CORDS - Continuous Representations of Discrete Structures
Authors: Tin Hadži Veljković
Erik J Bekkers
Michael Tiemann
Jan-Willem van de Meent
Venue: ICLR 2026 Poster (2026)
Keywords: Continuous set representations
Neural fields
Variable-cardinality prediction
Invertible encoding/decoding
Diffusion and flow matching
Object detection
Molecular generation
Simulation-based inference
Primary area: learning on graphs and other geometries & topologies
TL;DR: We turn discrete objects into continuous fields that implicitly encode their count, offering a simple way to handle variable cardinality across tasks and domains.

Abstract: Many learning problems require predicting sets of objects when the number of objects is not known beforehand. Examples include object detection, molecular modeling, and scientific inference tasks such as astrophysical source detection. Existing methods often rely on padded representations or must explicitly infer the set size, which often poses challenges. We present a novel strategy for addressing this challenge by casting prediction of variable-sized sets as a continuous inference problem. Our approach, CORDS (Continuous Representations of Discrete Structures), provides an invertible mapping that transforms a set of spatial objects into continuous fields: a density field that encodes object locations and count, and a feature field that carries their attributes over the same support. Because the mapping is invertible, models operate entirely in field space while remaining exactly decodable to discrete sets. We evaluate CORDS across molecular generation and regression, object detection, simulation-based inference, and a mathematical task involving recovery of local maxima, demonstrating robust handling of unknown set sizes with competitive accuracy.

Published as a conference paper at ICLR 2026
CORDS: CONTINUOUS REPRESENTATIONS OF DISCRETE STRUCTURES
Tin Hadži Veljković UvA-Bosch Delta Lab University of Amsterdam
Erik Bekkers
Michael Tiemann'
A.ILab University of Amsterdam
Tübingen AI Center
Eberhard Karls Universitär Tübingen
Jan-Willem van de Meent UvA-Boschh Delta Lab University of Amsterdam
ABSTRACT
Many learning problems require predicting sets of objects when the number objects is not known beforehand. Examples include object detection, molecular modeling, and scientific inference tasks such as astrophysical source detection. Existing methods often rely on padded representations or must explicitly infer the set size, which often poses challenges. We present a novel strategy for addressing this challenge by casting prediction of variable-sized sets continuous infer- ence problem. Our approach, CORDS (Continuous Representations of Discrete Structures), provides an invertible mapping that transforms of spatial objects into continuous fields: a density field that encodes object locations and count, and feature field that carries their attributes over the same support. Because the map- ping is invertible, models operate entirely in field space while remaining exactly decodable to discrete sets. We evaluate CORDS across molecular generation and regression, object detection, simulation- inference, and mathematical task involving recovery of local maxima, demonstrating robust handling of unknown set sizes with competitive accuracy.
INTRODUCTION
In problems where we wish to reason about discrete structure, we often need to reason about a set of objects without knowing the cardinality of this set in advance. Examples include object detec- tion (Tian et al. 2019 Wang et al. 2024a)), scientific inference tasks, such as reconstructing catalogs of astrophysical sources (Vafaei Sadr et al. 2019 Cornu al. 2024). or conditional molecular gen- eration, where the conditioned property does not uniquely determine the number of atoms (Faltings) al. 2025 Pham et al., 2022). Inferring cardinality directly from data is often difficult, making sampling in conditional generation or inference tasks inefficient.
Reasoning about unknown cardinality challenge that has been around for long time. Classic approaches include model selection with variational inference (Beal) 2003). reversible jump MCMC (Richardson & Green 1997), and Bayesian nonparametrics (Hjort et al., 2010). In modern ap- proaches based on deep learning, common strategy is to pre-allocate capacity beyond what is typically needed, and then suppress or ignore unneeded capacity (Xu et al. 2024). In the sciences, similar ideas appear when combining variational inference and empirical Bayes estimation, where learning the prior can serve to prune unneeded degrees of freedom (van de Meent et al. 2014). These examples reflect pervasive pattern: rather than modeling the distribution over cardinalities p(N) explicitly, many methods sidestep the issue by way of user-specified truncations or paddings.
In parallel, continuous representations have become increasingly common across domains, offering flexible ways to encode signals and structures and partly addressing the challenges of variable car- dinality. Neural fields and coordinate-based models (Mildenhall et al. 2020 Sitzmann et al. 2020
* Part of this work was carried out when Michael was still with the Bosch Center for Artificial Intelligence, Renningen, Germany.

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Published as a conference paper at ICLR 2026
R ROOL
R
p(r)
hgr)
Figure 1: An image with N MNIST digits (top) is encoded with CORDS into density field p(r) (middle) and per-class feature fields (x(r) (bottom). The number of objects is encoded directly in the density mass, N p(r) dr.
Xie et al. 2022) showed how images and scenes can be embedded in continuous domains, and this perspective has since been extended molecules and proteins Pinheiro al. (2024); Kirchmeyer et al. (2025); Faltings et al. (2025). These approaches remove the need to specify the number of objects advance, since atoms or components are represented as smooth densities that can, in prin- ciple, be sampled or inpainted flexibly Faltings et al. (2025). Yet cardinality is still only inferred indirectly, and object attributes are often added afterwards through auxiliary classifiers or peak- detection heuristics, rather than being built into the representation itself. As result, continuous fields provide flexibility but not unified treatment of both counts and features.
To address this gap, we introduce CORDS (Continuous Representations of Discrete Structures). Here, discrete objects are mapped into continuous fields, smooth functions defined over an ambient domain (c.g. space, time, grids), where density field encodes object counts and positions, and feature field carries their attributes. The total density mass then acts as a continuous, differentiable quantity that implicitly encodes the number of objects. The construction is bijective, models can operate directly in field space and still recover discrete sets without auxiliary mechanisms such as fixed slots or padding. This provides systematic alternative existing workarounds, offering single representation that applies across domains.
RELATED WORK
Generative modeling graphs. Graph generative modeling typically framed in either dis- crete or continuous terms. Discrete methods treat molecules as graphs of nodes and edges encoded through adjacency matrices, in DiGress (Vignac al. (202)) which applies discrete denoising diffusion, variational flow matching [Ejjkelboom et al.]2002)), which casts flow matching as varia- tional inference over categorical states, and MoFlow (Zang & Wang 2020). which uses invertible flows to generate atoms and bonds with exact likelihoods. Continuous methods instead generate atomic coordinates and features directly in 3D space. Examples include G-SchNet (Gebauer et al, 2020), which models molecular conformations with equivariant GNNs, ENF (Satorras et al. 2022 which extends normalizing flows with equivariant dynamics, and diffusion- based approaches such EDM (Hoogeboom et al. 2022 EDM-Bridge (Wu et al.) (022).. and GeoLDM (Xu al. 2023
More recent architectures such as Ponita (Bekkers et al. (024) and Rapidash (Vadgama et al.
2025
further improve scalability with multi-scale equivariant representations.
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Alongside these approaches, several works have explored representing molecular graphs in con- tinuous fields rather than explicit graphs. VoxMol (Pinheiro et al., 2024) represents molecules as voxelized density grids processed by convolutional networks, while Ragoza et al. (Ragoza et al. 2020) introduced one of the earliest 3D molecular generative models based on atomic density fields, reconstructing molecules through peak detection and bond heuristics. FuncMol (Kirchmeyer al., 2025 proposed a neural field parameterization of molecular occupancy, and ProxelGen (Faltings al. 2025) extended this idea to conditional generation and inpainting. While these methods re- move the need to fix graph size in advance, they ultimately recover atoms and features through thresholding or auxiliary classifiers, leaving cardinality and features only indirectly modeled.
Object detection. Two-stage detectors such as Faster R-CNN (Ren et al. 2016) generate region proposals followed by refinement, while one-stage models like YOLO (Wang et al. 2024a) and RetinaNet (Lin et al. (018) predict boxes and classes densely, balancing speed and accuracy. Effi cientDet (Tan et al.] 2020) further improves this trade-off via compound scaling. Transformer-based approaches, exemplified by Deformable DETR (Zhu et al.) (02T) and the real-time RT-DETR fam- ily (Wang al., 2024b), accelerate convergence and improve small-object handling. Methods that operate through heatmaps or density maps, c.g., CenterNet (Duan et al. (201)) crowd counting (Xu al., 2019), and microscopy detection (Li et al,, (022). are most closely related to our setting, as they localize objects from continuous spatial representations. However, unlike CORDS, these ap- proaches focus purely on localization and do not model or recover object-level features from the underlying fields.
Simulation-based inference. Simulation-based inference (SBI) is used in domains such cos- mology, astrophysics, and particle physics where the likelihood intractable but simulators are available. Early approaches such Approximate Bayesian Computation (ABC) (Beaumont al. 2002) relied on handcrafted summary statistics, while modern neural methods neural posterior es- timation (NPE) and neural ratio estimation (NRE) (Papamakarios et al. 2019. Lueckmann et al. 2021 provide flexible and scalable inference. Recent advances include flow matching posterior estimation (FMPE) (Dax et al., 2023), which leverages flow matching to improve scalability and accuracy, achieving state-of-the-ar results in gravitational inference. Yet, in detection, variable event cardinalities are still typically handled by padding, rather than modeled directly.
We provide more exhaustive survey of related work in Appendix E
CORDS: CONTINUOUS FIELDS FOR VARIABLE-SIZE SETS
Our goal is to establish bijective correspondence between discrete sets and continuous fields. This allows models to operate directly in the field domain, where learning and generation are often more convenient, while still ensuring that discrete predictions can be recovered exactly whenever needed. The construction applies uniformly across modalities: the only difference lies in the choice of the ambient domain Ω Rd (c.g. pixel grid for images, three- firee-dimensional space for molecules, the time axis for light curves).
We consider set S (r,,, x;),, of objects with positions E Q and feature vectors x; RRd, Let Ω x Ω + Rzo be a continuous, positive kernel with finite, location- innttionindeeendent mass Sn K (r:s) dr.
Encoding. In the CORDS approach, discrete set is transformed into continuous fields by su- perimposing kernels centered at the object positions. The resulting density field represents where objects are located, while the feature field aligns with by distributing the object attributes over the same spatial support:
N p(r) = [
N x, Krr
h(r) =
(1)
Our next objective is to establish conditions for exact invertibility so that (p. h) determine the set uniquely.
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Decoding. The inversion is made possible by three structural properties of the encoding. The total mass of the density determines the number of objects, since each kernel contributes the same constant integral a. The shape of the density identifies object locations, it must be explained by superposition of kernel translates. Finally, the feature field aligned with the density, so projecting it onto the recovered kernels yields the original features. We formalize each of these steps.
Cardinality, Each object represented via kernel whose integral a, so that
N = J. p(r) dr.
This makes variable cardinality straightforward: the number of objects is inferred directly from the density field.
Positions. Once the number objects known, their locations are encoded in the shape of the density field. Because is by definition a superposition of kernel translates, positions can be recovered by solving the kertel-matching problem
N Σ ((r:):))
/ (p(t))
min Eiy..ENN
the field truly originates from the forward transformation, the original centers achieve the global minimum. In practice, approximate solutions found with gradient-based optimization already suffice, and can be further refined higher accuracy required.
3. Features. Once the positions are fixed, the final step is to recover the object features. Because the feature field constructed from the with the recovered positions. For each position r; we define (r) = Kr;;r;;; these kernels span the subspace the feature field, so reconstructing the features amounts to projecting onto this basis. To make this concrete, we form the Gram matrix G E I N with entries
kernels the density field, its support aligns
Gú -/ (((()))(r))
and the projection matrix B e R with rows
B
h(()) ((()) dr.
The system B = GX then recovers the feature matrix X Nxd, whose the feature vectors Xg. Under mild assumptions on the kernel, is symmetric positive-definite ensuring that this system has unique solution
x
(4)
This solution exactly matches the features that generated the field during encoding.
With this construction, obtain a bijection between finite sets and their corresponding fields. The conditions that guarantee exact recovery, together with the formal results and proofs, are detailed in Appendix A In contrast, Appendix C.I focuses on how approximate decoding in practice.
