NeMo-map: Neural Implicit Flow Fields for Spatio-Temporal Motion Mapping
Authors: Yufei Zhu
Shih-Min Yang
Andrey Rudenko
Tomasz Piotr Kucner
Achim J. Lilienthal
Martin Magnusson
Venue: ICLR 2026 Poster (2026)
Keywords: Neural Implicit Representation
Spatiotemporal Dynamics
Human Motion Modeling
Maps of Dynamics
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: A continuous spatio-temporal map of dynamics using implicit neural representations, achieving more accurate and efficient motion pattern representation.

Abstract: Safe and efficient robot operation in complex human environments can benefit from good models of site-specific motion patterns. Maps of Dynamics (MoDs) provide such models by encoding statistical motion patterns in a map, but existing representations use discrete spatial sampling and typically require costly offline construction. We propose a continuous spatio-temporal MoD representation based on implicit neural functions that directly map coordinates to the parameters of a Semi-Wrapped Gaussian Mixture Model. This removes the need for discretization and imputation for unevenly sampled regions, enabling smooth generalization across both space and time. Evaluated on two public datasets with real-world people tracking data, our method achieves better accuracy of motion representation and smoother velocity distributions in sparse regions while still being computationally efficient, compared to available baselines. The proposed approach demonstrates a powerful and efficient way of modeling complex human motion patterns and high performance in the trajectory prediction downstream task. The code is publicly available at https://github.com/test-bai-cpu/nemo-map.

Published as a conference paper at ICLR 2026
NEMO-MAP: NEURAL IMPLICIT FLOW FIELDS FOR SPATIO-TEMPORA MOTION MAPPING
Yufei Zhu
Shih-Min Yang'
Andrey Rudenko
Tomasz P. Kucner
Achim Lilienthal
Martin Magnusson'
'reeboo University, Sweden
Technical University of Munich, Germany
3 Aalto University, Finland
ABSTRACT
Safe and efficient robot operation in complex human environments can benefit from good models of site-specific motion patterns. Maps of Dynamics (MoDs) provide such models by encoding statistical motion patterns a map, but existing representations use discrete spatial sampling and typically require costly offline construction. We propose a continuous spatio-temporal MoD representation based implicit neural functions that directly map coordinates the parameters of a Semi Soni-Wrapped Gaussian Mixture Model. This removes the need for discretiza- tion and imputation for unevenly sampled regions, enabling smooth generalization across both space and time. Evaluated on two public datasets with real-world peo- ple tracking data, our method achieves better accuracy of motion representation and smoother velocity distributions in sparse regions while still being computa- tionally efficient, compared to available baselines. The proposed approach demon- strates powerful and efficient way of modeling complex human motion patterns and high performance in the trajectory prediction downstream task. The code is publicly available uthttps:/// /ttttp:://ithus www. wttp:au....oottttttteeeeeeeeeeee http://eeee-------eeeeeeeeeeeeee map
INTRODUCTION
Safe and efficient operation in complex, dynamic and densely crowded human environments is crit- ical prerequisite for deploying robots in various tasks to support people in their daily activities. Ex- tending the environment model with human motion patterns using a map of dynamics (MoD) is one way achieve unobtrusive navigation, compliant with existing site-specific motion flows (Palmieri et al. 2017] Swaminathan et al. 2022).
Incorporating MoDs into motion planning provides benefits in crowded environments, since they encode information about the expected motion outside of the robot's sensor range, allowing for less reactive behavior. MoDs can also be applied to long-term human motion prediction (Zhu et al., 2024 2025a). MoDs help predict realistic trajectories that implicitly respect the complex topology of the environment, such as navigating around corners or avoiding obstacles.
Several approaches have been proposed for constructing MoDs. Early methods modeled human mo- tion on occupancy grid maps, treating dynamics as shifts in occupancy (Wang al., 2015 2016). These approaches struggle with noisy or incomplete trajectory data. Later, velocity-basee repre- sentations have been introduced, most notably the CLiFF-map (Kucner et al.) 2017). which models local motion patterns with Gaussian mixture models, effectively captures multimodality in human flows and has been successfully used both robot navigation and prediction tasks. The methods above are computed in batch, given set of observations. Online learning methods have also been explored to update motion models as new observations arrive (Zhu et al. (2025)), allowing robots to adapt changing environments without costly retraining from scratch. Temporal MoDs have also been explored, including STeF-maps (Molina et al.] 2022), which apply frequency- models to encode periodic variations in the flow.
