledge, methods such as Action Matching are more computationally expensive than what is presented in Algo 2. This is because Algo 2 remains simulation-free due to an unbiased stochastic estimate of the OT cost—rather than computing the entire cost using simulation. In practice, this is quite fast and is not a significant computational overhead in all the experimental settings we consider in the paper. At a broader level, approaches that do not connect marginals via OT run the risk of not being valid solutions to the Schrödinger Bridge problem, which is the thesis of our work. We hope that this answers the very interesting question raised by the reviewer.

Computational speed of TrajectoryNet

Thank you for your great suggestion. We have now included the computational cost of TrajectoryNet (and Curly-FM) in Table R1. We report these results for the 2D and 10D settings for the experiments reported in Table 4.

Table R1: Computational efficiency comparison between CurlyFM and Trajectory Net

As observed in the above table, we find that TrajectoryNet is considerably slower than CurlyFM and is not a scalable approach. This is evident in that training TrajectoryNet takes, on average, 11 times longer than Curly-FM in the 2D case, and 17 times longer in the 10D case. Moreover, as dimensionality increases from 2D to 10D, Curly-FM incurs an increase in computational cost by ~1.7x, whereas TrajectoryNet incurs at least a 2.5x increase in computational cost. We use (*) to denote runs that did not finish within allotted resource allocation time. We will include these results in the updated draft of the paper.

Concluding remarks

We thank the reviewer again for their time and effort during this review process. We hope that our rebuttal here was successful in answering all the great questions raised by the reviewer and allows the reviewer to continue positively endorsing our paper. We are also happy to answer any further questions the reviewer may have during the remainder of the rebuttal period.

References

[1] Tong, Alexander, et al. "Trajectorynet: A dynamic optimal transport network for modeling cellular dynamics." International conference on machine learning. PMLR, 2020.