Relation to kernel mean embeddings. Equation 1 is closely related to kernel mean embeddings (KMEs), which represent distribution (or empirical sample) as function given by kemel- weighted superposition of its support points (Muandet et al., 2017). While KMES are usually used to embed distributions for learning, CORDS uses the same supee--superpositionn principle represent finite sets of objects as continuous fields, with a constructive decoding back to the underlying set under mild identifiability assumptions.
PRACTICAL CONSIDERATIONS.
Sampling strategies. Fields defined continuous domain Ω, but training requires finite representation. We therefore sample a set locations {r,) evaluate the fields (p(r;), h(r,)) at those points, and feed the resulting tuples directly into neural networks. This differs from neural
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Importance sampling
Uniform sampling
QE ø-
hg(r)
QER
h2(r)
Ir pin), h(n))
Figure 2: Sampling strategies for evaluating fields. Left: Importance sampling draws coordinates in proportion the density p. concentrating samples where signal is present. Right: Uniform sampling evaluates fields on a fixed grid, covering the domain evenly. In both panels, the curves are isocontours of p(r). and the colors show the values of the feature fields (r)...
fields in the usual sense, where signals encoded implicitly inside network; here, work explicitly with sampled field values.
Two sampling approaches are possible: uniform sampling importance sampling (Figure 2. For molecules in 3D, uniform grids are inefficient: the signal occupies only small region of space, resolution grows cubically with grid size, and fixed boxes impose artificial boundaries. Instead, we adopt importance sampling, drawing locations r; with probability proportional to the density field and then evaluating the fields at those points. This concentrates samples where information is present, avoids the need for bounding boxes, and allows the model to learn coordinates and fields jointly for arbitrarily sized molecules.
For regular domains such as images or time series, however, uniform sampling remains natural: pixels form 2D grid images, and evenly spaced points define the ID domain of signal. In all cases, the model ultimately receives the sampled tuples {[r,, Ps. h.))
Neural architectures for field processing. We align the choice of neural architecture with the sampling scheme. For molccules, we discretize fields using importance sampling. which produces large unordered point sets. In this setting, we use the Erwin architecture (Zhdanov et al. 2025), hierarchical, permutation-invarian transformer that scales to thousands of points while preserving global context. For images and time series, inputs lie on regular grids; we use standard 2D and CNNs that exploit locality and run efficiently on uniform samples. This division lets CORDS tackle irregular 3D geometry with point-based transformer, and lean on compact CNNs where grid structure natural.
4 EXPERIMENTS
We apply CORDS in four settings where variable cardinality naturally arises: molecular gener- ation (QM9 and GeomDrugs), object detection in images (MultiMNIST with out-of-distribution counts), simulation-based inference astronomy (burst decomposition of light curves), and a syn- thetic benchmark for local maxima. These tasks span pixel grids for images, three-dimensional space for molecules, time series for light curves, and abstract continuous domains for mathematical functions, all within the same fieldbbsed representation. Results for QM9 property regression and for the local-mane maxima benchmark are deferred to Appendices[D22 [.] respectively; additional il- lustrations and domain-specific details appear in Appendix In all experiments, encode objects
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Vadgama et al.j Published as a conference paper at ICLR 2026
Model
QM9
GeomDrugs
Atom (%)
Mol (%)
Valid (%)
Unique (%)
Atom (%)
Valid (%)
ENF
85.0
4.9
40.2
98.0
G-Shneet
95.7
68.1
85.5
93.9
GDM
97.0
63.2
75.0
90.8
GDMAAG
97.6
71.6
90.4
99.0
77.7
91.8
EDM
98.7
82.0
98.7
81.3
92.6
Bridgee
98.8
84.6
92.0
98.6
82.4
92.8
GLDM
97.2
70.5
83.6
98.9
76.2
97.2
ALDMAUG
97.9
78.7
90.5
98.9
79.6
98.0
GeoLDM
98.9
89.4
93.8
98.8
84.4
99.3
PONITA
98.9
87.8
Rapidash 99.4
92.9
98.1
97.2
CORDS
97.9
82.3
91.0
97.1
78.4
94.6
Table 1: QM9 and GeomDrugs unconditional generation results, evaluated by the standard RDKit evaluation. Higher is better. Baseline results are adapted from 2025
(Xu al. 2023
with Gaussian kernel
r r 20²
K(r,r,) exp
4.1 MOLECULAR TASKS
Datasets. Two benchmarks considered. QM9 (Ramakrishnan al. 2014) contains small
ganic molecules (up to 29 heavy atoms) with DFT-computed molecular properties; it is used for both regression and generation tasks. GeomDrugs comprises larger, drug-like molecules covering a broader chemical space and larger atom counts; we use it for unconditional generation to assess scalability and robustness of the proposed framework higher cardinalities and a setting where modeling additional features, such as charges, is crucial.
Converting molecules to fields, sampling, and backbone. Atoms, described by their coordinates and type/charge features, mapped to density and feature fields p(r) and h(r) using Eq. equa- tion []] At each training iteration, we discretize these fields by sampling M spatial locations, yielding an unordered set of sampled points
(rr, (o(rs), hr:)) M
which forms the input the model (Erwin for all tasks). All learning and sampling steps per- formed purely in the field domain, with discrete graphs recovered via decoding only for evaluation metrics that explicitly require molecular graphs. Further details on sampling policies, normalization strategies, hyperparameters, and training schedules are provided in Appendix C.2
Unconditional generation. In the generative setting, both fields and the sampling locations must be modeled explicitly. To achieve this, CORDS learns joint distribution over coordinates and field values, denoising the entire [r,,, Ps; " After generation, fields are decoded molecular graphs using the decoding procedure from the CORDS section: the number of nodes N is esti- mated from the density mass, node positions are recovered by kernel center fitting, and features are reconstructed via linear projection.
We compare against two distinct groups of baselines, presented separately due to differences in evaluation procedures. Table in Figure compares CORDS to approaches that operate directly on continuous or voxelized representations, such as VoxMol (Pinheiro et al. 2024) and Func- Mol (Kirchmeyer et al. 2025), following their standard sanitization and post-processing evaluation steps. Table I provides results according to the standard evaluation criteria common recent dis- crete and continuous generative modeling literature (e.g., EDM: Hoogeboom et al. 2022), Rapidash; Vadgama et al.) (2025)), where validity, uniqueness, atom-level stability, and molecule-leve stability metrics reported.

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Conditional distribution p(Njc)
generated
Model
Atom
Mol
Valid
Unique
data
0.20
data
99.8
98.7
98.9
99.9
piN
0.15
VoxMol
99.2
89.3
98.7
92.1
0.10
FuncMolae:
99.2
88.6
100.0
81.1
FuncMol
99.0
89.2
100.0
92.8
0.05
CORDS
99.2
93.8
98.7
97.1
N
Figure 3: Left: Conditional generation QM9. Histogram of the predicted atom count distribution p(N]cc when conditioning on property ranges unseen during training. Right: Unconditional gen- eration results QM9, evaluated using OpenBabel postprocessing, following VoxMol. Bascline results are adapted from (Kirchmeyer et al. 2025).
AP
AP50
AP75
Model
In-dist
OOD
Drop (%)
In-dist
OOD
Drop (%)
In-dist
OOD
Drop (%)
DETR
81.2
65.4
19.5
84.0
71.7
14.6
74.2
55.1
25.8
YOLO
71.9
54.3
24.5
78.8
64.2
18.5
59.9
43.1
28.0
CORDS
76.8
64.2
16.4
81.5
71.8
11.9
68.0
53.7
21.0
Table 2: MULTINNIST object detection results in-distribution vs. OOD, Drop (%) is relative per- formance decrease.
Finally, evaluate generalization to larger molecules GeomDrugs. Here, modeling categorical atom features, specifically charges, essential: omitting them harms standard metrics such validity and atom stability. A strength of CORDS is that continuous features are repre- sented directly in field space and decoded back to graphs, which enables evaluation without post- processing, In contrast, prior continuous approaches (e.g., VoxMol, FuneMol) typically operate with one-hot atom types and resort to heuristics for charges, which limits comparability on this bench- mark; we therefore follow the usual GeomDrugs evaluation protocol. The generative setup mirrors QM9: training and sampling are done entirely in field space, with decoding only for evaluation.
QM9, CORDS matches or improves upon continuous/voxel baselines (VoxMol, FuncMol) and reaches the overall performance range of E(3)-equivariant GNNs (EDM, GeoLDM, Ponita), despite using a non-euuivariant domain-agnostic backbone. This the sense in which we describe the results as competitive.
Conditional generation (QM9). Most conditional generators are trained by conditioning on tar- get property values and are then evaluated by predicting the properties of generated samples with independent predictors, reporting MAE against the targets (Hoogeboom et al.) 2022). When condi- tioning property c, must also model the conditional size distribution p(N M cc.. Prior work typically discretizes e into bins and treats (N, c) as joint categorical variable over the number of nodes and the bin of the conditioning variable. This creates support gap: if bin is unseen during training. sampling N at that e becomes impossible.
Our approach conditions directly on continuous properties (here, polarizability a) without discretiz- ing either e or N. Cardinality emerges from field mass, so p(N [N[c) is learned as part of the conditional field distribution. To test generalization, we remove a range of during training and condition on that range at inference. Despite the holdout, we recover coherent conditional distribution over the number of atoms, as reflected in the induced atom-count histograms in Fig.
4.2 OBJЕCT DETECTION (MULTIMNIST)
Setup. We demonstrate CORDS on images where the discrete objects of interest are bounding boxes. Each bounding box instance is specified by its centre (st, y) E TR2 and carries types of

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Observed Predictive Median
Posterior
True N
Predictive 50%
200
0.5
Counts
150
0.4
100
50
0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
5
6
Time
Decoded number components
(a) Posterior reconstructions of light curve
(b) Posterior over component count
Figure 4: Simulation-barn based inference on light curves. (a) Observed light curve (blue) with re- constructions from posterior samples 6 ~ p(8 E). Each reconstruction is obtained by decoding sampled fields into component parameters 0 and simulating the resulting light curve. (b) Posterior over the number of components p(N) E).
features; class label (0-9) and box shape hw. We encode such sets into aligned fields on the image plane: density field p(r). per-class channels that carry one-hot information, and two size channels that store (w, h) where mass is present (Fig. 1 Discrete predictions are recovered with the decoding equations from Section Data is generated on the fly using an online MULTIMNIST generator, avoiding any additional augmentations and effectively yielding infinite-data regime. Each image contains up to Nmax digits (here =ma==15); digits are uniformly sampled per image, randomly rotated and rescaled, and placed black canvas. Ground-trut bounding boxes and classes are the targets for all experiments.
OOD evaluation. Beyond standard in-distribution evaluation (images with at most Niax objects), also construct ontrof-distribution split where the number of digits exceeds the training range. This tests whether a detector can handle variable cardinality without relying on pre-set capacity or dedicated counting head. Query-based models implicitly cap predictions via their slot budget; once the scene contains more objects than slots, additional instances are necessarily missed. In CORDS, the cardinality is encoded by density mass, so increasing scene density naturally reflected in the representation, and the decoding remains valid for larger scenes without changing the network.
Training objective. We train by minimizing a pixel-wise mean squared error (MSE) on both the density and feature fields, combined with a penalty on mismatched counts. The overall loss is
L = LMSE + A( - N)²,
where the predicted count N is given by the total mass of the density field,
Ñ = plu,y) dx dy.
This way, the number of objects is treated as continuous, differentiable quantity and optimized jointly with the other objectives.
Baselines and metrics. We compare to a DETR detector with a fixed query budget and a com- pact anchor-free YOLO variant. All methods are competitive in-distribution. Under OOD counts, the fixed-query baseline underestimates due to capacity limits, while CORDS continues to track the true cardinality more accurately. We report the standard detection metrics in Table For fair com- parison, we allocated all networks with a total of 8 million parameters. The exact implementation details of the haselines discussed in Appendix
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4.3 SIMULATION- BASED INFERENCE FOR FRBs
Simulation-based inference (SBI) deals with settings where the likelihood is unavailable but simula- tion from the generative process is possible: we draw parameters from prior, generate observations l, and train a conditional model approximate the posterior p(ole). We adopt flow matching for posterior estimation (Dax et al. 2023), which learns a time-dependent vector field that transports a simple base distribution to the target posterior, giving us amortized inference with tractable density.