"Correspondence yufei. zzhuocru.se

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However, existing MoDs require spatial discretization, with a manually selected map resolution for point locations and interpolation to estimate motion at arbitrary positions. This discretization intro- duces information loss, reduces flexibility, and complicates tuning across different environments.
To address these challenges, in this work, instead of representing motion patterns on a discrete grid, propose a continuous map of dynamics using implicit neural representation. We learn neural function that maps spatio-temporal coordinates to parameters of a local motion distribution. Implicit neural representations have emerged as powerful tools for encoding continuous functions, provid- ing compact and differentiable models with strong generalization. Leveraging these properties, this formulation allows the model to smoothly generalize across space and time, while maintaining mul- timodality in places where flows tend to go in more than one direction since it produces a wrapped Gaussian mixture model of expected motion given a query location and time.
We evaluate our approach on real-world datasets and show that continuous MoDs not only improve representation accuracy but can also be computationally efficient. Our method yields smoother and more consistent velocity distributions, resulting in more accurate representations of human motion patterns and higher performance in the trajectory prediction downstream task. contrast to baseline approaches that rely on time-consuming per-cell motion modeling, it trains a model that represents motion continuously over space and time and enables fast inference at arbitrary locations. Unlike the faster but discretised representation of STeF-map Molina et al. (2022), method preserves non-discretised directions, yielding results closer spirit to CLiFE.
In summary, the main contribution of this work is entirely novel representation of flow-aware maps of dynamics, named NeMo-map. In contrast to existing methods, NeMo allow continuous spatio-lemporal queries to generate location- and time-specific maltimodal flow predictions. As evidenced by our experimental validation on real-world human motion data, NeMo efficiently learns a highly accurate statistical representation of motion in large-scale maps.
2 RELATED WORK
A map of dynamics (MoD) is a representation that augments the geometric map of an enviromment with statistical information about observed motion patterns. Unlike static maps, MoDs incorporate flow information, allowing robots to reason about how humans typically move in given environment.
MoDs can be built from various sources of input, such as trajectories (Bennewitz et al., 2005 dynamics samples, or information about the flow of continuous media (e.g., air or water) (Bennetts et al. 2017). Furthermore, these models can feature diverse underlying representations, including evidence grids, histograms, graphs, or Gaussian mixtures.
Several MoDs types are described in the litera- ture, providing an efficient tool for storing and querying about historical or expected changes in environment states. Occupancy-bart ods focus mapping dynamics pancy grids, modeling motion as shifts oc- cupancy (Wang al. 20115] 2016). Trajectory- based methods extract trajectories and group into clusters, with each cluster representing typical path through the environment (Ben- newitz et al.) 2005). These approaches suffer from noisy incomplete trajectories. To ad- dress this, Chen et al. (2016) formulates trajec- tory modeling as dictionary learning problem and uses augmented cemi-nonnegative sparse coding to find motion patterns characterized by partial trajectory segments.
meth-
occu-
Figure 1: Probability density of a Semi-Wrapped Gaussian Mixture Model (SWGMM) with two components, visualized on a cylinder. Orientation 0 is wrapped around the circular axis, while speed extends along the vertical axis. The represen- tation allows joint modeling of angular (orienta- tion) and linear (speed) variables, capturing mul- timodality in motion patterns.
MoDs can also be based on velocity observa- tions. With velocity mapping. human dynamics can be modeled through flow models. Kuener al. (017) presented probabilistic framework for mapping velocity observations, named Circular-
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Linear Flow Field map (CLiFF-map). CLiFF-map represents local flow patterns as a multi-modal, continuous joint distribution of speed and orientation, as further described in Sec. [ A bene- fit of CLiFF-map is that it can be built from incomplete (de Almeida et af. 2024), without the need to store a long history of data or deploy advanced track- ing algorithms. CLiFF-maps are typically built offline, for the reason of high computational costs associated with the building process. This constraint limits their applicability in real environments.
spatially sparse velocity observations
When building flow models, temporal information can also be incorporated. Molina al. (2022) apply the Frequency Map Enhancement (FreMEn Krajník et al. (2017), which is a model describing spatio-temporal dynamics in the frequency domain, build a time- dependennd probabilistic map to model periodic changes in people flow called STeF-map. The motion orientations in STeF-map are discretized. Another method of incorporating temporal information is proposed by Zhi et al. (201)). Their approach uses a kernel recurrent mixture density network to provide a multimodal probability distribution of movement directions of typical object in the environment over time, though it models only orientation and not the speed of human motion.
