Flow Matching Neural Processes
Authors: Hussen Abu Hamad
Dan Rosenbaum
Venue: NeurIPS 2025 poster ()
Keywords: flow matching
neural processes
Primary area: deep_learning

Abstract: Neural processes (NPs) are a class of models that learn stochastic processes directly from data and can be used for inference, sampling and conditional sampling.  We introduce a new NP model based on flow matching, a generative modeling paradigm that has demonstrated strong performance on various data modalities. Following the NP training framework, the model provides amortized predictions of conditional distributions over any arbitrary points in the data. Compared to previous NP models, our model is simple to implement and can be used to sample from conditional distributions using an ODE solver, without requiring auxiliary conditioning methods.In addition, the model provides a controllable tradeoff between accuracy and running time via the number of steps in the ODE solver. We show that our model outperforms previous state-of-the-art neural process methods on various benchmarks including synthetic 1D Gaussian processes data, 2D images, and real-world weather data.

Flow Matching Neural Processes
Hussen Abu Hamad Department of Computer Science University of Haifa
Dan Rosenbaum Department Computer Science University of Haifa
Abstract
Neural processes (NPs) are class of models that learn stochastic processes directly from data and can be used for inference, sampling and conditional sampling. We introduce a new NP model based on flow matching, a generative modeling various data modalities.
paradigm that has demonstrated strong performance Following the NP training framework, the model provides amortized predictions of conditional distributions over any arbitrary points in the data. Compared previous NP models, our model is simple to implement and can be used to sample from conditional distributions using an ODE solver, without requiring auxiliary conditioning methods. In addition, the model provides a controllable tradeoff between accuracy and running time via the number of steps in the ODE solver. We show that our model outperforms previous state-of- offte-of-the-rt neural process methods on various benchmarks including synthetic ID Gaussian processes data, 2D images, and real-world weather data.
Introduction
Recent advances in generative machine learning are primarily driven by models capable of leveraging context information to enhance their predictions. To be effective across varying levels of contextual information, these models must handle multiple degrees of conditional uncertainty and therefore accurately capture conditional distributions at multiple levels of abstraction.
The neural process (NP) framework introduced by Garnelo et al. [13, 14] provides a principled approach to context-based predictions by formulating the problem as learning stochastic processes from data, essentially training generative models of functions. Similarly Gaussian processes, NP models can be used as priors over functions capable of generating samples of arbitrary points along the functions. These target points can be predicted unconditionally or conditioned on a context of observed points along the function. This formulation has made neural processes popular approach for modeling data in various domains.
In recent years, many different models have been proposed within the neural process framework, differing their architecture and stochastic mechanisms. Early models focused on stochastic latent variables and were trained by variational inference [14]. More recent approaches such as Nguyen Grover [32] leverage transformer-based architectures [40] an autoregressive setup, achieving improved performance in many scenarios.
Despite these advancements, several challenges remain unresolved. With the exception of autoregres- sive approaches, models tend to underfit the training functions, failing to capture intricate structures. On the other hand, autoregressive models, while more expressive, require sequential sampling of the target points one at a time and are therefore expensive to sample from. Additionally, these models lack global or hierarchical representation of the function and the uncertainty which can limit their applicability and harm their performance in some cases. For example, the first dimensions in the autoregressive order are constrained to simple distributions.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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elagg
transformer
Agg x
Ag
10%
ligh
ODE solver time
3/2
Figure I: FlowNP: we model a probability flow of a stochastic process where at each step our model takes the values of observed context points from the given function (ctx), and an intermediate value of the target points (tgt) at time and predicts the velocities of the target points. With an ODE solver this can be used as model of p (y*8t 1288t, {xxexx (ytx)) to generate samples or compute likelihoods.
In this paper, we introduce FlowNP, new neural process model based on flow matching a recent generative modeling paradigm that is closely related to diffusion and score-based models and has demonstrated strong performance in images and other modalities [27, 25, 1J. Flow matching operates by continuously transforming a simple and tractable initial distribution into the target data distribution by flowing through a continuous path of intermediate probabilities. The approach enables both generating samples and computing log- likelihood over data by solving an ODE.
Our model is built transformer architecture where the input tokens include both the observed context points and intermediate values the target points as generated across the flow path. At each step, the model outputs velocity vector which guides the sampler in updating the target point values for the subsequent step. This formulation allows FPowNP to capture complex structures effectively while enabling parallel sampling of all target points.
FlowNP offers several advantages over existing approaches: (1) it is simple to implement, train and use for sampling and inference, (2) it is capable of generating samples and computing likelihoods of any conditional distribution, (3) in contrast to autoregressive models, it generates all target points globally coherent and consistent representation of uncertainty, and (4) inference time is controlled by the number of ODE steps rather than the number target points. Extensive experimentation standard NP benchmarks demonstrate that FlowNP consistently outperforms prior NP models, achieving state-of-the-ar results across multiple datasets.
in parallel, providing
Our implementation is available at https: /ttp://ggttttt.tt www/danrsmmfllollll
2 Background
2.1 Neural Processes
Neural processes (NP) [13, 14), provide a framework for training generative models of functions, by modeling stochastic process using dataset of functions. Given such dataset, model is trained to predict an arbitrary target set of points along function, based on a context set of observed points
along the same function. More formally, if the functions are defined F [ X + }}, models are trained to predict a conditional distribution over the target set:
Po (1""" {xxxx
()*x))
(1)
where retx E M and bt E N the positions of the M context points and N target points
respectively, and y E Y and gtt E yw are the function evaluations of these points yi = (z,). X' and y can be of arbitrary dimensions. The context is placed in '()' brackets for better readability.
This approach of modeling stochastic processes follows from Kolmogorov's extension theorem [42] stating that stochastic processes can be defined via a collection of joint distributions over finite sets, Pæi... (,)..), it meets two conditions: exchangeability and consistency.
This condition requires that all joint distributions over the sets are equivariant with respect to permutations:
Exchangeability.
Pri. (Dim) Px(Z ( (H)L:)) P=*.).......)) (1x(1)
(2)

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for any given permutation æ. Models that directly predict conditional distributions such as in Eq. 1 should also be invariant to permutations in the context set.
Consistency. This condition requires that joint distributions are consistent with marginalizations:
Prism (1)mm)
Pai.. )dnm)dym+
AA S m n.
For models that directly predict conditional distributions as in Eq. 1, this translates to the following formulation of marginal consistency:
Po (yim Fimm f .y ] (yinn [risn fx ,y dym++oms 1A S m S n,
(4)
and additionally, models should hold conditional consistency, i.e. the chain rule:
(ys++ss [2,1-5², ....s...3) ( = pe (y., |Z {I,a U,a } pe (y.> (2,e (
(5)
for any two sets of indexes s and s². We note that while most NP models follow the exchangeability condition, they do not fully guarantee that the consistency conditions hold (see App. A).
2.2 Flow Matching
Flow matching [25, 27, 1, 2] a generative modeling paradigm that has recently achieved significant traction. It is closely related to diffusion modeling and its appeal stems from being both conceptually very simple and versatile.
The fundamental object in flow matching is a probability path (x) defined over a continuous time parameter i [0, 1]. This path defines smooth transition between the two distributions Po(x) and Pi (x).. Most commonly, (a)) corresponds to simple tractable distribution such as a Gaussian, and target distribution which is defined only through samples from the training data such as images.
Pi (x) corresponds
In its simplest formulation, which also adopt here, training is performed by generating samples of a conditional probability path pt (*, (z1) computed as an interpolation (x) + (1 - t))a between a
data sample 2y and noise sample Zo- Given this sample, model is trained to predict conditional
velocity un (x) = æ1 2o, using a squared loss:
[()) = Re,,, [[, - ( (rre(z1)) |x1)]]
can be shown that the expectation over 21 results the model approximating the unconditional velocity ur (at) = d' 25 defining an ordinary differential equation (ODE) which can be used for both sampling and likelihood estimation.
For sampling, we first generate a sample from the noise po (zo) and then use an ODE solver to transform it into sample from the data distribution P1 (æ1). To compute the likelihood of given sample F1, we employ the change of variable formula. This involves transforming 21 back a corresponding noise sample from the tractable distribution Po (zo), and then calculating the likelihood of Ze while correcting for the flow's volume change with an estimate of the Jacobian trace.
2.3 Related Work
Neural processes. A prominent approach to form distributions over continuous functions is by using Gaussian processes [36]. While they can be powerful, leveraging their capacity in modeling arbitrary functions is limited due to the complexity of training and evaluating them. To this end, neural processes were introduced, allowing the prior over functions be learned from data straightforward way. The first model in the NP class is the Conditional Neural Process (CNP) [13] which is deterministic and predicts an independent Gaussian distribution for each target point given the context. Later, the Neural Process (NP) [14] introduced a latent variable to allow capturing global uncertainty over the functions. Empirical evaluations of the approach was conducted by Le al. [22]. Following, different extensions to the model were proposed. These include using attention mechanisms (ANP) [21], translation invariance through convolutions [15], bootstrapping [23], and more [8, 18, 3, 4].