We demonstrate this approach on the problem of modeling Fast Radio Bursts (FRBs). FRBs are short, millisecond-sette scale flashes of radio emission of extragalactic origin, whose astrophysical mech- anisms are still not fully understood Petroff et al. (2022); (hang/(2023) They are typically modeled as a superposition of a variable number of transient components, each characterized by parame- ters such as onset time, amplitude, rise time, and skewness. Recovering the posterior over these parameters given noisy photon-count light curves is natural setting for SBI.
In our experiment we work with ID photon-count light curves with Poisson noise. We first sample the number of burst components N from a uniform prior. Given N, each component has parameters 0 = (to. A, Trise, skew) drawn from astrophysical priors. If N were fixed, this problem would be straightforward: for example, we could represent each component as token of dimension 4 and train a transformer with flow matching conditioned on f. The challenge is that N varies, so we also need to recover p( (N E), which not trivial. This is the motivation for our approach.
Training and inference. We map bursts into continuous fields on the time axis: density p(t) with peaks onset times and feature field h(t) carrying (A, Trise: skew) on the same support. Each light curve is discretized on uniform grid of K points (K (K=1000), where we evaluate (p. h) and append the observed counts {(t). We then train a flow-matching model to approximate p(p(t), h(t) | 4) directly in field space; at inference pendix C.4 details the encoding and decoding. Figure shows posterior light-curve reconstructions and the induced distribution p(N] {
sample fields and decode them into component sets. Ap-
5 DISCUSSION
Implications and takeaways. Our experiments show that field-based learning handles variable cardinality reliably across domains. In molecular generation, CORDS attains competitive results against well-established GNN baselines on QM9 and GeomDrugs, while outperforming prior contin- uous approaches such as VoxMol on QM9, A key advantage is that non-categorical atomic features, such as partial charges, can be modeled directly in the feature fields and decoded back. In object detection, CORDS cxhibits a smaller performance drop under out-of distriiiiint object counts, plausibly because cardinality encoded as density mass and can be regularized with simple count penalty, making the representation more stable as scenes become denser. For comulation-basee inference on light curves, the approach sidesteps explicit modeling of p(N]E): the posterior over cardinalities arises naturally from the learned field distribution, simplifying training and inference.
Limitations. CORDS incurs practical costs. High-fidelity reconstruction in molecules benefits from dense sampling (~ 10 points per molecule), which makes direct scaling to larger graphs com- putationally expensive. Accuracy also depends on precise kernel-center localization; refinements (e.g., L-BFGS) help but add latency, creating speed-accurace trade-off. In detection, overlapping kernels can hinder separation of nearby objects, requiring fine-tuning kernel widths. We further discuss and explain these limitations in Appendix C.2
Future work. Several directions follow naturally. On the detection side, evaluating CORDS on larger-scale benchmarks (e.g., COCO) would test robustness under heavy occlusion, class diversity, and crowding, and could explore learned, spatially adaptive kernels to separate nearby instances. For molecular modeling, extending conditional generation to richer tasks, such as pocket-conditioned ligand design, regional inpainting, or multi-property control would further probe the benefits of working in field space with continuous attributes (charges, spins, partial occupancies). More broadly, we expect CORDS to be most useful in hybrid settings that combine discrete and continuous modal- ities, for instance, molecules materials described as discrete atomic graphs alongside continuous

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DFT-derived electron densities (Elsborg et al. 2026), since CORDS provides way map the dis- crete structure into a continuous field representation that naturally aligns with the density modality.
CONCLUSION
We introduced CORDS, a framework for modeling variable-size sets through continuous fields, where positions and features are mapped onto density and feature fields and cardinality is recovered directly from total density mass. The appeal lies in its simplicity: single field-based representation captures counts, locations, and attributes, enabling learning entirely in field space while still allowing exact recovery of discrete predictions across molecules, images, and simulated astronomy data.
Compared to earlier continuous reeresenatnr methods in computer vision and molecular model- ing, CORDS achieves competitive performance while offering the flexibility to represent arbitrary features beyond predefined types. Overall, the results point to single field-based representation as an elegant and broadly applicable alternative: the same encoding (density and feature fields), decod- ing (mass for counts, kernel centers for positions, projections for attributes), and training objectives carry across tasks, providing unified solution modeling variable cardinality.
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A CORDS THEORETICAL FRAMEWORK
We collect precise assumptions, definitions, and proofs validating the decoding steps and the duality between finite sets and continuous fields. To readability, each formal statement is followed by brief intuition.
General intuition. A finite set of elements with positions and features is encoded as pair of continuous fields by superimposing a fixed kernel at each position: a scalar density p and a feature field h that uses the same kernels. Decoding proceeds in three deterministic steps: (i) read off the cardinality from the total mass of p. (ii) recover the positions by matching & with sum of kernel translates, and (iii) recover the features by projecting h onto the recovered kernel translates, which yields a small linear system with Gram matrix Uniqueness comes from standard identifiability of squal-weigh kernel mixtures and positive-definiteness of G.
A.1 CORDS DECODING PROOFS
A.1.1 RECOVERING THE NUMBER ELEMENTS
Proposition A.1 (Cardinality). Let S E Sn and ((S) = (p, h). Then / p(r) dr = (S).
Proof. By linearity the integral and the definition of o.
1 p(r) dr Σ / K(rar:) dr Σ N.
Intuition. Every element contributes to one kernel "bump" whose total mass is a. Adding them and dividing by gives the count.
A.1.2 POSTIVVE-DENNTENSS OF THE GRAM MATRIX
Lemma A.2 (Gram matrix SPD), For distinct centers (r:)m,, and m²(-) K((rr,), the matrix
Gij Sq (x( (r) dr is symmetric positive-defiitt.
Proof. Symmetry immediate. For any c E TR. ((0), let de(r) = C (r).. Then
T Gc = / (o(r)) dr 2 0.
Gc O, then de in I hence a.e.; by (A2) the translates are linearly independent, so 0, contradiction. Thus G is SPD.
Infuition. Think of { set of directions in Hilbert space. Their Gram matrix computes inner products. no nontrivial combination of the Ką cancels out, the quadratic form T Gc is strictly positive for all nonzero c.
A.1.3 EXACT RECOVERY OF FEATURES
Proposition A.3 (Feature inversion with correct a). Let S E Sn with ((S) (p. h) and distinct centers (ry). Define G, B from the recovered centers as above. Then
and so the recovered features equal the original features.
B - GX
X = aG-lB. B,
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Proof. From h(r) = xg
h(()) h(r)) dr = x, (r 1 - [GG Gijaj.
B₁ .2
Stacking rows gives B = G. Invertibility follows from Lemma A.2 yiclding = aG-101 B.
Intuition. Because h is linear combination of the same kernels that make up p. projecting h onto each kernel recovers the corresponding coefficient vector. The Gram matrix accounts for overlap between kernels; its inverse untangles that overlap.
A.1.4 RECOVERING POSITIONS FROM THE DENSITY
Proposition A.4 (Position recovery). Let S E Sn with (SS) (p, h) and N |S]. If (A3) holds, then the minimizers of equation S are exactly the ground-truth centers up to permutation, and the optimal value is 0.
Proof. For the ground-truth centers {r.}. the integrand in equation 3 vanishes pointwise by con- struction, so the objective equals Conversely, any minimizer with value 0 satisfies p(.) <E=K(<=> ((:r,), hence E; K(:r;;) = E, K(;rt) By (A3) the sets of centers coincide up to permutation.
The density must be explainable as sum of identical shapes (kernels) placed some- where in S2 If equal-weight mixtures are unique, there is only one way to place N such shapes to obtain exactly at the original centers (order irrelevant).
Intuition.
B CORDS FRAMEWORK FOR GRAPHS
In order to extend our work to graphs, or data with discrete relational objects (such as edges), we will define Field of Graph quadruple
f = (pa. Po. ha he)
where
P2 + R20 is the node density;
Pe : :2: x Ω + Rzo is the edge density;
h, Ω + Rd a vector-valued node feature field.
h, Ω
dy vector-valued edge feature field.
We specify concrete construction of the encoding map & Gn + F. This construction associates to each graph distributional representation over the domain ft using fixed spatial kernels to define the node density, edge density, and feature fields.
Let us fix two continuous, positive kernels:
kn Ω Ω + Rzo: The kernel k, determines how node mass distributed across space, while hi governs the represen- tation of edges.
ke (Ω x Ω2
R>0-
Given a geometric graph G (V, X, Y, Y,R. R) E Go. we define its field representation
(G) (prs Pe: hns he)
as
Pa(r) Σ (r:
pe(ri,r2)= Σ
Re((1,, (2): (Tus
NEW
(u.))E
(5)
ha(r) Σ Kn
(r(r)r2)) Σ
VEV
(ur))
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Special case: Dirac kernels.
Choosing
(r; = 8(r re),
(e((r1, r2): (Tus r.)) 8(ri - -ra) -(r) rv),
recovers the standard discrete graph structure distributional form. construction generalizes the original discrete graph while embedding it in continuous domain. With a suitable choice of interactions, we obtain the traditional message passing.
This limiting case shows that
B. CONTINUOUS GRAPH CONVOLUTION AND THE DISCRETE MESSAGE PASSING LIMIT
Throughout this appendix we work ambient domain Ω C Rd
geometric graph G
(V, E, R, X) with node positions rv and node features ((( E IR is encoded into continuous objects
Pn. Pe, (b) S2 - TR:
particular, for a node kernel kn and an edge kernel hig we have
Pa(r) = Σ
ka(r:ru).
wEV
Pe(ri, F2) = Σ
Me((1, (ru,
(u,o)E
h (r) Σ h(t) (r;r).))
wEV
B.1.1 CONTINUOUS CONVOLUTION
Given field feature h :Ω + Rd layer k. the continuous graph convolution introduced in
h (r.) = of (w hr) (((a)))))))
(r.)
where W E Rdd, is trainable linear map, any point-wise non-linearity (ReLU, SiLU,
and m (R) denotes the message field aggregated from all spatial locations.
B.1.2 DIRAC KERNELS AND THE DISCRETE LIMIT
We now take
(r:rru)) 8(r rw), ke (c((1, 12): (Fu) 5(r1 (()) ro);
i.e. each node (edge) is represented by Dirac delta of unit mass centred at its position.
Proposition B.1 (Continuous convolution - message passing). Let kin ke be the Dirae kernels above. Then, evaluating equation] 7 at the node positions r; r. yields
h (k+1) a(W
Σ Cuv h BEV
mu
where = 1f (u, v) E. and ( otherwise. That is, the continuous convolution reduces exactly the standard message-passing update with sum aggregation.
Proof. Using the encoding equation b with Dirac kernels,
(o(r.
Σ
8(1; 5p);(r -rg)
(P-g)EE
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Fix node and set r; Tu- Substituting into the message integral in equation 7 gives
/ h (1)
m
(16
(r.)
(r())))))
Σ
8 - rp) 8(1;))rr - rg) dr,
(n.O)EE
Σ
5(ru rp) h (rg)
(P.9)EE
Σ
Sup h (k)
Σ EEV
Cuv h!!
(P.9)EE
where Sup is the Kronecker delta (the Dirac delta evaluates to 1 iff r. = r., or equivalently = p). Finally, plugging (*) (rw) = m back into equation gives equation
n
Remark (Vanishing-width Gaussian kernels). If kin and ki are isotropic Gaussians of width (as used in Eq. (9) of the main text), then Knn ke _-0, Dirac distributions in the sense of tem- pered distributions. Therefore the continuous convolution converges the message-panee saaaggg--assnngggg update equation 8 as o - 0.