It important to note that map of dynamics is not a trajectory prediction model. Whereas trajectory predictors (e.g.. LSTMs) aim to forecast the future state of agents by propagating state information forward in time from an initial state, our goal is fundamentally different. We seek to construct spatio-temporal prior that encodes the distribution of motion patters in the environment itself. This prior can be queried directly at any spatial coordinate and any time of day, providing motion statistics that can support downstream tasks such as planning long-term prediction, but it does not by itself' generate trajectories for individual agents.
METHODOLOGG
3.1 PROBABILISTIC MODELING OF HUMAN MOTION
Our spatio-temporal map of dynamics produces probability distributions over human motion veloe- ities. A velocity v is defined by the pair of speed (a positive linear variable p e IR) and orientation (a circular variable 0 E [0, 2))
To capture the statistical structure of such data, we model human motion patterns with a Semi- Wrapped Gaussian Mixture Model (SWGMM), similar to the CLiFF-map representation (Kucner et al. 2017). While a von Mises distribution would be effective for purely angular variables, is not suitable when combining cireular and linear components. Roy et al. (012)) proposed the von Mises- Gaussian mixture model (VMGMM) to jointly represent one circular variable and linear variables. However, their model assumes independence between the circular and linear dimensions, which limits its ability in capturing real-world correlations. To overcome this, SWGMM (Roy et al. 2016) jointly models dircular-lineat variables and allows correlations between them.
An SWGMM models velocity v = [p. T] as mixture of J Semi-Wrappee Normal Distributions (SWNDs):
p(v () Σ j=1
WAASH (v),
(1)
where & = {& (wj, My Σ denotes finite set of components of the SWGMM. Each wy a mixing weight and satisfies S wj 5 1), μy the component mean, and E, the covariance, An SWND L
NSW is formally defined as
NEW Σ NNuss(p -
kEz
where is a winding number. practice, the PDF can be approximated adequately by taking l E (- (Mardia & Jupp 2008).
The SWGMM density function over velocities can be visualized as a function on a cylinder, as shown in Fig- D Orientation values 0 are wrapped on the unit circle and the speed runs along the length of the cylinder. This formulation yields a flexible and interpretable probabilistic representation of local human motion, capturing multimodality and correlations between orientation and speed.
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[x)
(x))
Ta.
MLP
SIREN
Semi wrapped
ft(
Gaussian Mixture Model
Figure 2: Method overview. A spatio-temporal query (x, 1) is mapped to parameters of a Semi- Wrapped Gaussian Mixture Model (SWGMM). The spatial coordinate is used to interpolate fea- tures from a learnable spatial grid G., and the temporal coordinate is encoded using a SIREN network. The spatial features (x), temporal encoding ft(t), and coordinates are fused via FiLM (Perez et al., 2018) and passed through an MLP, which outputs the parameters a SWGMM, providing continuous, multimodal probabilistic representation of motion dynamics at the queried location and time.
3.2 LEARNING CONTINUOUS MOTION FIELDS
Previous MoD approaches, such as CLiFF-maps and STeF-maps, rely on discretizing the environ- ment into cells and fitting local probability models. Discretization leads to information loss and prevents querying at arbitrary locations. We address this by introducing a continuous map of dy- namics parameterized by neural implicit representation. The method overview is shown in Fig. 2
To capture temporal variations in human motion, we condition the MoD on continuous temporal variable. Incorporating time into the input enables the model to capture temporal variations without requiring sequential rollouts, and enables efficient queries motion dynamics at arbitrary spatio- temporal coordinates (z, t).
Problem statement.
Given a dataset D of N spatio-temporal motion samples:
D {(x,, vi)
where x; E R is the spatial coordinate, t, R is continuous temporal variable, and v; [pt, 0.] is observed velocity, we learn a continuous function Da that maps a spatio-temporal coordinate (x, t) SWGMM parameters:
(x = wx t), (x, (x, j=l
(3)
where J is the number of mixture components, weights wj 2 0 and Σÿ=1 Wi 1. Each of the j components models the joint velocity v = [o, 0] with a Semi- Grapper Normal Distribution N/W At inference time, querying & at any coordinate yields the full set of SWGMM parameters, resulting in a continuous probabilistic representation of motion dynamics. This formulation enables the model learn smooth, continuous motion fields while retaining the multimodal characteristic of human motion.