3

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Among the extensions of NP, notable example is the Transformer Neural Process (TNP) [32] which treats the prediction of conditional distributions as sequence modeling. Of the three proposed variants in the TNP paper, the autoregressive model (TNP-A), implemented with a causal mask, achieves the highest scores on most common benchmarks, and here we refer to it simply as TNP. Autoregressive-CNN [7] shows that models can be effectively deployed as autoregressive NPs even if they were not trained autoregressively. However, the autoregressive approach has limitations that we address here, namely, the potential complexity of generating samples point-by-poine and the failure to capture complex distributions in the first points in the autoregressive order. An interesting model that combines autoregressive modeling with diffusion [6] treats the full sampling process sequence. Our model is implemented with a transformer, like TNP, however, rather than predicting the next single target point in an autoregressive fashion, flow matching velocity for all target points at once. We show that our model outperforms the TNP.
the transformer a predictor of the
Diffusion continuous spaces Following the success of diffusion models for data in discrete and finite spaces such as image grids [37, 38, 17, 39] different works have investigated their extension to infinite and continuous spaces [19, 34, 35, 41, 5, 30, 20]. These approaches use diffusion flow matching to model joint distributions with some spatial structure. Given a joint distribution several approaches be applied predict conditional distributions. Pidstrigachh et al. [35] and Bond-Taylor & Willcocks [5] use guidance term computed based on the conditioning context and Kerrigan et al. [19] and Dutordoir et al. [11] use replacement method [29] generate conditional samples.
Using joint distributions over continuous spaces to predict conditional distributions can be seen implemention of neural processes using the definition of conditional distributions p(zly) y)) p(z, v))(()). The Neural Diffusion Process (NDP) [11] makes this connection explicitly and uses this method to evaluate the model on NP benchmarks. In comparison, formulation, and is trained to capture both joint distributions and arbitrary conditionals directly, thus amortizing the computation of conditioning. We show that our model outperforms the NDP on standard benchmarks, and that amortizing the conditioning enables directly generating conditional samples without needing auxiliary conditioning methods such guidance replacement.
model uses a flow matching
3 The FlowNP Model
Our goal is implement model of conditional distributions defined by any arbitrary target and context sets of points coming from an underlying function as formulated in Eq. 1.
We implement our model using a transformer architecture [40] that predicts the velocities of the target variables at time in the probability path defined by the continuous flow. The model is depicted in Fig. 1. We perform full self-attention between all input tokens, whereby each of the tokens updated using an attention operator on all other tokens. The tokens we feed to the transformer are divided to context tokens and target tokens.
Target tokens:
These tokens represent the intermediate values of the N function points " evaluated at the N target positions ==*², For the evaluation time t in the probability path, each token is formed by concatenating a single target point position together with the time t and the intermediate value of the variable that time.
token tt, embedd[x t, V/. 1
These tokens represent the observed points along the function, on which the distribution is conditioned. They are formed in the same way as the target tokens, except that they always contain the true observed function evaluations yft* y and the time 1, equivalent to the data distribution pi (v).
Context tokens:
token
ctx; = embed[[x 1, yi 1)
(8)
The output tokens corresponding to the input target tokens are each projected to the original dimension of the function values dim())) and used as the velocity vector at time the continuous probability path defined by the model.
Output tokens:

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3.1 Training
We train our model following the standard NP training setup using a conditional flow matching approach. At each training step, a minibatch of functions is extracted from the training set, where for each function two random sets of points are used as the context and target sets. For each step we randomly define different sizes of these two sets. Given the context and target sets, we train our model as a velocity predictor by generating a random sample from an intermediate distribution along the continuous probability path yt ~ (Vt 188 (y) This achieved by interpolating the clean data with random noise:
Vt = tyi t0t + (1 t)yo
rgt
5gt
yo Fet ~ (0,,
(9)
The loss is computed using a squared error between the model's output and the conditional velocity of the target points (v tgt ( y/ 3/0 tgt Since the model has access to the clean values of the context points, it effectively has side information about the velocity of target variables. In summary, the training loss is computed by:
(()) EE|lu² (vt ,t,
t5² ,0 ctx v(tx)= (31
yo )12
(10)
where the expectation is over:
F, fr ggt xxxx yi ~ f.t ~ 21 tgs ~ N
The training process is presented in Alg. 1 in the appendix.
3.2 Evaluation
Once the model is trained, can be used both for generating samples and computing likelihoods over data. In order to use the model to generate samples, random values of the target set at time t = 0 are drawn from a Normal distribution 3o te ~ (O, /) and used as the initial condition when solving the ODE from 0 to 1. This can be done with various ODE solvers where each evaluation of the velocity 4n is performed by calling the model with the additional input of target positions and context positions and values, n4 Rs u (y) in the appendix.
ig: L, x: tgt etx yltx). The sampling process is presented in Alg.
In order to compute likelihoods over data, a similar ODE is solved in the reverse direction. Starting from the target values y) st t 1, samples are transformed back = 0 and their likelihood is evaluated with the standard Normal distribution. To accommodate for the change of volume, the change of variable formula is applied where the Jacobian is estimated across the probability path using the Hutchinson trace estimator.
Running time Our model is based on a transformer architecture, therefore is interesting to analyze its running time compared the TNP. First, the number of tokens used for predicting N target points given a context of M points in our model is N + M compared to the TNP model which uses 2N M in order to comply with causal masking. Second, generating samples with TNP requires N evaluations as it is an autoregressive model. For our model the number of evaluations depends on the ODE solver and is a parameter that can be tuned according the required accuracy, however it is independent of the number of target points N. The main disadvantage of our model compared TNP is in evaluating likelihoods, where TNP requires only one model evaluation, and our model, similar to sampling, requires multiple evaluations as part of the ODE solution. Wall-clock time comparisons for some of the experiments are reported in Fig. 2 and Fig. 3.
3.3 Exchangeabilityy and Consistency
In this section we discuss the properties of our model in relation to the requirements of the Kolmogorov extension theorem as presented in Sec. 2.1, namely the exchangeability and consistency properties.
Within given conditional prediction task p |x=,,, f2stx, ctx)), our model is guaranteed to be invariant with respect to permutations of both the context and the target, and therefore complies with the exchangeability property. This is due to the transformer architecture and the treatment of tokens as sets rather than sequences. Specifically, all tokens undergo the exact same processing and we do not use positional encodings that rely on ordering.
5

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On the other hand, the consistency property is not implied by the architecture of our model and therefore it is not guaranteed. However, even though is not guaranteed by the model itself, the training paradigm of NPs promotes both the marginal and conditional consistency properties. This is because in every training sample, the context and target sets are constructed randomly from a true underlying function, which is itself sample from the stochastic process defined by the training set. Therefore the training objective is a Monte Carlo estimate of the ground truth stochastic process.
We note that also in previous NP models, consistency is not fully guaranteed and holds only in a limited sense. Specifically, while models like CNP, NP and TNP are consistent over marginalization (Eq. 4), they are not consistent over conditioning (Eq. 5). On the other hand models like NDP are consistent over conditioning and not over marginalization. For more discusison on this see App. A.
4 Experiments
Our experiments are conducted on three distinct data domains: synthetic 1D Gaussian processes, image data from EMNIST [10] and CelebA [28], and real-world weather prediction data ERA5 [16].
We implement a single FlowNP model architecture across all experiments in the main paper, adapting only the input dimensions dim(X) and output dimensions dim()/). We use transformer with 6 layers of full self attention, 128 hidden dimensions and 4 attention heads. We use sinusoidal encodings for the input and flow time with 10 frequencies per dimension, except for the ERAS experiments where we use 40 frequencies per dimension. We emphasize that the encoding is a function of x rather than the position in the sequence. For likelihood evaluation we use an ODE solver based on the midpoint method with 100 steps implemented by Lipman et al. [26], and for sampling we use the Euler method with 100 steps. See Alg. and Alg. in the appendix for the training and sampling algorithms respectively. The transformer architecture and implementation of our model are based on Nguyen & Grover [32]. All training. inference and sampling are performed with an NVIDIA RTX4090 GPU. The implementation of all experiments and models is available at https: / //g.tub...... com/ daarsmmfllsse
In the appendix we provide more examples of larger models trained on the CelebA image data and discuss modified sampling method that incorporates small noise within the sampling process.