B.2 EXTENDING CORDS TO NON-GEOMETRIC GRAPHS
The core construction of CORDS relies on the existence of an explicit geometric embedding p V + f2, which maps each node to position in continuous domain. However, many real-world graphs do not come equipped with natural spatial coordinates. To extend our framework to such non-geometric graphs, we propose using spectral embeddings derived from the graph's topology.
B.2.1 SPECTRAL EMBEDDINGS VIA GRAPH LAPLACTAN
Given graph G E, X) without predefined node positions, compute a spectral embedding based the graph Laplacian. Specifically, let A giv|iVV be the adjacency matrix and D the diagonal degree matrix with D = deg(i). The (unnormalized) graph Laplacian is defined as:
D A.
Let { be the eigenvalues of L, with corresponding eigenvectors (v, (v.) We define a spectral embedding
Psper (v) = (v2(v), (a(v),... va+ v (v)),
where (v) denotes the v-th entry of the i-th eigenvector. The first non- non-trivial eigenvectors capture global structural information, positioning nodes with similar topological roles close to each other in Rdd
This spectral embedding provides a continuous proxy for node positions, enabling us to apply the same CORDS construction as in the geometric case. Effectively, it allows to treat arbitrary graphs as if they were embedded a geometric space, lifting them into continuous field representation.
B.2.2 LIMITATIONS AND PRACTICAL CONSIDERATIONS
While spectral embeddings offer a principled way to introduce geometry into non non-geemetric graphs, they come with trade-offs. Specifically:
The embedding dimensionality is a design choice. Using fewer dimensions provides a com- pressed view of the graph's 's topology, which be sufficient for downstream tasks like regres- sion or classification.
However, reducing d also means that the bijective property of the Greph-Field dual pair is lost, the original graph cannot be perfectly reconstructed from the continuous representation.
For applications where reversible mapping is not critical (e.g., predictive tasks on node-level graph-level properties), this trade-off is acceptable. On the other hand, generative tasks that require recovering discrete graph structure from fields would necessitate higher- digher-dimensionall embeddings or alternative encoding strategies.
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In summary, spectral embeddings allow to extend the CORDS general graphs without prede- fined coordinates, enabling continuous representations even in the absence of natural geometry.
C ADDITIONAL EXPERIMENTAL DETAILS
C.I APPROXIMATING THE INVERSE DECODING PRACTICE
The exact inversion formulas in Appendix A involve domain integrals and solving linear systems built from kernel inner products. When fields are only available finitely many sample locations (either on grid or from sampler) these integrals are approximated by Monte Carlo (MC). Below we describe the practical decoding we use all experiments. It proceeds in three steps: (i) estimate the number of elements N, (ii) recover their positions, and (iii) reconstruct their feature vectors. We emphasise the intuition at each step and show how importance sampling enters the feature inversion.
Notation and sampling setup. Let {r, ) Ω be the evaluation points (field samples). They are drawn either uniformly from a bounding box Ω or by importance sampling from proposal q(r) proportional kernel mixture centred near the (unknown) element locations. We write p(r,) and h(r.) for the sampled density and feature fields at those points. In all cases, MC weights are taken
1 W. S
Ws Sq(r,)
(importance sampling),
(uniform over box).
Remark: for uniform sampling the mathematically unbiased weight Volifan), in our implementa- tion use I and rely on the fact that the unknown constant volume multiplies both sides of the linear system in Step (iii) and cancels out in the solve.
(i) Estimating the number of elements Ñ and normalising p.
The subsequent position fit needs to know how many kernels to place. Hence we estimate N directly from the sampled density before any other step.
In theory, N = [g p(r) dr. With sampling r. the Monte Carlo (MC) estimator is
8
s Σ
p(rs)
NMC Σ p(x.) w. = 3 (r.))
where w, su(r,) for importance sampling and w, 3/4 for uniform sampling over box (up to an unknown constant volume factor).
For importance sampling, is typically only known up to a global constant (g & Σn K'rop (. D); for uniform sampling. the box volume enters q. In practice we avoid carrying these constants by working with calibrated density: during training the encoder & optionally rescales so that its sample mean equals the true cardinality. At test time we therefore set
5
N = round Σ p(r.))
Ep(rs))
Because of MC noise 1/ E. p(rs) will rarely be exact integer; empirically it concentrates within +0.5 of the truth, so rounding is appropriate
Kernel-quadrature alternative. When the density field is (approximately) superposition of known kernel translates, one can also estimate fę p(r) dr without explicit knowledge of by using kernel quadrature (a.k.a. Bayesian / GP quadrature). Let K(+--) be the kernel used by the encoder, and assume its mass is (center-) Jindeeendent,
[ K(r;s) dr for all Ω.
a calibration pass is not available, one can use the generic Niwc above with explicit w. sq(r,3² for importance sampling, the unknown proportionality constant cancels after the normalisation step below.
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Define the kernel matrix on the sample locations r, by
gt == (r) s, E (I....,S).
We compute quadrature weights w E IR by matching the integrated kernel sections,
(K + EI) w a,
with small ridge > 0 for numerical stability. The resulting estimator
Nko = Σ
Ws p(r.)
a=1
exact whenever lies in the span of {K(::ra)>==> and provides a principled, kernel-native ap- proximation otherwise. This option removes the need to track proposal normalisation constants while still leveraging the known kernel shape. We use the calibrated-mest estimate the main experiments for simplicity, and kernel quadrature available or when sample distributions are highly non non-uniform.
optional alternative when calibration is not
(ii) Recovering positions.
Initialisation by a mixture fir. Given the samples {(r,, p(r,)) and the estimate Ñ, we fit an Ñ N-component isotropic Gaussian mixture model (GMM) to the rs, initialised with k-means++ coordinates. The resulting means {Fw) are coarse location estimates. (Optionally we search over N + 6 components by BIC and pick the best, but we keep N unless BIC strongly prefers neighbour.)
Here BIC denotes the Bayesian information criterion, BIC(K) = -2 log La + Pk log S, where L, is the maximized likelihood k-component GMM with pk free parameters fitted S samples.
Refinement by kernel matching. We then refine the centres by minimising the squared discrepancy between the observed density and superposition of kernel translates. Writing x(r; ru) K (r; r.) and allowing a global amplitude > 0 to absorb small normalisation mismatches (e.g., boundary truncation, unknown a), we minimise
S
pos (fra)aa)
S (pr.)) p(r,)
1
1 a Σ k(r.iru))
We run IBFGS for at most 50 iterations starting from the GMM means. When the dynamic range large, we minimise the same objective in log space (i.e., replace both terms by their log, with small floor), which stabilises the fit of a and the centres near sharp peaks.
Intuition. Step (ii) exactly mirrors the theoretical position recovery (Appendix AA: we seek the unique set of centres whose kernel sum reproduces the observed The GMM gives good basin of attraction; IBFGS then snaps the centres onto the mode locations determined by p.
(iii) Reconstructing feature vectors from h.
Theory recap. For the recovered centres {r define (r) = K(rrra). The ideal (I?²) projec- tion used in Appendix AeeendixArreade reads
Guv 1. Ki (T)()) Ko (r) dr,
Bu: = / (T))) dr,
= a
In code we approximate both integrals by MC with the same weights w.. Monte Carlo feature inversion. With samples r, q and weights w, "q(r.)' define
g
Guv =
Σ
Ru (Is) No (Tx) Wa.
(10)
==1
Bu:
h(r,)
Nig (xs) Ws-
(11)
s=1
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We then recover the feature matrix by solving the N Ñ system
C - B, and setting
X = a
(12)
This expression valid for any a >> and reduces to the usual solve GR= when a In our implementation we experimented with three standard radial kernels: Gaussian, Laplacian, and Epanechnikov, each normalised to unit mass, that 1 and simply solve GX B. Concretely, these kernels have the usual forms
KGauss (r) o exp( -1111///00"
(r) exp( exp(-11r|12/o).
Kipan (r) max(0, 1- |111)//0))
where a is a bandwidth parameter. All three are positive, radially symmetric, and compactly ef- fectively compactly supported, and in our experiments they lead to very similar decoding behaviour; we therefore use Gaussians by default and view alternatives.
kernels as interchangeable
For numerical robustness we add tiny diagonal eI & (s is 10-4 times the average diagonal) and fall back to least squares if a direct solve fails. When element types are categorical (e.g., one-hot), take argmax over the feature channels of each row of X.
Proposal normalization. g is known only up to constant (importance sampling) or includes an unknown box volume (uniform), Wi is known up to the same constant factor c. Both G and B in equation 10 equation 11 are multiplied by which cancels in the linear system G G = B. This why using W/ a under uniform sampling sufficient practice, and why we can implement importance weights with unnormalised mixture q xx En Kiprop ru).
Summary.
1. Cardinality: estimate N from the sample mean of p. round to the nearest integer (non-integral values within =0.5 are expected).
2. Positions: fit an N-component isotropic GMM to the sample coordinates and refine the means by minimising equation with LBFGS (optionally in log-space, with global amplitude a).
3. Features: form G B by the MC formulas equation 10-equation 11 using the same weights w, as for the integral, solve G X = B (or a G-1 if non-unit kernels are used), and post-process categorical channels by argmax.
Practical notes (hyperparameters). We initialise the GMM by k-means++, search over Ñ components with 8 0.15 Ñ unless N is very small, and run LBFGS for at most 50 iterations with a Wolfe line search. Importance sampling uses the same kernel family the density with of
proposal bandwidth (s ample simma) close to the encoding bandwidth; cemperature-sharpenn the proposal probabilities helps focus samples near density peaks; overall complexity per instance OSS Ñ 3.
C.2 MOLECULAR GENERATION IN FIELD SPACE WITH EDM
Fields and importance sampling. Following CORDS, molecule with atoms at {r.) and per-atom features {Φ.) (atom type logits and, when used, charge) mapped continuous fields on g3; a density p(r) and feature channels h(r) built by placing normalized isotropic kernels of width
each atom:
N
(())=
No (r-rr)
h(r) Σ
Du Ni (r-ra)))
==1
At training time we do not operate on graphs. Instead, we discretize fields by importance sampling M query locations from proposal q(r) proportional p(r), and read out the field values those points:
((r,, (r.),,, (r:)))
where "n" denotes channel-wise normalization (coords, density, features). Densities represented either in log-space (log p.)) or in raw space (pn), controlled by a flag; all learning is carried out directly these fields.
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Figure 5: A molecular graph (top) is encoded with CORDS into a density field p(r) (middle) and feature fields h2 (r) (bottom), which correspond to atom types here. The number of atoms is encoded directly in the density mass, N p(r) dr.
(b)
Figure 6: Visualization of node feature fields in the continuous domain. We depict the node fea- ture field (r) as 3D iso-contours over the spatial domain, with each color representing a different feature channel. (a) An intermediate state of h2 during denoising diffusion, where node features fields are still dispersed and overlap spatially. (b) The final denoised field after model inference, where node contributions are well-separated. Superimposed are the reconstructed discrete graph nodes obtained by applying the inverse mapping decoding, illustrating how the continuous field en- codes node positions and features, enabling recovery of discrete graph structure.
EDM preconditioning and losses. We train a score network in the Elucidated Diffusion Models (EDM) framework to jointly denoise coordinates and field channels. Let
|pn, hn]
(13)
denote the stacked field features at the sampled locations, and let p be the corresponding coordinates.
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For each molecule, we draw log normal noise level by sampling (0, 1) and setting
exp(E E Patd + Pmean).
(14)
We then add isotropic Gaussian noise to all three channel families:
Pr center(p) + Upp
p Pr + a Upp
h" hn + Mh
(15)
and define the noisy field stack
x² == 10" .h].
(16)
The network operates preconditioned inputs
(Piny Fin) = (Cin Gint"),
(17)
and predicts residuals (Ap, Ar). To couple coordinate, density, and feature channels, use single EDM preconditioning scale based on the geometric mean
data (r or, oT)
(18)
The resulting EDM scalings are
Jaata 0² + data
data
Cskip
Cout Vo² + dasta
Gin
Vodata + +o²
(19)
We form denoised estimates via
p Cakip Po Cout (Pin Ap),
(20)
Cakip + Cout (Fin Ar).