Architecture. In our neural representation, we parameterize 4 with a fully connected multilayer perceptron (MLP), conditioned on both spatial and temporal features:
f, (x)
E C.C.
R
spatial features
temporal cocoding
For spatial features, learnable grid G, E #R is queried at location by bilinear interpola- tion, producing f, (x). This captures local variations in motion patterns while remaining continuous in space.
For temporal encoding, we encode with SIREN, the sinusoidal representation network (Sitzmann et al. 2020), which uses periodic activation functions throughout the network.
The MLP input concatenates the raw coordinates and the spatial and temporal features, z = [x, t, f, (x), ]. and outputs SWGMM parameters. This ceature-conditioned representation ables the model to flexibly encode local variations in motion dynamics while maintaining global smoothness across both space and time.
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Likelihood and training. the predicted SWGMM is
For spatio-temporal coordinate (x,, the velocity likelihood under
N = Σ wy wy(x,, (4) (5(0))) (x(x) j=1
plve P (x,,,
where A/SW denotes the semi-wrapped normal distribution that wraps the angular component (see Eq. (2). The model is trained by minimizing the negative log-likelihood of motion samples from the dataset under the probability density function (PDF) produced by the model:
((0) -
log p(v: (P((x;,4))
RESULTS
To evaluate spatio-temporal maps of dynamics that capture changes of human motion patterns over time, it is essential to use datasets that span multiple days and reflect variations in human motion patterns. Our experiments were conducted using real-world datasets, ATC (Brščié et al. 2013) and ETH (Pellegrini et al. 2009)/UCY (Lerner et al.] 2007). The ATC dataset provides sufficient multi-day coverage for evaluation, consisting of long-ter real-world people tracking data collected in a shopping mall. The ETH/UCY dataset contains pedestrian trajectories captured in outdoor environments such university campuses, with multiple unique scenes. Details are in Appendix B
4.1 BASELINES
Circular-Lineal Flow Field Map (CLiFF-map). CLiFF-map (Kuener et al. 2017 represents motion patterns by associating each discretized grid location with an SWGMM fitted from local ob- servations. The environment is divided into set of grid locations, and each grid location aggregates motion samples within fixed radius. The SWGMM parameters each grid location are estimated via expectation-maxxiizatiooo (EM) (Dempster et al. 1977). with the number and initial positions of mixture components determined using mean shift clustering (Cheng 1995). When training the CLp, the convergence precision is set to Ic-5 for both mean shift and EM algorithms, with a maximum iteration count of 100. The grid resolution set to 1 m. To evaluate different hours, we train separate CLiFF-mapp for each hour using the motion samples observed during that time.
Online CLiFF-map. Online CLiFF-map (Zhu et al. (2025b) extends the static CLiFF model by updating the SWGMM parameters incrementally Each grid location maintains an SWGMM, which is initialized upon first receiving observations and subsequently updated using the stochastic pnpectation-mxxmmizatin (sEM) algorithm (Cappé & Moulines (2009). sEM, the expectation step of the original EM algorithm is replaced by stochastic approximation step, while the maximization step remains unchanged. Like the static CLiFF-map, Online CLiFF-map outputs SWGMM parameters at each grid location, but supports follow a spatio-temporal setting by gener-
motion observations become available.
continuous adaptation over time. In the experiments, ating an online CLiFF-map for each hour. Observations collected in an hour interval are treated as the new data batch for updating SWGMMs, producing a temporally adaptive motion representation.
Spatio-Tempore Flow Map (STeF-map) STeF-map (Molina et al. 2022) is spatio-temporal map of dynamics that models the likelihood of human motion directions using harmonic functions. Each grid location maintains katel temporal models, corresponding to kytef discretized orientations of people moving through that location over time. By modeling periodic patterns, STeF-map cap- tures long-term temporal variations in crowd movements and can predict activities at specific times of day under the assumption periodicity in the environment. Following Molina et al. (2022), we set Kitee 8 in the experiments, and the model orders for training STeF-map, i.c. the number of periodicities, is set to
4.2 QUANTITATIVE RESULTS
To quantitatively evaluate the accuracy of modeling human motion patterns (MoD quality), the negative log-likes likelhhodoo (NLL). An MoD represents human motion as a probability distribu- tion velocity conditioned spatio-temporal coordinate (æt, y, implemented either an
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SWGMM (our method and CLiFF-maps) histogram (STeF-map). To evaluate representation accuracy, we use test data consisting of observed human motions in the same environment. For each test sample (z, y/ t), we query the MoD to obtain the predicted distribution and compute the likelihood of the observed motion under this distribution. A higher likelihood indicates that the pre- dicted distribution better aligns with the observed data. We report NI.L for numerical stability and easy comparison, so lower NLL values correspond to more accurate motion representations, i.e., higher quality MoDs. Table D and 2 report the accuracy results for the ATC dataset and the ETH/UCY dataset, respectively. Our method achieves the lowest NLL, outperforming all baselines. Online CLiFF-map, CLiFF-map, and STeF-map exhibit higher NLLs, with paired t-tests showing p < 0.001 under the null hypothesis that baseline NLL is less than or equal to ours.