Baselines We compare to the baselines CNP [13], NP [14), ANP [21] and TNP [32] using the implementation in Nguyen & Grover [32] and Lee et al. [23]. In order make fair comparison of the main conceptual differences between the methods, we base our model on the transformer architecture in TNP, and reimplement NDP [11] using the same architecture. Therefore, both TNP and NDP models share our network architecture and differ only in the training objective and evaluation method. Specifically, TNP uses the same transformer as we do but is trained by autoregressive maximum likelihood using a causal mask as implemented by the authors, and for the NDP baseline we implement the model with the same transformer used in FlowNP and TNP, omitting the bi-directional attention used in the original NDP. Therefore, NDP differs from our model only by training on unconditional joint distributions, using a linear variance- presenrviigeegggggggge. noise schedule and applying a diffusion loss based on predicting the denoised data rather than the flow velocity. In order to generate conditional samples from NDP we use a guidance-barr based method, and we evaluate the NDP likelihood by the probability flow ODE with the same ODE solver used for FlowNP. We note that for benchmarks that also appear on the original NDP paper (namely Fixed-Noisy RBF and Fixed-Noisy Matern) our implementation of NDP results in better log- likeliiod than the original one.
4.1 Synthetic 1D functions
We start with the standard benchmarks of NPs generated from synthetic ID Gaussian processes (GP), We follow the evaluation protocols of both Lee et al. [23] and Bruinsma et al. [8]. which are used respectively in the TNP [32] and NDP [11] papers. In the first [23], we generate data from three different GP kernels: RBF (also called squared exponential SE), Matern- 2 and Periodic. For each kernel, functions are generated using parameters which are randomly chosen for each sampled function separately. Evaluation is done on held-out data using a random number of context and target points sampled uniformly between 3 and 47 where the total number is constrained to be equal or less than 50. In the second protocol [8] we use two GP kernels: RBF and Matern- 2. using a fixed set of parameters, and additional Gaussian observation noise with variance 0.052. Evaluation is done on

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Table I: Comparison of log-likelihood computed on the target for various 1D GP datasets. We provide mean and standard deviation over five runs. FlowNP consistently achieves performance, outperforming matching all other models across all datasets.
MATERN
EXIID-NGIS
BINID-NOISY MATERN. B
CNP
0.31 0.04
0.12 0.02
-0.63 = 0.01
-1.00 + 0.02
-1.09 0.00
NP
0.31
0.13
-0.61 0.01
-1.13
-1.18
ANP
1.10 0.04
0.85 + 0.04
-0.89 + 0.04
-0.87 + 0.03
-0.98 + 0.02
1.29
NDP
1.20 = 0.04
0.94
=
-0.54 0.01
0.71 = 0:01
0.27 0.02
FLOWNP (OURS)
1.69
0.02
1.30
0.02
30.50 = 0.00
0.71 + 0.01
0.30::00.01
RBF
Matern 5/2
Ground
Truth
ANP
se002sec
TNP
0.8sec
NDP
0.5eec
FlowNP
0.2sec
Figure 2: Samples from models trained on an RBF kernel (left) and Matern- kernel (right), ANP captures the uncertainty mostly through its predicted local variance (shaded area) while TNP, NDP and FlowNP can generate coherent samples that cover the global uncertainty. In contrast NDP, FlowNP generates conditional samples directly without needing auxiliary conditioning methods such as guidance. contrast TNP that generates the samples point-by-point FlowNP generates all points in parallel, resulting in faster and smoother sampling.
held-out data with the same parameters, a uniformly random context size between and 10, and fixed number of 50 target points. These datasets denoted by Fixed-Noisy RBF and Fixed-Noisy Matern- 3. Likelihood for the latent variable models NP and ANP are computed with variational importance sampling [9] using 200 samples, and for the NDP and FlowNP using the midpoint ODE solver with 100 steps.
Results are summarized in Table 1, where for each dataset we report the mean and standard deviation of five repeated experiments, each using different random seeds and evaluation sets. Note that performance of baseline models is improved compared to the previously reported results [32, 8] due to our hyperparameter tuning (see App. B). Our model, FlowNP, consistently outperforms all other models while in some cases achieving comparable results the next best models, TNP and NDP.

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Table 2: Target log-likellng lokeliillhoodd mean standard deviation for the EMNIST and CelebA image datasets and the ERA5 weather dataset. FlowNP consistently outperforms all othe models.
MODEL CNP
EMNIST 0-9
EMNIST 10-46
CELEBA
ERA5
1.27 0.04
0.73 0.04
2.10 + 0.01
4,06 + 0.06
NP
1.17 2 0.03
0.92 : 0.02
2.53 0.03
0.03
3.35 : 0.08
ANP
1.35
1.18 1.00
0.01
7.76 0.08
TNP
2.08 1: 0.02
1.80 1: 0.05
3.95 0:01
11.32 1: 0.09
NDP
1.58 0.02
1.47 0.01
4.28
0.04
6.76+ 0.07
FLOWNP (OURS)
2:50 + 0.01
2.42+:0.01
6.37
0:01
12.79 0:00
GT Context
TNP 72.6sec
NDP 40.4
FlowNP 4.6sec
9
9 9 9 9
9 9
9 9
9
9 9
9
7
7
9
7
7
7
7
7
7
9
7
7
7
3
3 0 8
8
8
3
3
3
8
3
3
5
9
0
555
5
5
5
8
5
55
2
2
2 2
2
7
2
2
2
2
Figure 3: Conditional EMNIST samples generated by TNP, NDP and FlowNP that were trained only subsets of pixels. For each model we show 4 samples. FlowNP generates sharp and diverse samples faster than other models.
Qualitative results are provided in Fig. 2, showing samples from models trained on the RBF and Matern kernels with fixed parameters. We compare ANP: a latent variable model trained with variational inference; TNP: an autoregressive model; NDP: diffusion model of joint distributions; and our proposed model, FlowNP. ANP largely fails to capture the global uncertainty with the latent variable, and tends to attribute large variances to local uncertainty, depicted as the shaded area corresponding to the predicted variance of each point. In contrast, TNP, NDP and FlowNP generate plausible samples that cover the global uncertainty. However while TNP generates samples in an autoregressive manner sometimes resulting in jumpy behavior, NDP and FlowNP generate all points in parallel mostly resulting in smoother samples. An advantage of FlowNP over NDP is that the conditional samples are generated directly from the model outpus rather than relying on guidance method as used for NDP. For each method we note the wall-clock time for generating single samples, showing that FlowNP runs faster than TNP and NDP. For visualiztion of the sampling process see Fig. 6 in the appendix.
4.2 Images
We follow with experiments two image datasets, EMNIST [10] and CelebA [28], where the input X' is 2-dimensional and represents the spatial position in the image, and the output y is either a ID grayscale value 3D RGB value of the pixels. The NP models are trained image pixels. For EMNIST, which contains images of characters, we train on the 10 first classes (digits 0-9) and evaluate both the in-distribution performance, and the out- out-of-distribution performance on the rest of the characters. Since the image data is generated from discrete representation of the pixels, the likelihood of continuous models can grow arbitrarily (see e.g. App. B. in [24]). Therefore we add small noise with variance 0.012 2 to the data.
random subsets of
Results are provided in Table 2. FlowNP outperforms all other models both for the in-distribution evaluation of EMNIST and CelebA, and for the ouitt--distributtin evaluation of EMNIST classes 10-46. We show samples generated by TNP, NDP and FlowNP of EMNIST images, using 70 observed pixels as context. Samples from FlowNP are sharp and diverse, and are generated by seven times less network evaluation and using less tokens compared to TNP. FlowNP samples also do not require computing guidance signals for conditioning like NDP. This leads to faster sampling.
In the appendix we show samples of CelebA images (Fig. 7) using larger FlowNP model based on DĪT-B [33] and compare to samples that are generated by adding small noise during the sampling process (App. C).
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Temperature
Prodicted Temperature Error
15.0
12.
5.0
25
6.0
-2.5
Carrape:: Ground truth Predictions
Contrst
-5.5
Predicted Temperature
Predicted Temperature Error
Wind
12.5
10.0
74
25
00
-25
Figure 4: Visualization temperature and wind prediction using conditional samples generated by FlowNP on two held-out data points from ERA5. The model generates coherent samples based on context points. As more context points are given (bottom row) predictions become more accurate.
4.3 Weather data
To assess the performance of the FlowNP model in real-world dynamic and multi-dimensional tasks, we use meteorological data from the ERA5 global dataset [16]. We follow the setup in Holderrieth al. [18] and extract data from a circular region with a 520km radius centered around Memphis, USA. Each sample represents weather snapshot at a single point in time during the winter months in the years between 1980 to 2018 and consists of 1245 grid points multiple metcorological variables: temperature, pressure, and two wind components (eastward and northward). We divide the years to 34K training samples and 17.5K evaluation samples. Training is performed using the NP setup where X' represents the 2D longitude and latitude, y represents the 4D meteorological variables and random subsets out of the 1245 points are used as the context and target sets.
The performance of FlowNP and the baselines are compared in Table 2. where the target set log- likelihood reported for each model. FlowNP outperforms all other models.
For a qualitative evaluation, Figure provides conditional samples of temperature and wind directions generated by FlowNP the ERA5 weather data. The figure demonstrates how prediction quality scales with the size of the context set: the top row, using fewer context points, shows predictions with greater degree of error compared to the bottom row, which is conditioned larger context set.
5 Analysis and Discussion
We provide analysis of FlowNP compared to the autoregressive approach of TNP and on different differentiating aspects compared to NDP. For more analysis on running times, different ODE solvers and the number of ODE steps for log- liglliiel computation see App. D.