(21)
Training minimizes weighted sum of MSE losses, using per-family daata values (i.e., Cc Tr, af) define the EDM weights. Concretely,
L Acvonds Levords + Ap Lp Areuts Lieats:
(22)
where
Levands |lu!/2(p (p center(p)) 2
(23)
Cp |u 12 (p Pu) 2
(24)
2
Lians w 1/2 (h- hn)
(25)
and the EDM weights are
o2 We 00
0²+ o2 Wp (a o
o² + oj wy = (ao,)²
(26)
Optionally, we add mass regularizer that penalizes the squared error between the predicted and true total density mass. Writing (N) for the mass computed by averaging p over the point set, this term penalizes
(N) - (N),
(27)
and it compatible with both log-density and raw-density parameterizations.
Sampler (Euler-Maruyama with Karras schedule). At test time we draw initial Gaussian noise (Po, 2o) and integrate the EDM Euler-Mane MarruaMaauyyaaa sampler along Karras o-ladder to T 1K =0 with optional "churn":
(P&+1. + (p) + (tR+1 - fx)) (P& Pi. 2h Di with fo = tx + Ytz,
fx
and Heun correction on every step. we enable message passing (> steps), we rebuild a radius graph on the current coordinates after each update; translation is removed by centering coordinates at the end. The sampler produces set of points and denoised fields ((r, (r.). (r:))))
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Decoding back to molecules (evaluation only). All training and sampling happen in field space. For metrics that require graphs, we apply the same decoder in CORDS: (i) estimate N from the density mass, (ii) fit atom centers by kemel matching to Pu. and (iii) reconstruct per-atom features by weighted linear solve from hn When charges are modeled, they are carried as continuous channels in h and decoded directly.
Evaluation Metrics. To assess the quality of generated molecules, we report the following standard metries:
Validity: the percentage of generated samples that correspond to chemically valid molecules, i.e., those that can be parsed and satisfy valence rules.
Uniqueness: the proportion of valid molecules that are unique (non-duplicates) within the generated set.
Atom Stability: the percentage of atoms in each molecule whose valence configuration is chemically stable.
Molecule Stability: the percentage of molecules in which all atoms are stable, i.e., no atom violates valence constraints.
These metries computed using standardized chemistry toolkit and follow established bench- marks for QM9 generation. Together, they reflect both structural correctness and chemical diversity of the generated samples.
The network acting unordered field samples is an Erwin-based encoder-decoder (Fieldformer head), receiving both coordinates and field channels. We apply con- sistent channel normalization: coordinates
Backhone and normalization.
scaled by fixed factor, features by another, and the density channel is either log p or p with its own scale. All ddota values are estimated from the RMS of these normalized channels on training data; the EDM ladder (Amin, mmxx) is set to yield wide SNR range on coordinates.
RUNTIME AND SCALABILITY FOR MOLECULAR GENERATION
Asymptotic costs. For a batch of molecules with at most N atoms, spatial dimension d, feature dimension dz, and M field samples per molecule, the CORDS encoding used during training scales as
(MMdd,)
to evaluate density and feature fields at the sampled locations. The Erwin-EDe EDM backbone then scales linearly in M (number of points), our overall per-step complexity is ((M.d,). Decoding inference time follows the analysis in Appendix C. given decoded molecule with Ñ atoms and S evaluation points, the position/feature reconstruction scales as
O(SÑ No),
where the 3 term comes from the small Gram solve. In our regimes (Ñ < 29 QM9, moderate N on GeomDrugs), the A term is negligible compared to the SN kernel evaluations; the feature solve becomes dominant only when Ñ is very large and S is small, clarifying the remark in Appendix C.1
Measured training and inference times. Table summarizes the measured runtimes for QM9 generation on single NVIDIA H100 GPU (batch size 128, 750k training steps). "Encode" refers to CORDS field construction during training, and "Decode" comprises position reconstruction (GMM + LBFGSS) and feature recovery.
Discussion. Encoding used only during training and contributes less than 1% of the per-step wall-clock time relative to the diffusion backbone. Decoding is used only during sampling and adds a moderate overhead: on QM9 the CORDS decoding time is on the same order as a single pass of the generative network, with position refinement currently dominating this cost. Our prototype im- plementation does not yet fully vectorize the position fitting; we expect a further order-of -fgggniudde speedup from optimizing this step in the public release. Overall, for small molecules (QM9) and
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Table 3: Approximate runtimes for QM9 molecular generation on single H100 GPU, Times are reported per molecule.
Phase component
Time [ms]
Description
Training: NN forward+hackwann
4.32
EDM Erwin backbone
Training: CORDS encode
0.014
atoms - fields
Inference: NN sampling
58.8
30 EDM steps in field space
Decode: position reconstruction
19.98
GMM search + IBFGS refinement
Decode: feature recovery
0.57
atom-lype feature Gram solve
Decode total
20.8
full CORDS decoding sampling + decoding
End-to-end total
79.6
medium- sized drug- like molecules (GeomDrugs), the additional overhead of CORDS remains prac- tical, while applications to very large macromolecules would require additional engineering and are left for future work.
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EDM PIPELINE
( (Pseudocode (RDM training, sampling, decoding)
# --- Encode atoms -> fields, sample M points
def aeteriize_and_saapll_ (atons):
rho, make Hields (atoms, keree (signa))
r_ q(I) rho (x)
# importance sampling
dens log (rho (rho if use_log_rho else rho (r_i)
coords r_i norm_coords
feats h(r_i) norm_feats
dens dens / norm_rin rho
return coords, stack (dens, feats], -1)
# [B, 31, M,+]
EDM training
batch in loader:
coords, feats = rasterize_and saapie___nd_aameeeeeeeee (batch)
sigra = (P_meen + P_std randn (18, $ per-molecuie noise pos_noisy center (coords) sigma randn_like (coords)
dens_noisy feats :1] sigma randn_like (feats[.... :11)
h_noisy feats 1:l sigma zanda like (featsl.... 1:1)
x noisy concat ( _oisy, b noisyl, -1)
preconditioning
c_skip, c_in, cLout edn_scalings (sigma, sigma_data- [lgma_data-_______._c pos_in, x_in c_in pos_noisy, c_in X_noisy dpos, dx FieldModel (pos_in, xiin, graph-radius_graple (pos_in) if mp>0 else None)
pos pos_hat hat c_skip pos_noisy clout (pos_in dpos)
x_t c_skip x_iiss + c_out (X_in dx)
# weighted losses
1. _oords mse_w (pos_hat, center (coorda), w_c(,_))
rho mse_w (x,, feats [;, (_r(--))
+ feats mxe_N(__hat
feats 1: w_f(-_f)
if mass>0: L -- _mss (mass [x,,
mass (feats I;, )+2 L.backward opt.step
zerozeroograd
# Sampling (BulerMaruyama + Karras ladder)
pos = ( for t_cur, t_next in karras_schedule (_max, min, K);
randn ( 31); pos center (pos)
x_hat, hat churn pos, t_eur, $_churn, S_noise) x_d, P_d FieldModel (c_in(t_hat) @__intt_hat)-po__hatt dx = (x_hat x_d) / t_hat; dp (pos_hat - P_d) / hat = heun_update (x, pos, dx, dp, t_cur, t_next) if mp>0: graph radius pralllssgrappppppppp pos center (pos)
(t_
# --- Decode (for metrics only) --- N_hat integgal (x=pos,, (r=pos, rho=x :1] K=N_hat] features gram gram_prooection (b=x (b=x centers=t0)
# count from mass
= fit_keeree_centers
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In the previous code, the FieldModel is based on Erwin, and can be summarized as follows.
---- FieldModel Erwin block ---)
Inputs: pos [8, 31, feats [8, M, 1+01, a sigma [8, 11, batcc [R+M], cond [H,K] or []
Outputs: dpos [8,M, 31, dfeat [8, 1+C]
def fieldformer steepeerrreeeeeeeeeeeeeee (pos, feats, sigma, batch, cond=Nore)
# 1) embedding (log scaled) log_sigma = log (sigma) 4.0 log_sigme_nodes broadcast_to (log_aigma, batch) [B+M, 1]
# 2) Encode sampled points
z_pos RFF (pos.vier) view (-1, 3))
coords fieids
2_feat FeatMLP (feats. veeatssvvew(-se (-1, 1-C))
2 sigigma SigmaEmbed
if cond is not None:
2_nond Condi t cinmmbed (broadcast_cond (cond, batch))
iii concat ([a_pos, 2_feat, 2_sigma, z_cond], -1)
else:
z_i concat ([z_pos, f_ae,,, z_sigmal, -1)
$ 3) Fuse encodings Linear (2_in, out_dim=H)
# 4) FILM modulation by
h0 SigmaFiIM (H) (h0, log gog_sigma_noder
# 5) Rrwin/Transformer trunk over field points main_mode (h0, node_positions-pon weepoosvvvewwwweee.ee.eeeeeo (-1,3), batch idx=batch)
# 6) Prediction head per-point residuals (h)
dpos, dfeat split (y, sizes- (3, 1-C), dim=-1)
7) Reshape back return dpos. view M, 3), dfeat. view (B,M, 1+C)
C.3 OBJECT DETECTION (MULTIMNIST)
Goal and idea. We compare three detectors that differ only in representation principle while keep- ing capacity and engineering comparable: (i) field-based (CORDS) detector that predicts aligned continuous fields ($3) (fi) a YOLO-like anchor-free detector (stride 8), and (iii) a DETR-like query-based detector. The YOLO/DETR baselines are deliberately minimal (no large-scale tricks or post-hoc stabilizers) so that the comparison focuses on the core ideas: density mass for counting (CORDS), cell-wise anchors (YOLO), and slot/query capacity (DETR).
Dataset and splits. We use an on-the-fly MULTIMNIST generator (black background, no extra augmentations). Each image is 128 x 128, with a uniformly sampled number of digits per image. Digits are randomly rotated (+:25) and rescaled to side length in [18, (2] pixels and pasted on the canvas with a small border margin. The training range is N E [1,N Nmax] with N Nmax=11 15. We report (i) in-distribution (ID) metrics on held-out images with at most Nmax objects, and (ii) an OOD split with exactly +1 objects to probe robustness to variable cardinality.
Targets and what the models predict. A scene is set of discrete objects (bounding boxes of digits). Each instance is (x, y, w. h, c) with center (x, y) E RR,, size (w, h)) and class E {O....,9) 9). CORDS encodes this set into fields
the image plane: p(r) (density), p(r) (x) (per-class mass channels), and p(r) p(r) E R2 (size mass). The YOLO/DETR baselines predict class probabilities, boxes, and objectnessnoo-ojjett their usual forms.
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Training objective (shared outline). All models are trained from RGB images (standard mean/std normalization) to their respective targets. For CORDS we minimize pixel-/sample-wit wise MSE on the field channels plus mass-bars based count penalty (as in the main paper):
C = CMSE(p, (p. pm, pH) + X (N - N)², Ñ = =J p v) dr dy.
(The model learns to make N an accurate, differentiable count.) For YOLO we use objectness BCE, class CE (positives), and box LI+GIoU. For DETR use the standard Hungarian matching with class CE (with a no-object weight), LI on boxes, and GloU.
Backbones and heads (capacity parity). We keep capacities comparable (x/8M parameters total) by adjusting base widths:
CORDS fields: a light Con-NeXt-like FPN (stride 8 feature map; full-resolution output). head predicts 1+K++2 channels: P. pITIKK. PN (with μ 1].). We also implement tiny UNet head; both behave similarly.
YOLO-like: a stride-8 CNN/ConvNeet-lite backbone feeding minimal anchor-free head that regresses cell-relative (Cas Cy w, h) and predicts objectness and K- K-way class logits.
DETR-like: small CNN/ConvNeXt-Ii backbone (H/ /8 or H/ 16 stride) followed by 3-layer Transformer encoder and 3-layer decoder with T Nmax learned queries; heads predict K+1 class logits (incl. no-object) and normalized box via 3-layer MLP.