Compared with Saps- methods based on SWGMM, such as ours and CLiFF-map, offer two key advantages. They jointly model speed and orientation, whereas STeF-maps do not include speed information. In addition, SWGMMs represent orientation continuously rather than through a discretized 8-bin histogram in STeF-map. These aspects lead to more accurate representation of human motion and contribute to the improved performance.
Limitations of CLiFF-maps are from discretizing the environment into grid cells, with each cell storing a locally fitted SWGMM. This grid-based design limits spatial resolution and introduces discontinuities cell boundaries in both space and time. In particular, dividing time into hourly intervals a coarse approximation that can produce abrupt changes, since human motion patterns do not necessarily shift exact hour boundaries. In contrast, continuous neural implicit representation. This enables smooth generalization across space and time, supports queries at arbitrary spatio-temporal coordinates, and provides a compact representation that avoids the memory overhead of storing distributions for every grid cell.
method models the MoD
We also compare the map building time as shown in Table 3 For the baselines, the training time corresponds to convergence on all grid cells, while for our method it corresponds to the neural network training time. Our method computationally efficient, training on a full day of data in under 20 minutes, while achieving higher accuracy. These results highlight the practicality of continuous MoDs for real-time applications, combining both accuracy and efficiency.
Accuracy evaluation the ATC dataset using average negative log- ligleiildhood (NLL), where lower values indicate better accuracy. We report mean + standard deviation, together with the reduction in NLL relative method and the corresponding 95% confidence interval (CI).
Table I:
Method
NLLJ
NLL reduction (vs Ours)
95% CI
Ours
0.775 + 2.052
Online CLiFF-map
1.527 + 4.156
+0.752
[0.749, 0.755] [1.185, 1.192]
CLiFF-map
1.964 + 4.953
+1.189
STeF-map
5.576 + 9.314
+4.801
[4.794, 4.809]
Table 2:
Accuracy evaluation the ETH/UCY dataset using NLL, where lower values indicate better accuracy. We report mean + standard deviation, together with the reduction in NLL relative to our method and the corresponding 95% confidence interval (CI) of this reduction.
Method
ETH
HOTEL
UNIV
ZARA
Ours
-0.384 +:2 2.051
-0.838 +:2 4.043
0.404 +:: 1.902
-0.342 + 2.152
CLiFF-map
0.112 4.005
0.701 + 4.533
0.518 + 2.125
0.068 4.265
Reduction vs Ours
+0.496
+1.539
+0.1 14
+0.410
95% CI
[0.344, 0.648]
[1.287, 1.791]
[0.049, O. 0.18
[0.215, 0.604]
Online CLiFF-map
0.086 + 4.451
1.241 + 7.142
0.577 + 2.548
0. + 5.477
Reduction Ours
+0.470
+2.079
+0.173
+0.528
95% CI
[0.307, 0.633]
[1.744, 2,412]
[0.098, 0.247]
[0.264, 0.792]
STeF-map
2.315 + 6,016
3.349 + 7.660
10.932 + 12.771
2.784 + 7.014
Reduction vs Ours
+2.699
+4.187
+10.528
+3.126
95% CI
[2.472, 2.926]
[3.820, 4.554]
[10. 187, 10.868]
[2.779, 3.472]

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Table 3: Training and inference times for map building on the ATC dataset. Lower values indicate faster performance. Experiments were conducted on a desktop computer equipped with an Intel 19-12900K CPU and an NVIDIA GeForce RTX 3060 GPU running Ubuntu 20.04.