FlowNP vs. TNP We consider the ID step function, where the output changes from y 0 to = 1 at a random point of æ. This function was used in Neal [31] and Dutordoir et al. [11] as an example that cannot be captured by Gaussian processes. Here we show that an autoregressive transformer- based model like TNP also cannot capture this function accurately. In Fig. we show samples from TNP and FlowNP. While TNP samples are noisy in the transition area, FlowNP samples are sharp. The reason is that even though TNP can model complex distributions via the autoregression, is constraind
Gaussians at every step. This is evident when looking at the marginal distribution at = 0, shown the right of Fig. 5 for both models. Comparing the marginal distribution of single point with samples from the entire function indicates that, even without a formal guarantee, FlowNP's training approximately preserves the consistency property (Eq. 4).

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Table Target log- likelihood for ablations on three aspects differentiating FlowNP from NDP.
Model
network output
noise schedule
cooditioning
RBF
EMNIST
clean: y1
linear-yp: de 9t vI
unconditional
1.20:0.44
L5820.02
diffusion:elean
cleam: y1
ope oz = 8t = vI
conditional
1.38
1.640001
diflusion:noise
noise:
lineur up: = 8, v1 82
conditiona
1.33+:004
1 1.56=0001
flow:linvvp
velocity: Mr yo
linear-yp: dr 1, vI
conditional
0.41:
0.48
Bow-poly2
velocity: yn Mo
polynomial 2: a, = f,, A, = t)
conditional
1.88
1.60
flowoosiie
velocity: Mo
cosine: sinl f ( (T))
conditional
1.22::0.22
I. 1.80
Bow:joint
velocity: yı yo
cond-ol: ae 8,
unconditional
L.730004
2.54:0005
FlowNP
velocity: yn yo cond ot: c2 = 8, =
conditional
1.6900002
2.50
Samples
p(y 2 0, 0,
FlowNP
TNP
FlowNP
Figure 5: FlowNP vs. TNP for a random step function. FlowNP samples capture the sharp transition occurting in random positions, while TNP cannot model this function as each step in the autoregressive prediction is Gaussian. On the right: the marginal distribution of y predicted for single point where = 0 further demonstrates this and highlights FlowNP's capacity to capture multimodal distributions.
We conduct an ablation study, summarized in Table 3, to investigate three key aspects differentiating FlowNP from the NDP baseline:
PI vs. NDP
1. The output of the network at each step (clean target data y1. noise yo, or flow velocity y1 yo)
2. The noise schedule used during training y, = a;y1 + 3:300
3. Amortizing context conditioning vs. modeling joint distributions only and computing condi- tional likelihoods through the joint/context ratio:
p (y 1t| fæ=tx, (xx)) = Pe ctx | /po (1ftx|/2
Our results show that using the flow velocity as the network output and conditional optimal-ttanspor noise schedule (cond-ot) yields superior performance across the models. We note that training FlowNP only joint distributions (flow:joint) and using it compute conditional likelihood through the joint/context ratio, achieves a slightly higher likelihood than a FlowNP model that is trained conditioned on contexts directly. This is perhaps expected a model trained unconditionally dedicates its entire capacity to modeling the joint data distribution, a less demanding task than amortizing the conditioning on all possible context sets. However, this slightly higher likelihood comes at cost of requiring auxiliary methods to generate conditional samples. In contrast, conditional training of FlowNP provides slightly lower maximum likeeliddd but is inherently capable of direct, auxiliary-free free conditional sampling.
Conclusion We presented FlowNP, a new model which implements neural processes using flow matching. This model is simple, efficient and outperforms previous neural processes models. We showed results on standard benchmarks such as ID GP data and 2D images as well real-world weather prediction data. Compared to NDP, we find that using a flow matching objective is favorable over diffusion and that modeling conditionals enables direct computation both for conditional likelihood and conditional sampling. Compared to TNP, FlowNP can efficiently capture non-Gaussian and multimodal distributions and generate samples for a set of points in parallel.
Limitations
The main limitation of FlowNP is the iterative sampling and likelihood computation. One possible approach to improve this could be based on methods like shortcut models [12] which explicitly train the model to require less ODE solver steps.
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A Consistency of NP Models
We discuss the consisteney property of different models more detail. The original CNP and NP models use architectures that ensure that given context, all target points are predicted independently. This implies that consistency over marginalization (Eq. 4) holds. However, the model structure does not guarantee consistency over conditioning (Eq. 5) because computing the model conditioned on different contexts cannot be guaranteed to lead to consistent results that comply with an underlying joint distribution.
TNP, which is based transformer performs best when is used autoregressive model. In that case it does not comply neither with exchangeability nor consistency, since even when using clever masking make the model invariant to the context order, the joint distribution of the target still depends on the ordering of the autoregression in the target points. While the authors of TNP propose different variants of the models that make it exchangeable and consistent with marginalization, they lead to a significant drop in performance, and are still not guaranteed to be consistent in terms of conditioning.
The reason that NDP is consistent over conditioning is that it only models joint distributions and therefore computes conditionals using the conditional definition:
(y
15t
yx*z)) pa (xxxxxx /pe (y (xx)
However the full self-attention architecture leads to predictions of target points that cannot be separated into independent factors and therefore cannot ensure consistency marginalizations.
B Training and Sampling
The algorithms for training and sampling are provided in Alg. and Alg. 2.
Fig. 6 provides demonstration of the sampling process for ID GP data by visualizing the transfor- mation from random noise to samples from the conditional distribution of Myy**11=** {2
yf**)).
The hyperparameters of the baselines are tuned to improve their performance. Namely, we find that for CNP and TNP no bound on the output variance is required, while for the models trained by variational inference, NP and ANP, we set the output variance bound to 0.05² and use 8 importance samples for training. The implementation in https: ////tttttt////// com cww/danreeeeeoooooocccoocwcoa wwwrsmmmflllloooo floWNP contains the configurations of all the models.
Algorithm Training
input: dataset of functions
repeat
ft F
Sample batch of functions, here shown for Random context and target sizes Sample positions
M.N U watx XM 228 2t0t X
yerx +- fi(x 28t (x
~ 1]
Sample flow time
My
ggt ~N(0,)) 1)
Sample noise
y, ge
+ t + (1 1--t)) t)b tgt
ctx (embed ([roxx 1, y ))x)))
M tokens
tokens tgt (embed ([rtet, de 1
N tokens
d ([tokens tokens'8t))
(()) | tg
y0 )112
0 - update 1 ((O)
until convergence
output:
trained parameters
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Algorithm 2 Sampling
input: context yotx so and target positions g
tokens = {embeed (1x=xx, 1, y y=*)))
M tokens
y'8e ~ NO /)
Random initialization
for 0 in 1/n s/n_steps steps increments do
tokens 'gt (embed ((x'8) y à - u ([tkens t))) y²8 + y Öû,
N tokens
end for output: predicted sample y*
1=0 0
f == 0.25
== 0.5
1 0.75
t= 0.9
0.95
0.98
t 0.99
t=1
Figure 6: A visualization of the sampling process, showing the intermediate values y Fgt for different time steps. Each column is conditioned on different context and each color represents different random sample generated by FlowNP.
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Ground-truth
Observed context
Deterministic samples
Stochastic
samples
Figure 7: Generating conditional samples of CelebA data using the FlowNP model trained only on
subsets of pixels.
C CelebA Images
We train larger FlowNP model using the DÝT-B architecture [33] CelebA images with resolution of 64 x 64. The model consists of 12 transformer layers with 768 hidden dimensions and 12 attention heads. We train this model using the same approach in the main paper, by sampling random subsets of context pixels and target pixels every time an image is drawn. We the use this model to generate samples conditioned on context 5% observed pixels as shown in Fig.
In some cases, we find that adding small levels of noise and scaling the predicted velocity during sampling process results in samples that look more coherent and less noisy. We implement this by replacing the update of st in each step of Alg. 2 to the following:
y gg + y ge + daçū, + δō,v
(1))
Where 10, is random noise, a are values greater than I and oe are small values. In our implementation we use + t(1 - t) and o; = t)?
A comparison between generated samples using Alg. 2 (Deterministic samples) and samples generated with the noise modification (Stochastic samples) provided in Fig. 7. While the results are very similar, it can be seen that the deterministic samples sometimes appear noisier.
This approach did not make any difference for the GP data which suggests that effective when the capacity of the model is limited relative to the complexity of the distribution.
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Tig-Likelhoodd
ODE-solver
log-likelihood
Euler
1.7941
Midpoint
1.6890
RK4 Dopri5
1.6856
1.6857
Sampling steps
Table 4: Comparing different
Figure Log-likelihood as a function of number of ODE steps, computed using the midpoint ODE solver on the RBF GP data.
ODE solvers on the RBF GP data.