How counting differs. DETR is slot-limited (at most outputs). YOLO grid limitedi but flexi- ble in count after NMS. CORDS learns N from the mass of p. so increasing scene density produces larger P without architectural changes; decoding scales seamlessly OOD counts.
CORDS (HELDSS) MODEL: FORWARD, TRAINING, AND DECODING
The fields head outputs unnormalized logits which are mapped p softplus( )), π softmax(()) μ a-). and then [p, pi, pH].
Forward prediction.
Training with Monte Carlo supervision. We train against sparsely sampled field targets built on-the-fly to avoid full-image integrals. For each image we draw S-4096 points {r,} as mix- ture of uniform and importance sampling (fraction -0.6;; those points and regress with an MSE weighted by MC weights w, (optionally unbiased). This is practical implementation of the integrals used for feature inversion in $3} uniform sampling cor- responds to constant while importance sampling uses w, x 1/g(r.) with proportional to kernel mixture around object centers (see code).
evaluate the analytic p, pm pn
Decoding at test time. We decode with simple seed-anddrefinne routine operating the predicted fields (no heavy post-processing) In brief: (1) compute per-class mass maps pTy; (2) infer how many seeds extract either per-class globally from Σ p; (3) take local maxima (optionally subpixel refinement); (4) read (size) and x (class) at seed locations; (5) score seeds by conf = TTk (1 - exp(- and apply NMS. fixed per per.image.topp cap (= Nmax) is used for fairness.
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(Illustrative pseudo-code (CORDS) }
def fooldd_forward
feat
1B,C,8/8, W/8] ->
FPN -> [B, H, Wj
logits - 'xi (feat)
[8, 1+k+2, W]
rhe softpl (logits[;, :1])
density 0
pi softmax (logital;; 1:14K], dim=1) mu sigmoid[Logits(:: 1+R:1+R+2]]
per-class probs size in 10,11 "2
return torch.cat ([rho, rhospi, rho-mu), dim=1)
def fields_decode (maps, K, H, W, alpha=1.0, nms 1ou=0.00.5. topk=15)
rho, cls_mass, size_mass split (maps) pi normalize mcasmmas, by=rho)
# [1], [K], [2]
# pi = cas_mass / rho
clamp (size_masa / rho, 0, 1)
how many per class? use density mass:
(alpha (rho [None, ;, p sum (-2, -1)) # [K]
dets []
for in range (
seeds Lopk_ (xho=pi [k)) squeeee (0), Nkik]
optional subpixel refine (soft-argmax in small window) wh bilinear_aample (mu, seeds)
conf bilinear_sample (pi[k,, seeds) (1
exp (-bilinear_samplle (rho seeds)))
boxes seeds_ xyxy (seeds, wh,
dets += rms rms_select select (boxes, conf, class_id-k, Lou=rms_iou return srune_topk (dets, topk)
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YOLO-LIKE BASELINE (ANCHOR-FREIE STRIDE 8)
tiny CNN or ConvNeXt-lite backbone produces stride-8 feature map. The head predicts for each cell: objectness, class logits over K digits, and a cell-relative box (Cz: Cy. w. h)) with
Backbone/head and parameterization.
tato(t)) W. Cy
to(t)) H,
= o(w h = h
where (gx, 9y) is the integer cell coordinate and H,, W are stride-8 sizes. This prevents "teleport- ing" boxes from distant cells.
Assignment and loss. We assign each GT to its nearest cell (or the nearest; k = by default). Losses: Cobj Logtss on all cells with negative down-weight, Cels CE on positives (optional label smoothing), Loox = L1 on (cx, Cy. w, h) GloU in pixels.
Eval. Scores are obj max; Pk- per.image.topk.
apply NMS (class-agnostic or per-class) and cap to
(Illustrative pseudo-code (YOIC).)
def yolo_forward (x)
= backbone_ strccboon_sszide@11 (x)
[B, C, H5, Wa]
logits_cis, bogits_obj, b_oox = head(f)
[B, K, Hs, Wal,
1B,1, Hs, Wa], [B, H,, Ns]
ex, сy, w, return dict (cls_ (oot(ccccoollltaflatte (logits_cls), abj_logits=flatee (logits_abj), pred boxes=flatten ([ex, cy, w,,),)))))
vecode_cell_eelattivelll (t_box)
normalized [0, !]
yels decode (out, H, W, conf Sonf_thr=0.22 nms ams__ooust iou=0...... 4, topk=15):
prob softmax (out ("cls_logits"" dim=-1)
[B, N, K]
obj sigmoid (out ["obi_logits"
[B, N]
scores, labels prob.
[B, [), [B,N]
scere obj
boxes_xyxy
boxes"]) ("pred_boxes"))
["pred_
8-11, W-1, H-1]
keep score >= conf_thr
dets _ms__er_________e per_class___ orgoosssso__aaootiiiccccoc_eecee (boxes_ xyxy, score, labels, iou=nms_iow
return prune_topk (dets, topk)
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DETR-LIKE BASELINE (MINIMAL)
Backbone/ranssormeer A compact backbone produces d-dimensional feature map, augmented with coordinate channels and sine positional encodings. A 3-layer encoder and 3-layer decoder operate on T' learned queries. We set T = -Nmax to reflect a "budget" comparable to the other models.
Matching, loss, and eval. Hungarian assignment (SciPy) is used match predictions to GT; if SeiPy is unavailable a greedy fallback is used. Losses: class CE with a reduced weight for the no- object class, LI on boxes (in (cz, Cy, w,h) hi),, and GloU in pixels. At inference compute scores as score (1 Pnoohi) max p(c=k), optionally apply NMS, and cap to per.image.topp
(I1 IIllustrativ pseudo-code (DETR).)
def detr_ rntr_forwarddc
$ [B, C, Rs, Wa] pos = sine_posenc (f); coord coord_channela (E) sre project (cat ([f, coord))) pos
backbone (x)
# [B, C, Rs, Wa]
8,2,C (Es*Ws), x.size (0), src.size
encoder (fiatten (arc) flatten (pos))
# [8,
tgt zeros (T, C) qpos query_embar (T,
hs decoder(tgt. + qpos, mem + flatten (pos)) ) [7,
(1 return diet (pred ( logits=class head (hs),
pred bwwes=siggiii (box mlp (hs)))
def detr_decode (out, H, W, conf_thr-0.4, nms ams_iou=0, 6, topk=15)
prob softmax (out (out(""reede ("pred_loggits"" dim=-1) $ [..., K+1j
F_nc probl I.... K]; p_cls, labels probl..... :K] max (-1)
scere (1.0 P_no) P_cls
boxes boxes_xyxy
(out ("pred_rreee begred_boc boxes"])
keep score >= conf_thr
dets optional ums (boxes (boxes_xyxy, score, Labels, iou=nms iou)
return prune_ topk (dets, topk)
HYPERPARAMETERS AND FAIRNESS GUARD
Common. Image size 128 x 128; classes K=10; Nmax=15. Optimizer AdamW (Ir 1 10-4 weight decay 5x 10- batch size 128, 200 epochs. We cap detections to per.image. 15 in all methods and allow optional class-agnostic NMS for fairess.
CORDS (fields). Backbone: Light ConvNeXt-FPN (stride 8). Head width "base" = 64 (chosen so the total params x YOLO/DETR). Density activation: softplus (or softplus() to ensure p > 0. Sampling: $-4096 points/image with importance fraction 00.6 Gaussian kernel bandwidth Tuom =0.02 (fraction of min H, W)). Training loss weights: Wp=4, Wes=1, Wsize=2 Optional weak count supervision on a random fraction of the batch (weight Wcount -0.5; off by default). At decode: seed_radius- 0.03, decode_alphe
1.0. NMS loU 1.0 (disabled unless stated), and per.image.topk= 15. We rescale training fields by constant R for numerical stability (#ho.rescale=10.0) and undo it before decoding and metrics.
YOLO-like. Backbone: tiny CNN ConvNeXt-lite stride 8 feature map with dmodee=55 Loss
weights: 11 1,==1, Wbox =2, =c No-object down-weight = 0.5. Label smoothing 0.0 (unless specified). Assignment: center cell (or k-nearest cells with k= by default). Inference: confidence threshold 0.25; NMS loU 0.40; optional class-agnostic NMS; per.image.topp- 15.
DETR-like. Backbone: tiny CNN or ConyNeXt-FPN: Transformer d-256, Aheads =8 #enc/dec layers 3/3, FFN 1024. Queries T Nmax unless noted. Loss weights: class 1.0 (no-object coefficient 0.5), L, on boxes 5.0, GloU 2.0. Eval: score = (1 - Prootj) max, Peš
threshold 0.40; optional NMS with loU 0.60; per image topk= 15.
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Metrics. We report AP at loU 0.50, 0.75, 0.90, the mean over [0.50:0.93] in steps of 0.05 (mAPs0 and a headline mAP50.7-- Count MAE #preds #GT averaged over images, where #preds is (i) f p for CORDS and (ii) the number of post-NMS detections for YOLO/DETR (both capped to per image topk for fairness).
OOD protocol. For the OOD split necessarily under-count when N. In contrast, the field-based model increases f p naturally with scene density and decodes with the same routine, without changing the network its capacity.
the test cardinality N= N=Nmax+ Query-based models
C.4 SIMULATION BASED INFERENCE FOR FRBs (IMPLEMENTATION)
Problem setting. We consider ID photon-count light curves with Poisson noise generated by a variable number of transient components (FRB bursts). Each component is parameter- ized by onset time fo. amplitude A, rise time Trises and skewness skew. The latent cardi- nality N is unknown and changes per observation. Our goal is amortized posterior inference p( p(((fo.u) Aus Tuy skew ] E).
Representation. As in CORDS, we map sets of components continuous fields on the time axis t EE [o, 1]: a density p(t) that places unit mass around each to, and a feature field h(t) that carries the remaining parameters over the same support. We show this in Figure Practically, p(t) and h- channels are built by convolving Dirac impulses with normalized Gaussians bandwidths a, and f dl N when evaluated continuously. We discretize uniform grid of 7 points (here T= 1000) and optionally downsample by average pooling to Tar=7 /7ddwwssample
Fieat, respectively, that
FRB Timeseries and Fields
50
0
-50
100
0.00
hrse
hakem
0.50
hamp
Time
1.00
Figure 7: An example of lightcurve f, and corresponding burst components encoded into density and feature fields the time domain.
Conditioning and model. Given light curve & RT, we standardize it (per-sample or us- ing a precomputed global normalizer) and append it as conditioning channel. The network is a ID conditional residual stack (CondTemporal 1Net) with grouped norms, safe dilations, optional lightweight self-attention, and learned sinusoidal time embedding used by all residual blocks. The model maps fields l fields (same channel count).
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Flow matching objective (FMPE). We use flow matching for SBI Dax al. (2023). For each training pair we sample t (0, 1) (power-law schedule), form noisy interpolation ft 14 + ore
with 14, tæı and o, = 1 (1 min (here 21 target fields and =~N((0 I)), and supervise the time-dependent velocity field d4 that transports fa to 21:
Σ1 (1 min) ft
LEMPE =1 (o t,t) 24 11
Up
min)
This yields an amortized posterior over fields p(p, h (E). We also maintain an optional EMA of parameters for stable sampling.
Decoding back to components (V). Given predicted fields on the grid:
1. Cardinality N. Estimate N round At E, p(t;)). where At the grid spacing. Because p comes from smooth network and finite grids, the is rarely an integer; round to the nearest integer (40.5 rule). This gives the number of kernels to fit next.
2. Onset times to- Extract N seeds by greedy NMS p with minimum separation of roughly 20, (in pixels/ime-steps), refine each seed by a quadratic sub-pixel step, and finally run a short IBFGS that minimizes in 4
>,, x(.- - to..) - pll w.r.t. { with the normalized Gaussian used
3. Features (A, Trise, skew). Solve the weighted normal equations (Gram projection) GX = B with G.v =, Wi k(t; fo..) <<(; - to,e), B,: = E, w/ s(t,, 1 to,u) H(t,) where H stacks the feature channels of h and wi are quadrature/MCC weights. On uniform grid, w; At (constant); with importance sampling, w; 3g(tz) are standard unbiased MC weights. We add small Tikhonov term to G and solve for X.