Method
Train time (minute))
Inference time (second))
Ours
19.26
1.363 x 10 10-6
Online CLipF-map
23,859
1.914 10
CLiFF-map
1831
1.914 10-3
STeF-map
0.815
5.665 x 10-5
4.3
QUALITATIVE RESULTS
Fig. and Fig. present qualitative examples of NeMo-map compared with baseline MoDs on the ATC dataset. NcMo-map captures multimodal human motion patterns, which is important in open spaces with intersecting flows. Fig. presents the central open area of the ATC shopping mall. In the region highlighted by a black circle, a dominant horizontal flow intersects with a vertical flow. NeMo-map preserves both motion modes, maintaining the full continuity In contrast, CLiFF-map fails to capture this multimodality consistently. It shows the vertical flow only at the top and bottom the marked region, while cutting off the middle section, resulting in an incomplete representation of the crossing motion. As result, NeMo map achieves lower NLL values and provides accurate representation, as shown in the NLL heatmaps in the bottom row.
the crossing traffic.
Fig. 4 shows the NeMo-map and baseline MoDs, together with corresponding NLL heatmaps, in the east corridor of the ATC shopping mall across different times of the day. In this corridor (right side of the ATC map), human motion patterns vary throughout the day. The dominant flow is directed left/upwards in the morning, shifts direction around noon, and turns into right/downwards in the evening. The flow fields generated by NeMo-map capture such temporal variations and implicitly align with the environment's 's topology, even though no explicit map was provided during training. For instance, pedestrian speeds decrease near resting benches, motion flows pass through exits, and flows follow the corridors. Same to the middle-region observations in Fig. [ NeMo--ap not only captures the dominant following flow but also preserves the flow that crosses the corridor, which baseline MoDs fail represent consistently. The richer representation leads to lower NLL values for NeMo-map, demonstrating its higher accuracy in modeling time-varying human dynamics.
Fig. 5 presents generated MoDs in ETH/UCY datasets. The first row shows the ZARA scene at frame 600, along with the pedestrian trajectories for reference. The dominant motion trend in ZARA is along the horizontal direction. While baseline MoDs mostly conform to this dominant flow, NeMo-map adapts to an emerging upper-left directional trend. The second row shows NeMo-map results for three other ETH/UCY scenes: ETH, HOTEL, and UNIV. Pedestrian trajectories from the entire sequences are shown. Ne.Mo-mp captures diverse motion patterns. For example in the HOTEL scene, pedestrians tend keep right when encountering oncoming flows.
4.4 DOWNSTREAM TASK VALIDATION
We evaluate NeMo-map in the downstream task of long-term human motion prediction (LHMP). Accurate LHMP is essential for various applications, include optimized motion planning, advanced automated driving, and improved human-robot interaction. While short-term predictions can often rely on current state and immediate interactions, long-term predictions demand more comprehensive modeling of human motion patterns. Maps of Dynamics (MoDs) provide way to encode such patterns and can act as a prior that guides motion prediction.
Experiments of LHMP are conducted on ATC using the same testing split in Sec. 4.2 The predic- tion horizon is up to 60s. s. We compare NeMo-map with MoD-based baselines: -HF- (Zhu al. 2023) and STeF-LHMP (Molina et al. 2022). diffusion-basse model (Gu al. 2022 and a transformer- provided in Appendix D
model (Shi et al. 2023 Implementation details and evaluation metrics are
Table[4presentt LHMP performance on the ATC dataset with the prediction horizon of 60 s. Results for other prediction horizons from 10s to 60 are detailed in Fig. NeMo-map achieves the best
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NeMo-map
Online-map
CLipF-map
STeF-mma
Omentallonn (llg))
-10
123
10000 dA <A <Aa
Figure 3: NeMo-map and baseline MoDs in the central open area of the ATC shopping mall. NeMo- map is continuous in space, but for visualization, we query it on uniform grid, while baseline MoDs defined only on discrete grid cells. The top row shows the velocity distributions generated by each MoD. At each location, arrow color encodes motion orientation and arrow length encodes speed. The black circle highlights region where horizontal and vertical flows intersect. NeMo- map preserves both motion modes, while baseline MoDs break the vertical flow continuity. The bottom row shows the corresponding NL.L heatmaps, where lighter values indicate lower NI.L, iee. more accurate representations.
09:00
12:00
15:00
18:00
09:00
12:00
15:00
18:00
STeF-mi -map CLiFF-man Online-man NeMo-map
Figure 4: NeMo-map and baseline MoDs the east corridor, located on the right side of the ATC map (Fig. 7). at different times of the day. The left four columns show the generated velocity fields. Nemo-map captures the temporal variation of pedestrian flows. The right four columns show the corresponding NLL heatmaps. Compared with the baselines, NeMo-map map preserves both the dominant corridor following flow and the crossing flow, leading consistently lower NLL values.
performance among MoD-based predictors, with paired t-tests showing p 0.01, under the null hypothesis that our method yields equal or higher mean error than the baseline.