D Analysis
ODE solver We compute the likelihood using several different ODE solvers for the FlowNP model on the RBF experiment. The results, presented in Table 4, show that except for the Euler method which overestimates the likelihood, all other solvers compute very similar values.
Number of ODE steps We run an ablation on sampling and log- likelihhoodd evaluation for different numbers of steps in the ODE solver using the midpoint method. For GP-RBF data, the mean of random seeds with different number of steps is presented Fig. The std of for all 0.014. This shows that the evaluation plateaus after T=60 steps. We observe similar behavior for sample quality.
Running time Comparing the sampling time between FlowNP, NDP and TNP using an NVIDIA RTX4090 GPU and 100 sampling steps for FlowNP: the time to generate sample in the GP experiment with 200 target points is 0.2sec for FlowNP, 0.5sec for NDP and 0.8sec for TNP. The time to generate EMNIST sample with 748 target points is 4,6sec for FlowNP, 10.4sec for NDP and 72.6sec for TNP.
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Reviews:
[{"review_id": "ukM5GW51vW", "reviewer": "Reviewer_MZUP", "summary": "This paper proposes a probabilistic model called flow matching neural process that trains a neural process with the flow matching objective.\nThe main idea is to train a conditional flow matching model that maps from a Gaussian to the desired distribution at the target location.\nThey use a transformer architecture to parameterize the vector field so that the flow model is able to condition on arbitrary number of context tokens.", "questions": "1. As mentioned by the author, diffusion neural processes have already been published before.\nGiven the connection between diffusion models and flow matching models (in fact some would argue they are equivalent), I wonder if the method proposed in this paper apply to diffusion neural processes.\nAlso, the authors perhaps should elaborate a bit more on the key difference between this paper and diffusion neural processes.", "limitations": "NA", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "1. The idea seems to be very clean---combining transformers and flow matching objective for neural processes.\n1. The empirical performance seems quite strong.\nThe proposed flow matching neural process outperforms all baselines consistently.", "was_revised": true, "final_justification": "I've read through the rebuttal and other reviews. I maintain my current score.", "rebuttal": "We thank the reviewers for their thorough review and constructive feedback, and we are happy that most reviewers recommend accepting our work.  \n\nWe address all the issues raised by the reviewers and as requested, include more detailed explanations and additional results that answer reviewers’ questions and further support our claims (see summary at the end of this response). We hope that the reviewers will acknowledge this.  \n\n## FlowNP vs. Neural Diffusion Processes\n\nIn the paper we describe the differences between FlowNP (our method) and Neural Diffusion Processes (NDP) in several places (e.g  lines 134-139 and in the appendix lines 465-471). In the revised version of the paper we will emphasize this in a clearer way. \nThe differences can be summarized in the following three points:\n1. FlowNP uses a linear conditional optimal transport noise schedule $x_t = t x_1 + (1-t) x_0$ and a model that predicts velocity, while NDP uses a variance preserving schedule and a model that predicts the noise. The ablations in table 4 directly compare this difference and we also add more noise schedules as requested by reviewer KAXg (see additional results #1 below).\n2. FlowNP models joint and conditional distributions of points while NDP only models the joint distribution. The approach in NDP is to model the joint distributions of arbitrary number of points in the signal which allows computing also conditional distributions using Bayes' law. However, in order to sample from a conditional distribution, this requires adding auxiliary methods such as posterior sampling guidance. In our method, computing the conditional distribution is amortized during training and therefore the model can be directly used for probabilistic inference or sampling from any arbitrary conditional distribution.\n3. While both FlowNP and NDP use transformers, the architectures are different. We explicitly leave this difference out of the paper's scope and in order to make a fair comparison we use the exact same architecture in both models (and in the TNP model) for all experiments. \n \n# Summary of additional results (repeated in all responses):\nAs requested, we have produced additional results to answer the reviewers’ questions and further support our claims. Some results include images that demonstrate qualitative aspects of samples, and will be added to the revised paper.\n\n\nWe summarize the additional results:\n \n### 1. Training with different flow schedules and computing likelihood with different ODE solvers.\n\nWe have trained our model using various schedules for the RBF GP data. Each model is then evaluated on held out data by computing its log-likelihood. Our results are as follows (using the general schedule of $x_t = \\alpha_t x_1 +  \\beta_t x_0$ with different choices of $\\alpha_t$ and $\\beta_t$):\n\n| Name        | Schedule                                     | Log-Likelihood |\n|:------------|:---------------------------------------------|---------------:|\n| CondOT      | $\\alpha_t = t, ~~ \\beta_t = 1- t$            |         1.6890 |\n| Polynomial2 | $\\alpha_t = t^2, ~~ \\beta_t = (1- t)^2$     |         1.0955 |\n| Cosine      | $\\alpha_t  = \\sin(0.5 t \\pi), ~~ \\beta_t = \\cos(0.5 t \\pi)$            |         1.2441 |\n| LinearVP    | $\\alpha_t = t, ~~ \\beta_t = \\sqrt{1- t^2}$            |         0.4112 |\n\n\nThe results show schedules other than the standard CondOT perform worse in this setting.\n\nWe also experiment with different ODE solvers for computing the log likelihood. For this we use our model trained with CondOT on RBF. We use a few different solvers using the implementation in the flow_matching package (based on torchdiffeq). The resulting  log likelihoods are:\n\n| ODE Solver | Log-Likelihood |\n|:-----------|---------------:|\n| Euler      |         1.7941 |\n| Midpoint   |         1.6890 |\n| RK4        |         1.6856 |\n| Dopri5     |         1.6857 |\n\nThe results show that except for the simple Euler method, other methods result in very similar values.\nWe will add both results above together with the same results on EMNIST to table 4 in the appendix to complete the ablation results.\n\t\n### 2. Higher resolution CelebA samples  \n\nWe have trained our model on higher resolution CelebA images $64 \\times 64$ and generated high quality conditional and unconditional samples. We will add these results to figure 6 in the appendix  (in the paper we showed results only for $32 \\times 32$ resolution, following the experimental protocol of prior models).\n\n\n### 3. Deterministic vs. stochastic sample generation\n\nWe have added a qualitative comparison of deterministic sample generation (using an ODE solver), vs. a method that adds small amount of noise in the sampling process. As we mention in the paper, for the GP and ERA5 datasets we did not see any difference between the methods, however for image datasets (especially CelebA) the samples are a bit noisier in the deterministic case. This effect is even less visible in the larger model trained for resolution $64 \\times 64$ (see additional results #2) where the deterministic and stochastic samples are very similar. \n   \n### 4. Error bars for ERA5\n\nWe have repeated the model training for ERA5 data with 5 different seeds, and obtained a mean and std of 11.6647 +- 0.2452."}, {"review_id": "RuV0b2uRZT", "reviewer": "Reviewer_qfUz", "summary": "- The authors use Flow Matching to improve sample generation from Neural Processes.\n- The authors use the known ability of transformers to aggregate context data points very effectively. \n- The authors use Flow matching as a method to generate coherent samples with more expressive densities than Transformer Neural Processes and better guidance than Neural Diffusion Processes. \n- The authors benchmark their method against existing approaches on both synthetic and real world data sets,  where  their method outperforms alternatives.", "questions": "- How did your results look without the added noise in the sample generation? \n- Why are ERA5 results not reported with standard deviation?", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n- Well written paper with a good background for the field.\n- Good benchmarking, building from simpler to more complex examples.\n- NPs are an exciting field which show great promise in meta learning priors and performing inference and the proposed approach significantly improves the ability of the proposed method to make coherent predictions at large numbers of points as well as providing accurate predictions.\n- While proposed method did not seem to require significant theoretical advances on previous methods, the two selected techniques compliment one another very well and solve a significant issue with Neural Processes.\n\nWeaknesses \n- The authors refer to their method as using an ODE solver for sampling, however they add noise in their sampling method, so their method more closely resembles sampling using Langevin dynamics. This addition to flow matching raises two concerns for me: \n\t- Empirically, I would have liked to see an ablation study comparing performance without this additional element.\n\t- Theoretically, this significantly changes how I would think about the proposed method, to something closer to an energy based model sampled with Langevin dynamics. I believe that this detail merited significantly more discussion in the main body of the paper.", "was_revised": true, "final_justification": "The authors present a technically solid paper which addresses a signigificant shortcoming of prior work in neural processes. The research is well presented, and with the additional discussion and studies which they have indicated they will include for the camera ready version I am happy to recommend acceptance.", "rebuttal": "We thank the reviewers for their thorough review and constructive feedback, and we are happy that most reviewers recommend accepting our work.  \n\nWe address all the issues raised by the reviewers and as requested, include more detailed explanations and additional results that answer reviewers’ questions and further support our claims (see summary at the end of this response). We hope that the reviewers will acknowledge this.  \n\n## Deterministic vs. stochastic sample generation\n\nWe agree that adding noise in the sampling process resembles Langevin dynamics. Though this can also be thought as a way to regularize the ODE solver, conceptually it is also equivalent to solving an SDE. We will add this discussion in the paper.\n\nEmpirically, we add results that compare between stochastic and deterministic sampling as requested, both for 1D GP functions and for 2D images. The results for 1D are indistinguishable (as also noted in the paper). For images (specially for CelebA) we observe that the deterministic samples are a bit noisier while the stochastic samples are smoother.  When repeating this for the larger models trained for higher resolution CelebA (as requested by reviewer ZPkJ) this effect becomes even smaller and the deterministic and stochastic samples become very similar. \nThis suggests that when the model is better, there is less need for stochasticity in sampling.\n\nWe believe that adding this discussion and empirical results to the paper will improve its contribution and we thank the reviewer for pointing this out.  \n\n## ERA5 error bars\n\nWe have now run trained our model for ERA5 on 5 different random seeds and obtain a mean log likelihood of 11.6647 with a standard deviation of 0.2452.\n\n  \n# Summary of additional results (repeated in all responses):\nAs requested, we have produced additional results to answer the reviewers’ questions and further support our claims. Some results include images that demonstrate qualitative aspects of samples, and will be added to the revised paper.\n\n\nWe summarize the additional results:\n \n### 1. Training with different flow schedules and computing likelihood with different ODE solvers.\n\nWe have trained our model using various schedules for the RBF GP data. Each model is then evaluated on held out data by computing its log-likelihood. Our results are as follows (using the general schedule of $x_t = \\alpha_t x_1 +  \\beta_t x_0$ with different choices of $\\alpha_t$ and $\\beta_t$):\n\n| Name        | Schedule                                     | Log-Likelihood |\n|:------------|:---------------------------------------------|---------------:|\n| CondOT      | $\\alpha_t = t, ~~ \\beta_t = 1- t$            |         1.6890 |\n| Polynomial2 | $\\alpha_t = t^2, ~~ \\beta_t = (1- t)^2$     |         1.0955 |\n| Cosine      | $\\alpha_t  = \\sin(0.5 t \\pi), ~~ \\beta_t = \\cos(0.5 t \\pi)$            |         1.2441 |\n| LinearVP    | $\\alpha_t = t, ~~ \\beta_t = \\sqrt{1- t^2}$            |         0.4112 |\n\n\nThe results show schedules other than the standard CondOT perform worse in this setting.\n\nWe also experiment with different ODE solvers for computing the log likelihood. For this we use our model trained with CondOT on RBF. We use a few different solvers using the implementation in the flow_matching package (based on torchdiffeq). The resulting  log likelihoods are:\n\n| ODE Solver | Log-Likelihood |\n|:-----------|---------------:|\n| Euler      |         1.7941 |\n| Midpoint   |         1.6890 |\n| RK4        |         1.6856 |\n| Dopri5     |         1.6857 |\n\nThe results show that except for the simple Euler method, other methods result in very similar values.\nWe will add both results above together with the same results on EMNIST to table 4 in the appendix to complete the ablation results.\n\t\n### 2. Higher resolution CelebA samples  \n\nWe have trained our model on higher resolution CelebA images $64 \\times 64$ and generated high quality conditional and unconditional samples. We will add these results to figure 6 in the appendix  (in the paper we showed results only for $32 \\times 32$ resolution, following the experimental protocol of prior models).\n\n\n### 3. Deterministic vs. stochastic sample generation\n\nWe have added a qualitative comparison of deterministic sample generation (using an ODE solver), vs. a method that adds small amount of noise in the sampling process. As we mention in the paper, for the GP and ERA5 datasets we did not see any difference between the methods, however for image datasets (especially CelebA) the samples are a bit noisier in the deterministic case. This effect is even less visible in the larger model trained for resolution $64 \\times 64$ (see additional results #2) where the deterministic and stochastic samples are very similar. \n   \n### 4. Error bars for ERA5\n\nWe have repeated the model training for ERA5 data with 5 different seeds, and obtained a mean and std of 11.6647 +- 0.2452."}, {"review_id": "5PffNyex8z", "reviewer": "Reviewer_KAXg", "summary": "The paper introduces FlowNP, a novel Neural Process (NP) model that uses flow matching for learning stochastic processes from data. FlowNP is designed to address limitations of existing NP models, such as underfitting, sequential sampling inefficiencies in autoregressive models, and lack of global uncertainty representation. The model uses a transformer architecture to predict velocity vectors for target points, allowing parallel sampling via an ODE solver. Experiments are provided to demonstrate that FlowNP outperforms existing NP models in terms of log-likelihood and sampling efficiency.", "questions": "- Can FlowNP be used to provide quantification of epistemic uncertainty? \n- Is FlowNP robust to noise perturbation in the training datasets?\n- Does FlowNP suffer from spectral bias (over-smoothing high frequencies)?\n- What are the limitations of the proposed method? For what data regimes the method excel and fail?", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The integration of flow matching with NPs is interesting, providing a new way to model conditional distributions without relying on autoregressive or auxiliary conditioning methods \n- The proposed parallel sampling method improves efficiency during generation, as it avoids sequential generation of autoregressive approaches \n\nWeaknesses: \n- Lack of consistency guarantees, since consistency (marginal and conditional) is not formally guaranteed\n- Lack of scalability evaluation, as it is not clear how well the method would scale to settings with very large context/target sets (e.g., data with higher resolutions)\n- Lack of ablation study on the design of the flow matching framework (sensitivity to choice of noise schedule/probability path, choice of ODE sampler, etc.)", "was_revised": true, "final_justification": "My final decision is a borderline accept. Overall the paper is a neat contribution to NeurIPS, providing a principled method to integrate flow matching with NPs. The authors have also sufficiently addressed my concerns and included additional results to help strengthen the paper. Provided that these additional results are included, I am inclined to go with a borderline accept.", "rebuttal": "We thank the reviewers for their thorough review and constructive feedback, and we are happy that most reviewers recommend accepting our work.  \n\nWe address all the issues raised by the reviewers and as requested, include more detailed explanations and additional results that answer reviewers’ questions and further support our claims (see summary at the end of this response). We hope that the reviewers will acknowledge this.  \n\n## Choice of scheduling and solver \nWe thank the reviewer for mentioning this. The simplicity of switching between different noise schedules and ODE solver is one of the benefits of the flow matching framework. We agree that demonstrating results with more choices will improve the paper. We therefore added more experiments by training models with various noise schedules, and evaluating the model likelihood with different ODE solvers. See additional results #1. The results will be added to table 4 in the appendix to complete the ablation analysis. In general the results show that the standard conditional optimal transport schedule works best in our 1D and 2D settings (although for more specialized settings other schedules might work better). Regarding the ODE solvers for likelihood computation, we find that all tested solvers behave very similarly, except for the Euler solver. \nFor sampling, we qualitatively compared samples generated with the Euler and midpoint solvers and could not find visible differences between them.\n\n## Lack of consistency guarantees \n\nConsistency guarantees is one of the big open challenges in NP models. While some simple models can achieve some level of guarantees, developing a high capacity model (e.g. without independent assumptions like CNP) is an active area of research. We hope our paper can serve as a good starting point for this.\n\n## Lack of scalability evaluation\n\nWhile we provide running time analysis that shows FlowNP is favorable compared to the baselines, we agree that our experiments are relatively small scale. Scaling up NPs is an important next frontier in NP research, and the results we add for higher resolution of CelebA as requested by reviewer ZPkJ (see additional results #2) is perhaps a first small step in this direction.  We believe that weather data provides an excellent opportunity to develop and evaluate large scale modeling.\n\n## Answer to questions\n\n**Epistemic uncertainty in NP:**\nWe are not aware of methods to compute epistemic uncertainty in NP models. This could be an interesting future research direction.\n\n**Robustness to noise:**\nWhile some of our datasets add observational noise we do not explicitly test the effect it has. This could also be an interesting future research direction.\n\n**Spectral bias:**\nWe do not test this quantitatively but we do observe that samples (specially in images) are generally smoother than the GT signal. We believe this is usually the case for models trained with mean-seeking objectives such as maximum likelihood and flow matching.\n  \n**Regimes where FlowNP excels / fails:**\nCompared to TNP, FlowNP would work better when the data is highly non Gaussian and there is no natural order of points. Otherwise, autoregressive models that use Gaussian distributions would be better.  Compared to NDP - while FlowNP  outperforms NDP in the settings that we test, we are not sure how both models would behave in different settings. Related also to the above discussion about different noise schedules, it would be interesting to find specific aspects in the signals that make different schedules work better than others.