Posterior use at test time. To estimate p(N E) sample many fields from the learned flow, decode each to Ñ, and histogram the outcomes. For predictive curves (Fig, each sample to parameters, render a Poisson rate curve via the FRB generator, and aggregate (median and quantiles). For corner plots we restrict to samples where Ñ equals the ground- truth count, align components by nearest-onset in 1D, and visualize the empirical posterior over aligned parameters.
decode
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PSEUDOCODE (TRAINING, SAMPLING, DECODING)
[Pseudocode]
$ --- Encode set -> 10 fields
def encode incode_fields fields rho =t)= N(E; (t)= _ N(t;to, rho, H = 0, |]
(params, time, sigma_rho, sigma_feat)
for u in range :
caramssactivv... [u]
# 0/1
k_ gauss (time params t0 [u], sigma_rho)
k_ gauss (time params. c0mma.t0lin [u], sigma_feat)
rho += m k_
feats_u = stack (_params.ampp(uu [u], params rise [u],
params [erams.skew[u]]] [u]]
a.append (m feats_u k_
H sum (H)
# [3, 7] # [C, T]
return stack [[rho, H[O,, H[1), H[2]), axis=0)
PMPK training step
def fmpe_step (model, fields_tghe lightcurve, sigma_min, alpha_t):
8, C, T - helds_tgt..saap
t sample saaplea_aseeeeeeeeeee (wwpaaaaaaaaaaaappaaaaaaaaaaaaa. shape=
# [0,
t_ No:,None,,Noi None]
mu mu
t_ fields_tto
sigma_t 1.0 (1.0 sigma_min) * t_
eps = randn_like (fleldds_e)
t mu_t + sigma_t eps
(fields_tgt
(1.0 sigma_min) f_t) / (1.0 (1.0
sigma_min) min) * t_)
sigma
v vahat = model (f_t, aandsttandardiaaiiiittttut time-t *
return mse (v_hat, w_t)
Sampling decode
def sample @amplee_nd_decode (model, lightcurve, S, ode_steps, cfg): cond standardizen (1ightcurve) / [None, None, [,,i,__eff]
/
F integrate ode (model, #0=randn ( (S, [_ff),, condscond.repeat (s, 1)
staps=ode_staps)
Ns, t0s, [], [], :1
for s range (S)
rhe clamp_min (F 01, 0)
_at (dullade
sho.sum()
mass ->
count
seeds ons_oon__e on_ X=N_hat,
min sep=2+sigma tho/delta_t)
t0_ = Ibfgs refine
Lbfgs__reiin (seeds, rho, Bigma_rhop
x @oive_weighteddgram grimmmmmmmm,,m (time, H-F's, 1:4], t0-t0_ref,
sigma__fattsigga__eee
w-delta_t)
Ns. (N_haaa t0s t0s. atpen ap.end return Ns, t0s,
ARCHITECTURAL DETAILS
Temporal backbone (CondTemporalNeet) A ID conv stack with grouped norms and resid- ual connections; dilations grow geometrically but are capped by the effective signal length to keep reflect padding valid. Optional squeeze-exetis excite improves channel calibration. Self-attention blocks can be inserted at chosen depths. Inputs are [fields, E (concatenated along channels), outputs are residual updates in field space (same channel count).
Channels. We use C=4 channels by default: p. hamps brises hakew- The amplitude and rise channels are logarithmic by default (base- 10); optional hig channel can be added.
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Normalization. Either per-sample standardization (zero mean, unit variance per sequence) global normalizer estimated over a large pool simulated field tensors (per-channel mean/- variance). The light-curve condition is always standardized.
HYPERPARAMETERS (USED IN ALL REPORTED FRB RESULTS)
Simulator. T=1000 points, background Vbkg=6. 6 (during training, N (nif Nmax) D) Priors: to ~ Unif log A Uni 2.477), log Trise ~ Unif(-3, -.2222), skew ~ Unif 6).
Fields. op=000, Ieat (on [0, 1)): optional downsample factor E (1,2, ..
Model. Base channels 192-384 (we use 192 for the main runs), 8-12 residual blocks, dilation base 2, group norm with 8 groups, optional SE (reduction 8), optional attention heads at few blocks.
FMPE. =0.011 t~ power(aa(=) (uniform), ODE steps 250 for sampling, init Gaus- sian scale 1.0.
Optimization. Adam W. Ir x 10 with cosine decay to 2 x 10 batch size 128, grad-clip 1.0, EMA decay 0.999 (activated after 1000 steps).
Posterior evaluation. S E [200, 1024] samples per observation, mini-batches of size 32-66 66 for the sampler.
PRACTICAL NOTES AND DIAGNOSTICS
Counting and rounding. On a uniform grid, f p dt is approximated by At Σ, p(t)). Because the encoder/decoder operate on smoothed fields and finite T, the sum is rarely exactly an inte- ger. We round to the nearest integer (+:0.5 rule). This determines how many kernels we fit in the subsequent location/feature steps; without it we would not know how many components to decode.
Feature reconstruction and weights. The Gram solve above is discrete version of the inte- gral equations in Appendix A With a uniform grid the weights wy equal A/ (constant). Under importance sampling (not used in our FRB runs but supported by the decoder). the same equa- tions hold with unbiased weights wa = L/(S g(t))). This is the practical Monte-Carlo imple- mentation of the feature inversion integrals.
Peak finding and refinement. We use greedy 1D NMS with a minimum separation propor- tional 20, (conservative for overlapping kernels), then one-step quadratic sub-pixel adjust- ment on p. and finally a short LBFGS that directly minimizes the reconstruction error w.r.t. {fu}.
Evaluation protocol. For each observation we (i) sample posterior fields and decode to get P(N E); (ii) produce predictive light-curve quantiles by rendering the generator decoded parameters; and (iii) make corner plots only for samples with N matching the ground-truth N (alignment by nearest lo on the line). We save panels. (N) histograms, median fields, and CSV/NPZ dumps for offline analysis (see code paths under FRa_results/).
Ablations. We implemented a DDPM objective for sanity but report only flow-matching (FMPE) results this work; switching to DDPM leaves the rest of the pipeline unchanged.
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(Minimal end-to-end sketch;
Training (FMPE)
for step in range wtsstees_ee_eppp___ * epochs): (B)
y_counts, params, =
fields encode_fields (parama, time, feat)
fields_n normalize
mode=field_rorm)
y_n
normalize (avg_pool p (y_counts, downsample)) loss = impe step (model, fields_An n, y_ sigma min=m.in.1
t=0.0)
loss. basskaawwa)d))) elip grad (1.0); opt.step() opt.zero_gradi
if use_ema: ema.updato (model)
with Posterior sampling for one observation
-
cond_y (ormalizeaavq_pont (y_obs, downsample))
F F_samples Empe_samp (ema_ore _r__o__mee prmme,,l, cord_y, steps-250) integrate ODE [l, []
Ns, tSs, Xs for F in samplees:
N_hat, t0_ref, X decode_ (F, time_eff, feat) # Ns Ns.append (N.aapend (N_hat); t0s. (t0_ref); xs xs.append (X) ((N),, parameters
D ADDITIONAL EXPERIMENTS
D.1 PREDICTING VARIABLE NUMBER OF LOCAL MAXIMA
To showcase more general and abstract mathematical task, we consider detecting local maxima of a scalar function f: R + R from irregularly sampled observations. The number of peaks is unknown and varies per sample, which poses challenges for models with fixed-size outputs. A visualization of this problem can be seen in Figure 8
We can easily cast this problem into our framework, by treating local extremas of the function as discrete objects, with positions corresponding to peak locations. Applying the CORDS encoding, the set of local maxima is transformed into continuous node density field. The model's 's task is to predict this field from irregular samples of f, after which the decoding recovers both the number and coordinates of peaks. A prediction is labeled as correct if it recovers the exact number of peaks, with each predicted peak lying within an e-neighbourhood of its true position. Since straightforward baselines address this specific setup, our goal is not to outperform existing methods, but to showcase how our framework naturally handles variable cardinality and infers structured information from sparse observations.
We generate fully-annotated training examples by drawing single realisation
2 ((r) ((r) aa vv D Σ cos (ws[TTr ba), r [-3,33
from Gaussian process with squared-exponenter kernel
k(r, r') = a² exp (-1|r r113/2(2
approximated by D 150 Random Fourier Features (RFF). Unless stated otherwise we amplitude a = 1.5 and length-scale 0.9.
Cosine envelope. To avoid pathological peaks on the domain boundary we multiply the raw field by separable taper E(r) (x(2 (v) E [0, 1] with margin m 0.8. where ex( 1 [[1 cos (IT clip([æ] (3 - m), O, m)) /m] (and analogously for ey). The envelope smoothly decays to 0 in a 0.8-wide frame, guaranteeing that all local maxima lie strictly inside the open square (-3 + m, m2
Ground-truth peaks. We sample 181 181 grid, apply the envelope, and locate peaks with 0.05, min_distance = 3). For each example we keep at most 50 peaks, padding with zeros if K M.
peak_lock laakklocal_max(hhie m
Training samples. From the same GP realisation we draw P 4096 i.id. points {ri uniformly from the domain. Their coordinates, the scalar value f(r:). and the list of peaks
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(h)
Figure 8: (a) Visualization of two-dimensional input function as surface plot. (b) The cor- responding density field p(x, y). obtained by interpreting the function's 's local extrema as discrete objects and applying the CORDS encoding.
constitute one training instance. During optimisation we randomly subsample K = 2048 of the P points to form a mini-batch.
Validation splits. For robustness we evaluate on GP length-scales scales {0.9, 1.1]. keeping the same envelope and amplitude.
Accuracy criterion. Given the set of predicted maxima P and ground-truth maxima P, we
greedily assign each E P to its closest E P. A sample is correct |P] = |P] and all assigned pairs satisfy
|| pll E. E T
A₂ = max I. min Zt, T 25.
with the default domain (A, = 6) this yields e = 0.24.
Additional training details. All models were trained on single NVIDIA A100 GPU using batch size of 96. Each model took approximately 5 days train across all experiments. We employed the AdamW optimizer with a learning rate 5x 10-5, coupled with a cosine annealing x 10-6 over the course of training.
schedule that reduced the learning rate minimum
We observe slightly higher accuracy on the longer length scale f 1.1, which we attribute to the smoother underlying function having fewer local maxima. This setup intentionally evaluates generalisation: the model trained on gigh-freqeeccy fields 0.9) and tested on both the same and smoother fields (f = 1.1) to assess robustness across varying levels of peak density.
Accuracy (2048 points)
0.9
92.7%
1.1
94.2%
Table 4: Local Maxima Prediction Accuracy. Accuracy of local maxima detection on different GP length scales. The model was trained on == using 2048 sampled points per example.
D.2 QM9 PROPERTY REGRESSION
With CORDS, we predict molecular properties directly from the continuous field representation, without decoding back to discrete molecular graphs. Since per-node features are not needed, predictions are obtained by pooling the final representation produced by the Erwin model. We compare CORDS against representative GNN baselines (e.g.. DimeNet++. SEGNN) on standard QM9 regression tasks. The aim here is not to surpass highly specialized architectures, but to show that continuous field-bsee representations already achieve competitive performance this well- established domain. Results reported in Table
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Table 5: QM9 regression results across different models.
Model
A
C11
C1
Ce
NMP
0,092
69
43
38
0.030
0.040
Schnet
0.235
63
41
34
0.033
0.033
Cormorant
0,085
61
34
38
0.038
0.026
LINet
0.088
68
46
45
0.038
0.030
LieConv
0.084
49
30
25
0.032
0.038
DimeNet++4*
0.044
33
25
20
0.030
0.030
TFN
0,223
58
40
38
0.064
0.104
SE(3)-Tr.