The improvements arise from NeMo-map's ability provide smooth and accurate motion fields across continuous space and time, in contrast to CLiFF-LHMP, which relies on a grid of SWGMMs and search for nearby cells during prediction. By modeling the velocity distribution continuously, NeMo-map captures more precise local flow patterns that result more accurate long-term pre- dictions. In addition, unlike STeF-map which represents orientation using 8-bin histogram and
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NeMo-map
Online-map
-aip
STeFFmap
ZARA
ETH
HOTEL
UNIV
Figure 5: Top row: NeMo-may map and baseline MoDs in the ZARA scene at frame 600. Pedestrian trajectories are shown as reference. NeMo-map adapts to the changes from dominant horizontal flow to emerging flow toward the upper-left entrance, while baseline MoDs remain aligned with the horizomtal movement. Bottom row: NeMo-map results for the ETH, HOTEL and UNIV scenes with trajectories from the entire sequences. captures multimodal human motion dynamies across different environments.
25
CLIFF-LHMP
NeMo-Map
50
NeMo-Map
CLIFF-LMPP
Obuervaiion
STeF-LHMP
40
STeF-LHMP
Ground
15
MID
E000
MID
NeMtce Map MID TUTR
ADE
20
30
40
60
10 20
40 50
60
Prediction horizon
Prediction horizon
Figure 6: Left: ADE/FDE (mean + one std. dev.) in the ATC dataset with planning horizon 10-60 Right: Human motion prediction with a 60 horizon. The red line represents the ground truth tra- jectory and the green line represents the observed trajectory. Prediction with NeMo-map achieves more accurate and realistic predictions than baseline methods. While the trajectories predicted by TUTR (Shi et al. 2023) and MID (Gu et al., 2022) are unfeasible (e.g.. crossing the walls), predic- tions guided with MoDs follow human flow and implicitly respect the structure of the environment over long horizons.
ignores speed, NeMo-map jointly models both speed and orientation in continuous manner, offer- ing more expressive motion prior. A qualitative example for 60 prediction is shown Fig, 66) Human motion prediction using NeMo-map map generates realistic trajectories that follow the complex topology of the environment, while deep learning-basee based results are unfeasible by crossing the walls.
Table Long-term human motion prediction results on the ATC dataset with a prediction horizon of s0 We report ADE/FDE (mean + one standard deviation), where lower values indicate better performance.
Metric
Ours
CLiFF-LHMP
STeF-LHMP
TUTR
MID
ADE +
5.08 4.28
5.45 + 4.54
5.88 5.55
12.10 + 8.20
21.20 5,28
FDE $
10.97 + 9.92
11.45 + 10.20
12.24 + 12.16
27.26 + 19.23
44.47 + 10.91
4.5
ABLATION STUDY
In this section, we analyze alternative methods for temporal encoding and further include hyper- parameter ablations on the SWGMM component number J and the spatial feature grid resolution H x W in Appendix
In our method, we use a SIREN network to process the temporal input. For comparison, we evaluate two alternative mappings of time into temporal feature vector f,()):

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Temporal grid. A learnable grid G2 E RxxC that captures daily periodicity, where K is the number of discretized time bins (set to 24). The grid feature corresponding to each time bin is concatenated with the spatial feature and passed through an MLP with hidden sizes [128, 64] and ReLU activations.
Fourier features. The time input é is mapped into a periodic embedding using Fourier features Tancik al. (002); Mildenhall et al. (2020). For F frequencies, we construct
f.(()) = 2it))" cos
cos(2"2T))) F-1
This representation enables the model to capture time-dependent variations at multiple res- olutions. The implementation is identical to the temporal grid variant, except the time grid is replaced by Fourier features with F = 4. yielding an 8-dimensional temporal embedding.
Table 5 summarizes the results of the ablation study on temporal encoding. Replacing SIREN with a temporal grid or Fourier features results in higher NI.., confirming the advantage of using SIREN for modeling continuous temporal dynamics. Among the alternatives, Fourier features outperform the temporal grid, but both remain less accurate than SIREN. The reductions in NLL relative to our method are 0.082 for the temporal grid and 0.063 for Fourier features, with 95% confidence intervals.