\n\n\n# Summary of additional results (repeated in all responses):\nAs requested, we have produced additional results to answer the reviewers’ questions and further support our claims. Some results include images that demonstrate qualitative aspects of samples, and will be added to the revised paper.\n\n\nWe summarize the additional results:\n \n### 1. Training with different flow schedules and computing likelihood with different ODE solvers.\n\nWe have trained our model using various schedules for the RBF GP data. Each model is then evaluated on held out data by computing its log-likelihood. Our results are as follows (using the general schedule of $x_t = \\alpha_t x_1 +  \\beta_t x_0$ with different choices of $\\alpha_t$ and $\\beta_t$):\n\n| Name        | Schedule                                     | Log-Likelihood |\n|:------------|:---------------------------------------------|---------------:|\n| CondOT      | $\\alpha_t = t, ~~ \\beta_t = 1- t$            |         1.6890 |\n| Polynomial2 | $\\alpha_t = t^2, ~~ \\beta_t = (1- t)^2$     |         1.0955 |\n| Cosine      | $\\alpha_t  = \\sin(0.5 t \\pi), ~~ \\beta_t = \\cos(0.5 t \\pi)$            |         1.2441 |\n| LinearVP    | $\\alpha_t = t, ~~ \\beta_t = \\sqrt{1- t^2}$            |         0.4112 |\n\n\nThe results show schedules other than the standard CondOT perform worse in this setting.\n\nWe also experiment with different ODE solvers for computing the log likelihood. For this we use our model trained with CondOT on RBF. We use a few different solvers using the implementation in the flow_matching package (based on torchdiffeq). The resulting  log likelihoods are:\n\n| ODE Solver | Log-Likelihood |\n|:-----------|---------------:|\n| Euler      |         1.7941 |\n| Midpoint   |         1.6890 |\n| RK4        |         1.6856 |\n| Dopri5     |         1.6857 |\n\nThe results show that except for the simple Euler method, other methods result in very similar values.\nWe will add both results above together with the same results on EMNIST to table 4 in the appendix to complete the ablation results.\n\t\n### 2. Higher resolution CelebA samples  \n\nWe have trained our model on higher resolution CelebA images $64 \\times 64$ and generated high quality conditional and unconditional samples. We will add these results to figure 6 in the appendix  (in the paper we showed results only for $32 \\times 32$ resolution, following the experimental protocol of prior models).\n\n\n### 3. Deterministic vs. stochastic sample generation\n\nWe have added a qualitative comparison of deterministic sample generation (using an ODE solver), vs. a method that adds small amount of noise in the sampling process. As we mention in the paper, for the GP and ERA5 datasets we did not see any difference between the methods, however for image datasets (especially CelebA) the samples are a bit noisier in the deterministic case. This effect is even less visible in the larger model trained for resolution $64 \\times 64$ (see additional results #2) where the deterministic and stochastic samples are very similar. \n   \n### 4. Error bars for ERA5\n\nWe have repeated the model training for ERA5 data with 5 different seeds, and obtained a mean and std of 11.6647 +- 0.2452."}, {"review_id": "cRh1MOQlyX", "reviewer": "Reviewer_ZPkJ", "summary": "The authors propose to train a model in the neural process framework with the flow matching algorithm. In their experiments, they outperform other neural process models on in-painting tasks for images and weather data and in terms of test-set likelihood.", "questions": "1. Line 190: Why does TNP require $2N + M$?\n2. Figure 2: Which kernels are in which column? What do true samples look like for reference?\n3. Do I understand Appendix A correctly that FlowNP is conditionally consistent but not marginally consistent?\n4. What is $\\alpha_t$ in Algorithm 2?\n5. How well does your model perform on unconditional image generation? Would it reduce to flow matching?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The paper is easy to follow and the authors evaluate their proposed model on a variety of data types.\n\nWeaknesses:\n1. The abstract claims to outperform state-of-the-art methods on various benchmarks but only includes neural process models.\n2. Line 192: The claim that \"it is independent of the number of target points $N$\" is slightly misleading. While the number of function evaluations is independent of it, the run time of each individual function evaluation does scale with $N$.\n3. Line 273: The authors claim that FlowNP is the fasted method in Figure 2. However, ANP is actually the fastest by a factor of 100.\n4. Table 2 and Line 281: You report continuous log-likelihoods, which is why they can grow without bounds. However, the standard for images would be to report negative log-likelihoods of the discretized data in bits per dimension, see for example Appendix B in [1] for an explanation. Then, your numbers could also be compared to other models directly, e.g. diffusion models or neural ODEs.\n5. Figure 4 does not include true data for reference.\n6. Both data and samples of CelebA in Figure 7 are much blurrier than they should be. Did you downsample the data? If so, did you make sure to plot the images at the same resolution that they actually are? If I zoom in, it looks like a blurry high-resolution image.\n\nRemarks:\n1. In Section 2.1, all $y_{i:n}$ should be $y_{1:n}$, I believe.\n2. The caption of Figure 2 has a typo in \"condinting\".\n\n[1] Lienen, Kollovieh and Günnemann. \"Generative Modeling with Bayesian Sample Inference.\", https://arxiv.org/abs/2502.07580", "was_revised": true, "final_justification": "Why not higher: While the method is interesting, I don't classify it as \"high impact\" at this point.\n\nWhy not lower: The paper reads well and has no major flaws.", "rebuttal": "We thank the reviewers for their thorough review and constructive feedback, and we are happy that most reviewers recommend accepting our work.  \n\nWe address all the issues raised by the reviewers and as requested, include more detailed explanations and additional results that answer reviewers’ questions and further support our claims (see summary at the end of this response). We hope that the reviewers will acknowledge this.  \n\n## Comparison to non-NP models (W1)\n**W1:** We only claim our method is advantageous against previous Neural Processes (NP) models. We agree that this can be presented more clearly and will update the abstract and introduction accordingly. As NPs are models of signals over continuous spaces, it is not directly clear how to compare them to models of gridded data such as standard diffusion models for images. We experiment with images (as is standard in NP papers) because they present an interesting example of complex functions over 2D spaces. The fact that the values of these functions (images) are only available in a regular grid is actually misleading from the main motivation of NP models (which is modeling continuous space signals). Note that during training, NP models only access a random number of pixels and never see a full image. We will emphasize this point further in the paper.\n\n## Running time analysis (W2, W3, Q1)\n**W2:** We agree that the number of target points affect the running time of each feed forward through the network (network evaluation). We separated our analysis to (1) the running time of each network evaluation (by comparing the number of tokens) and (2) the required number of network evaluations. When comparing to the TNP baseline which is implemented with the same transformer architecture as ours, we explicitly state the number of tokens used in each feed-forward (lines 189-190). Our model uses N+M tokens (number of target points + number of context points) vs. 2*N+M for the TNP. **Q1:** The reason TNP has an additional token per target is because of the method they apply to incorporate causal masks among the target points - each target point is presented once with the coordinate only, $x$, and once as a pair of coordinates and values $x, y$.  \n\n**W3:** In figure 2 our method is the fastest among the top performing methods, TNP and NDP, which form the main baselines we compare to. We apologize for the mistake (we only included ANP at a later stage) and have fixed line 273 to state “FlowNP is faster than TNP and NDP”.\n\n## Continuous log likelihoods (W4)\n**W4:** The reason we do not measure the log-likelihood in bits per pixels is that NP models (ours and the baselines) are models of signals over continuous spaces (see also answer to W1). This means we effectively model infinite pixels, even when we measure the performance on regular-gridded images.\nWhen the values in the data are obtained from an original discrete representation (like the EMNIST and CELEBA images) we add small amount of noise to prevent arbitrary growing log-likelihood. We will further emphasize this point and include a discussion that compares this to the setup described in [1].\n\n\n[1] Lienen, Kollovieh and Günnemann. \"Generative Modeling with Bayesian Sample Inference.\", https://arxiv.org/abs/2502.07580\n\n\n## CelebA qualitative results (W6, Q5)\n**W6:** All the CelebA experiments including the images presented in the appendix are for a downsampled $32 \\times 32$ resolution, because we follow the experimental setup of the previous NP methods used as baselines. We have now trained a FlowNP model on higher resolution $64 \\times 64$. The generated samples are of high quality and we will include them in the appendix of the revised paper.\n\n**Q5:** Using this model, we have also generated unconditional images (by giving zero context points). The quality and diversity of the samples demonstrate the capacity of the model to capture both conditional and unconditional distributions of complex signals. This model still does not reduce to standard flow matching on images as it models signals on continuous spaces as demonstrated on the 1D GP data in the paper and discussed in W1 in this response.  \n\n\n## Other points:\n\n**W5:** The blue arrows are the ground truth values at each point and are to be compared with the red arrows which are the model’s predictions. \n\n**Q2:** In figure 2 the RBF kernel is on the left and Matern Kernel is on the right. We will add this to the caption and include the GT samples using Gaussian Processes. Thank you for pointing this out.\n\n**Q3:** In appendix A we discuss previous NP models. In section 3.3 we discuss the conditions of Kolmogorov Extension Theorem (defined in Sec. 2.1) for our model. In short - exchangeability is guaranteed and both conditional consistency and marginal consistency are not guarnateed.\n\n**Q4:** $\\alpha_t$ is a scalar close to 1 that adapts the step size in the samplng process. See lines 438-440, and discussion with reviewer qfUz about sampling.  \n\n**Remarks 1 & 2:**  We will fix these typos. Thank you for pointing them out.\n\n \n# Summary of additional results (repeated in all responses):\nAs requested, we have produced additional results to answer the reviewers’ questions and further support our claims. Some results include images that demonstrate qualitative aspects of samples, and will be added to the revised paper.\n\n\nWe summarize the additional results:\n \n### 1. Training with different flow schedules and computing likelihood with different ODE solvers.\n\nWe have trained our model using various schedules for the RBF GP data. Each model is then evaluated on held out data by computing its log-likelihood. Our results are as follows (using the general schedule of $x_t = \\alpha_t x_1 +  \\beta_t x_0$ with different choices of $\\alpha_t$ and $\\beta_t$):\n\n| Name        | Schedule                                     | Log-Likelihood |\n|:------------|:---------------------------------------------|---------------:|\n| CondOT      | $\\alpha_t = t, ~~ \\beta_t = 1- t$            |         1.6890 |\n| Polynomial2 | $\\alpha_t = t^2, ~~ \\beta_t = (1- t)^2$     |         1.0955 |\n| Cosine      | $\\alpha_t  = \\sin(0.5 t \\pi), ~~ \\beta_t = \\cos(0.5 t \\pi)$            |         1.2441 |\n| LinearVP    | $\\alpha_t = t, ~~ \\beta_t = \\sqrt{1- t^2}$            |         0.4112 |\n\n\nThe results show schedules other than the standard CondOT perform worse in this setting.\n\nWe also experiment with different ODE solvers for computing the log likelihood. For this we use our model trained with CondOT on RBF. We use a few different solvers using the implementation in the flow_matching package (based on torchdiffeq). The resulting  log likelihoods are:\n\n| ODE Solver | Log-Likelihood |\n|:-----------|---------------:|\n| Euler      |         1.7941 |\n| Midpoint   |         1.6890 |\n| RK4        |         1.6856 |\n| Dopri5     |         1.6857 |\n\nThe results show that except for the simple Euler method, other methods result in very similar values.\nWe will add both results above together with the same results on EMNIST to table 4 in the appendix to complete the ablation results.\n\t\n### 2. Higher resolution CelebA samples  \n\nWe have trained our model on higher resolution CelebA images $64 \\times 64$ and generated high quality conditional and unconditional samples. We will add these results to figure 6 in the appendix (in the paper we showed results only for  $32 \\times 32$ resolution, following the experimental protocol of prior models).\n\n\n### 3. Deterministic vs. stochastic sample generation\n\nWe have added a qualitative comparison of deterministic sample generation (using an ODE solver), vs. a method that adds small amount of noise in the sampling process. As we mention in the paper, for the GP and ERA5 datasets we did not see any difference between the methods, however for image datasets (especially CelebA) the samples are a bit noisier in the deterministic case. This effect is even less visible in the larger model trained for resolution $64 \\times 64$ (see additional results #2) where the deterministic and stochastic samples are very similar. \n   \n### 4. Error bars for ERA5\n\nWe have repeated the model training for ERA5 data with 5 different seeds, and obtained a mean and std of 11.6647 +- 0.2452."}, {"review_id": "xuemPVmCOS", "reviewer": "Reviewer_KkGa", "summary": "This paper proposes a flow matching neural process (FlowNP), which combines the neural process with flow matching by using flow matching to generate target value given target position and context. The proposed FlowNP is simple to implement and can sample from conditional distributions without auxiliary methods. Moreover, the model provides a controllable tradeoff between accuracy and running time via ODE solver steps. Empirically, they show that FlowNP outperforms SOTA in 1D GP, 2D image and weather data.", "questions": "See my question in weakness: what’s the gain for overlaying FM with NP? Can it significantly beat FM, or give us some insight/ inspiration for new methods?", "limitations": "The paper already discussed their limitations: iterative sampling and likelihood computation.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strength**: the idea is simple and show improvement from previous NP methods. The writing is clear and very easy to follow.\n\n**Weakness**: The proposed method combines FM and NP in a straightforward way, which may raise questions regarding novelty. However, if it works fine, I’m happy to solve problems in a simple way. But I’m not sure if overlaying FM with NP is necessary: what’s the benefit of FlowNP, comparing to using FM directly (FM is flexible enough, and can be used for inference, sampling and conditional sampling)? If the improvement is not very significant but can potentially give insights/ inspirations to new methods, I’m also OK with it.", "was_revised": true, "final_justification": "The author adjusted main issues I proposed.", "rebuttal": "We thank the reviewers for their thorough review and constructive feedback, and we are happy that most reviewers recommend accepting our work.  \n\nWe address all the issues raised by the reviewers and as requested, include more detailed explanations and additional results that answer reviewers’ questions and further support our claims (see summary at the end of this response). We hope that the reviewers will acknowledge this.  \n\n## Flow matching + NP vs. Flow matching\nThe main motivation for using flow matching within the neural processes (NP) framework as we propose is to have a flexible model of signals over continuous spaces. In contrast standard diffusion or flow matching models that are trained on data with specific structure and dimension (e.g. a grid of pixels for images), NP models operate on continuous signals and can be used to generate samples or compute probabilistic inference involving arbitrary points within the signal's space.  In that regard, the experiments over images, although a standard NP benchmark, are a bit misleading since they demonstrate the performance on functions that only have values in regular gird intervals. Note that in contrast to standard diffusion or flow matching models of images, NP models only have access to a random number of pixels per image during training and never see a whole image. We will make this point clearer in the text (see also response to reviewer ZPkJ weakness 1).\n\n We believe that within the framework of NPs, our work of introducing flow matching provides significant contributions compared to previous models including the implementation and conceptual simplicity and the ability to capture more complex distributions without relying on autoregressive modeling, \n \n# Summary of additional results (repeated in all responses):\nAs requested, we have produced additional results to answer the reviewers’ questions and further support our claims. Some results include images that demonstrate qualitative aspects of samples, and will be added to the revised paper.\n\n\nWe summarize the additional results:\n \n### 1. Training with different flow schedules and computing likelihood with different ODE solvers.\n\nWe have trained our model using various schedules for the RBF GP data. Each model is then evaluated on held out data by computing its log-likelihood. Our results are as follows (using the general schedule of $x_t = \\alpha_t x_1 +  \\beta_t x_0$ with different choices of $\\alpha_t$ and $\\beta_t$):\n\n| Name        | Schedule                                     | Log-Likelihood |\n|:------------|:---------------------------------------------|---------------:|\n| CondOT      | $\\alpha_t = t, ~~ \\beta_t = 1- t$            |         1.6890 |\n| Polynomial2 | $\\alpha_t = t^2, ~~ \\beta_t = (1- t)^2$     |         1.0955 |\n| Cosine      | $\\alpha_t  = \\sin(0.5 t \\pi), ~~ \\beta_t = \\cos(0.5 t \\pi)$            |         1.2441 |\n| LinearVP    | $\\alpha_t = t, ~~ \\beta_t = \\sqrt{1- t^2}$            |         0.4112 |\n\n\nThe results show schedules other than the standard CondOT perform worse in this setting.\n\nWe also experiment with different ODE solvers for computing the log likelihood. For this we use our model trained with CondOT on RBF. We use a few different solvers using the implementation in the flow_matching package (based on torchdiffeq). The resulting  log likelihoods are:\n\n| ODE Solver | Log-Likelihood |\n|:-----------|---------------:|\n| Euler      |         1.7941 |\n| Midpoint   |         1.6890 |\n| RK4        |         1.6856 |\n| Dopri5     |         1.6857 |\n\nThe results show that except for the simple Euler method, other methods result in very similar values.\nWe will add both results above together with the same results on EMNIST to table 4 in the appendix to complete the ablation results.\n\t\n### 2. Higher resolution CelebA samples  \n\nWe have trained our model on higher resolution CelebA images $64 \\times 64$ and generated high quality conditional and unconditional samples. We will add these results to figure 6 in the appendix  (in the paper we showed results only for $32 \\times 32$ resolution, following the experimental protocol of prior models).\n\n\n### 3. Deterministic vs. stochastic sample generation\n\nWe have added a qualitative comparison of deterministic sample generation (using an ODE solver), vs. a method that adds small amount of noise in the sampling process. As we mention in the paper, for the GP and ERA5 datasets we did not see any difference between the methods, however for image datasets (especially CelebA) the samples are a bit noisier in the deterministic case. This effect is even less visible in the larger model trained for resolution $64 \\times 64$ (see additional results #2) where the deterministic and stochastic samples are very similar. \n   \n### 4. Error bars for ERA5\n\nWe have repeated the model training for ERA5 data with 5 different seeds, and obtained a mean and std of 11.6647 +- 0.2452."}]

Author rebuttal:

Meta-review: This paper introduces Flow Matching Neural Processes (FlowNP), which integrate flow matching into the Neural Process framework to enable efficient conditional sampling over continuous spaces. The method provides reasonable results across Gaussian processes, image inpainting, and weather data, with reviewers highlighting clarity after the rebuttal. While questions remain about scalability, positioning against non-NP baselines, and the extent of novelty, the paper is well presented justifying acceptance.
Decision: Accept (poster)