0,142
53
35
33
0.051
0.054
EGNN
0.071
48
29
25
0.029
0.031
PaiNN
0.045
45
27
20
0.012
0.024
SEGNN
0.060
42
24
21
0.023
0.031
CORDS
0.085
50
32
30
0.086
0.039
Resampling as strong regularizer. Molecules are encoded to fields, and we have two options: either resample fields using importance sampling at each training step, or for each molecule, evaluate sampled fields once and save them. To evaluate the role of sampling form of regularization, we compared two variants of CORDS on QM9 regression: one which spatial evaluation points are resampled at each epoch (our default), and another in which the field encoding is computed once per graph and fixed throughout training. As shown in Table 6 disabling resampling leads to dramatic increase in MAE across all targets, more than dou- bling the error in most cases. This confirms that stochastic sampling during training acts as a strong regularizer, promoting generalization by exposing the model to diverse field realizations. Conceptually, this consistent with the interpretation of the model as learning an underlying continuous function that is only ever observed through finite sampling process. Without sampling, the model risks overfitting to a specific discretization. With resampling, however, we gain robustness spatial variation- enabling the use of large models (100M+ parameters) even on small datasets like QM9. In Table we show the results on all targets compared to other baselines, with resampling at each training step.
QM9 Regression results (CORDS only)
Model
Resample
Q
A=
Eh
EL
μ
Cp
CORDS V
0.085
50
32
30
0.086
0.039
CORDS
x
0.350
99
72
70
0.240
0.142
A (%)
+311.8%
+98.0%
+ +125.0%
+133.3%
+179.1
+264.1%
Table 6: Comparison of CORDS performance on QM9 regression with and without resampling of evaluation points. The third row shows the relative increase in error when disabling resampling.
E EXTENSIVE RELATED WORK
Neural fields and continuous representations. Neural fields, or implicit neural representa- tions, model data as continuous functions of coordinates, Early works like DeepSDF (Park et al.)
2019). NeRF (Mildenhall et al. 2020), and SIREN (Sitzmann al. 2020) established neu- ral fields as flexible signal representations across 3D shapes and visual data. Building on this,
Functa and COIN+ (Dupont et al. 2022, 2023) explored generative modeling and cross-modal compression via neural fields. More recently, Generative Neural Fields (You et al. 2023 and
Probabilistic Diffusion Fields (Zhuang et al. 2023) extended these ideas to scalable generative modeling of continuous signals. Equivariant Neural Fields (ENFs) (Wessels et al.. 2025 fur-
ther enhance neural fields with geometry-ave aware latent variables, enabling steerable, equivariant representations that support fine-grained geometric reasoning and efficient weight-sharing.
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Smoothed Fields in Computational Physics. The formalism for continuous graph represen- tations that we develop here bears similarities to classical methods in computational physics, such smoothed-particle hydrodynamies (SPH) and particle-in-cell (PIC) methods (Price 2005 Rosswog, 2009). These approaches compute scalar, vector, or tensor fields by convolving parti- cles with smoothing kernels, typically for force computation and simulation tasks.
LLM USAGE
Large language models (LLMs) were used to revise sentences and correct grammar, to generate visualization code for some figures, and to assist with the implementation of the MULTIMNIST dataset and corresponding baseline methods. All conceptual contributions, experiment design, analysis, and the writing of original content were carried out by the authors.
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Reviews:
[{"review_id": "lzUxofaZNl", "reviewer": "Reviewer_YEsV", "summary": "The paper proposes a representation for discrete data based on neural fields, named CORDS. The central idea is to define an invertible mapping between features and continuous fields. The authors show experiments with CORDS on different tasks/domains (molecular generation, object detection, simulation-based inference in astronomy).", "questions": "* The idea seems interesting, but the method is not showing particularly good results for the chosen tasks. Moreover, the results do not support the initial claims.\n\n* For molecular benchmarks, please clarify if baseline numbers were re-run or taken from earlier papers (if any).", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* The unification of continuous fields and discrete encodings is interesting.\n* The presentation is generally clear, and the theoretical derivations are also clear.\n* The idea is applied to different tasks/domains, which means that the method shows some generalization.", "weaknesses": "* In molecular generation (Table 1) the performance is considerably lower than other models. In molecular regression (Table 4) the performance is also significantly worse than other methods (e.g. EGNN, PaiNN, SE-GNN). In particular, the dipole estimation has the worst error (0.086) of all methods, several times higher than other methods (e.g. PaiNN: 0.012, SE-GNN: 0.023). Results for object detection are limited and have a similar issue. This does not match the \"competitive performance\" described in the abstract. Results for OOD are relatively better, but generally worse than other methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dGq7jBbUWn", "reviewer": "Reviewer_BLiG", "summary": "This paper introduces CORDS, a framework that maps variable-sized discrete sets (e.g., points with features) into continuous representations through density and feature fields. Each element contributes a kernel function centered at its position, whose integrated density encodes cardinality while the feature field carries attribute information. The author conducts experiments on diverse domains—molecular generation, object detection (MultiMNIST), scientific simulation inference, and detection of local maxima. Experiments show competitive performance and robustness to variable object counts.", "questions": "See Weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "CORDS provides an elegant formulation that continuously encodes cardinality, positions, and attributes in a single differentiable field, with provable invertibility. This offers a strong conceptual link between discrete structures and continuous implicit fields.\n\nThe framework is tested across four distinct domains, showing broad potential beyond a single application type.", "weaknesses": "The approach relies on dense sampling (∼10³ points per structure) for high-fidelity reconstruction, which can become prohibitively expensive for large-scale data (e.g., complex molecules or real images). The paper lacks quantitative analysis of runtime, memory footprint, or scaling trends with respect to the number of objects or spatial resolution. Without this, the practical feasibility on larger datasets remains unclear.\n\nWhile CORDS achieves performance comparable to existing models on molecular generation task, it does not clearly surpass them in generation quality. Moreover, as the number of atoms increases, the computational overhead of the continuous-field representation (especially the sampling and kernel-matching stages) grows rapidly, potentially making the approach less scalable than discrete or equivariant diffusion models. Given these results, it remains unclear what practical advantage CORDS offers for molecular generation compared to current SOTA methods—beyond the theoretical elegance of its continuous formulation.\n\nHow are weights for LMSE and count penalty (λ) tuned in detection, and how do they affect OOD-count robustness?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "enPXeyA5l7", "reviewer": "Reviewer_6aGq", "summary": "The paper proposes to represent discrete sets of arbitrary cardinality as continuous density and feature fields. This enables models to predict/generate sets of objects without fixing cardinality in advance. The authors formally show this bijection. The density field is simply a sum of Gaussians centered at each object, whereas the feature field is the same except each Gaussian is weighted by the corresponding feature vector. Recovering a discrete set requires drawing samples to integrate the density field to find the cardinality $N$. Then, positions are recovered by fitting a Gaussian mixture model to the samples and densities and refining object positions with LBFGS. Feature vectors are recovered by solving a linear system with coefficients and constraints defined by the recovered Gaussians and feature field. The framework of CORDS generalizes Gaussians to generic kernels. CORDS is validated with experiments across 3D molecules, images, astrophysical signals, and local maxima of functions.", "questions": "1. In 3D molecule generation, how much is the computational cost of integration-based decoding, both in absolute wall-clock timing and relative to the neural network part? How long does training take, in wall-clock time and number of training steps, and what compute resources are used?\n2. Is it possible to make the kernel learnable?\n3. Why is it that (line 1129) \"feature solve typically dominates cost only for very small $\\hat{N}$\"?\n4. The bijection between discrete sets and continuous fields seems more promising in the other direction: mapping high-dimensional signals down to sparse discrete sets, like tokenization. Alternatively, one could coarse-grain large, discrete sets into continuous fields with fewer points. Does the CORDS framework enable/motivate any approaches in this line of thinking? In physics and chemistry, there are many continuous fields that if represented as discrete but sparse objects could reap benefits: electron densities, wavefunctions, scalar and vector fields in quantum field theory and statistical field theory (e.g. $\\phi^4$ theory or solutions to a Landau-Ginzburg Hamiltonian).\n5. Future work in decoding spectra (NMR, IR) with overlapping signals could be interesting.\n\nSome potentially related works are Gaussian splatting for rendering images [1] and representing electron density as sums of Gaussians [2].\n\n[1] Kerbl, B., Kopanas, G., Leimkühler, T., & Drettakis, G. (2023). 3D Gaussian splatting for real-time radiance field rendering. ACM Trans. Graph., 42(4), 139-1.\n\n[2] Elsborg, J., Thiede, L., Aspuru-Guzik, A., Vegge, T., & Bhowmik, A. (2025). ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals. arXiv preprint arXiv:2503.08305.\n\n\nnit-picking:\n1. line 269: \"using Eq. equation 1\" has an extra word\n2. Several citations do not have parentheses around them.\n3. line 886: \"we will define Field of Graph is a quadruple\"\n4. line 1062: BIC is not defined\n5. line 1102: there is no mention of Laplacian/Epanechnikov kernels elsewhere in the paper\n6. line 1253: \"diffusion machinery\"\n7. When introducing kernels, it may be clearer to suggest that the reader imagine Gaussians as a typical example.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Motivation for modelling cardinality is clear in the introduction.\n\nContextualization with related works is extensive without being wordy.\n\nThe paper is beautifully written with both intuitive descriptions as well as formalism. Figures are clear and visually appealing.\n\nThe proposed method is validated with experiments across multiple domains, showcasing the generality of CORDS. In 3D molecules, CORDS achieves state-of-the art generation quality out of field/voxel-based methods.", "weaknesses": "The paper does not discuss its limitations enough. Representing 3D molecules as continuous fields still requires handling discrete samples as inputs to the neural network, so this blows up $N\\approx 20$ atoms into $M\\approx 1000$ points, and incurs extra cost in integrating densities, gradient-based recovery of positions, and a $O(N^3)$ linear solve to recover feature vectors. It is unclear how much overhead is added by each of these steps, as wall-clock timing, number of training steps, and compute resources are not reported for molecule generation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HTQQHgjX4n", "reviewer": "Reviewer_gPMT", "summary": "This paper tackles the problem of predicting the cardinality of object sets and proposes a method that utilize kernel based continuous field that can map to the finite set of spatial objects. In this case, the total density mass is the number of objects. The proposed CORDs provides an invertible mapping that enable the exact decoding from the field space to the discrete space. The authors demonstrate the effectiveness of proposed representation in various applications, including molecular generation, object detection, simulation-based inference and synthetic functions.", "questions": "- Now, the current representation is rather unified for all downstream tasks, do you think it might be valuable to calibrate current method to become some task-specific representation?\n- How the kernel choice and kernel parameter affect the construction?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed method demonstrates a clean and clear idea that enable the encoding to continuous fields and exact decoding to discrete sets, which also makes the prediction of object cardinality feasible\n- This problem it tackles itself is a rather fundamental issue that can be further applied to many downstream tasks as shown in the paper without being restricted to specific use case\n- The bijective construction make the transition between discrete and continuous space smooth without the need of additional module", "weaknesses": "- The computational cost including the encoding and decoding process might increase when dealing with larger samples\n- The experimental performance of CORDS may fall behind some baselines, for instance, the performance in the molecular tasks, so what could be the benefit of this method compared to those baselines in this case.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oGPQpTkps8", "reviewer": "Reviewer_7wBB", "summary": "The paper claims to develop CORDS (continuous representation of discrete structures), which provides a \"bijective\" representation that maps sets of spatial objects with features to continuous \"density\" and \"feature fields\".", "questions": "What is the mathematical/ML problem that the paper is solving?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 1, "contribution": 1, "strengths": "None.", "weaknesses": "The major problem with this paper is that it uses some jargon and terms that are not mathematically defined, nor are they standard in machine learning research. This hinders the ability of a reviewer to fully understand and judge this paper. For example, starting from the first sentence of the Abstract, what does \"number\" mean when the paper says \"their number in advance\"? The Abstract talks about existing methods in the Abstract, but no problem has been laid out. The Abstract then talks about a challenge that is not substantiated.  What does \"variable cadinality\" mean? What does \"bijective representing\" mean? What is \"Feature field\"? These are all non-standard terminologies that I have not seen before.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)