Table 5: Comparing alternative time encodings for NeMo-map, again using the ATC dataset and comparing average negative log- log-liieli (NLL) where lower indicates better accuracy. We report mean + standard deviation, together with the reduction NLL relative to our method and the corresponding 95% confidence interval (CI). All models are trained using the Adam optimizer with learning rate 10-3 for 100 epochs.
Method
NLL 4
NLL reduction (vs Ours)
95% CI
Ours
0.775 + 2.052
Temporal grid Fourier features
0.857 + 2.113
+0.082 +0.063
[0.081, 0.083]
0.838 + 2.105
[0.062, 0.064]
CONCLUSIONS
We introduced the first-ofitt-kindd continuous spatio-temporal map of dynamics representation NeMo-map, novel formulation of MoDs using implicit neural representations. In contrast to prior discretized methods such as CLiFF-map and STeF-map, our approach parametrizes continuous neural function, which outputs the parameters of Semi- Gaussian Mixture Model at arbi- trary spatio-temporal coordinates. The model enables smooth generalization across space and time, and provides compact representation that avoids storing per-cell distributions.
Through experiments on the large-scale ATC dataset and the outdoor ETH/UCY dataset, we demon- strated that NeMo-map achieves substantially higher accuracy (lower negative log-likelihood) than existing MoD baselines, while offering practical map building times suitable for large-scale datasets. We also demonstrate that the learned flow fields capture multimodality, temporal variations, and en- vironment topology without requiring explicit maps, and achieve higher performance in the trajec- tory prediction downstream task. Ablation studies confirmed the advantage of using SIREN-based temporal encoding over discrete or Fourier alternatives.
In summary. the results highlight continuous MoDs as a practical and scalable tool for modeling human motion dynamics. By combining accuracy, efficiency, and flexibility, the representation offers powerful prior for downstream tasks such as socially aware navigation, long-term motion prediction, and localization in dynamic environments. Despite its advantages, the proposed method has limitations in handling sharp spatial or temporal discontinuities, such as temporary barriers or sudden event-driven changes. Adapting to such changes requires additional observations, which is a limitation shared by existing MoD approaches. In future work, we plan to extend this formulation with online update mechanisms to adapt continuously to evolving crowd behaviors, and to better handle sharp, event-driven discontinuities in motion patterns, further bridging the gap toward long- term weal-wordd deployment.
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ETHICS STATEMENT
The dataset used for training are publicly available and fully anonymized, representing persons only 2D positions without identifiers visual data. Maps of dynamies further aggregate these trajectories into statistical motion patterns, no personal information are retained.
7 REPRODUCIBILITY STATEMENT
For reprodueibility, provide the full training and evaluation code together with detailed instruc- tions in the supplementary material. The package includes our main model as well as the variants used in the ablation study. In addition, provide scripts for generating and visualizing maps of dynamics (MoDs).
ACKNOWLEDGEMENTS
This work received funding from the European Union's Horizon 2020 research and innovation pro- gramme under grant agreement No 101070596 (euRobin), and was supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
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A THE USE OF LARGE LANGUAGE MODELS (LLMS)
Parts of this manuscript were edited with the assistance of LLMs to improve grammar and clarity. All scientific content was written and verified by the authors.
B DATASET
ATC Brrsii et al. 2013) This dataset provides sufficient multí-day coverage for evaluation. ATC was collected in shopping mall in Japan using multiple 3D range sensors, recording pedestrian trajectories between 9:00 and 21:00, over total of 92 days. ATC covers a large indoor environment, with a total area covered of approximately 900 m Because of the large scale of the dataset, we use first four days in the dataset (2012 Oct 24, 2012 Oct 28, 2012 Oct 31, and 2012 Nov 04) for experiments. The data from October 24 is used for training, while the other three days are used for evaluation. The observation rate is downsampled from over 10 Hz to Hz. After downsampling, the training set contains 717,875 recorded motion samples, and the test set contains 5,114,478 samples. ATC dataset environment layout is shown in Fig. 7
ETH (Pellegrini et al., 2009/UCY (Lerner et al., 2007) The ETH/UCY datasets contain pedes- trian trajectories captured in outdoor environments such as university campuses and shopping areas. We evaluate using all four scenes; ETH and HOTEL from the ETH dataset, and UNIV and ZARA from the UCY dataset.
entry/esit paints
hights revers thee exclaco meverro
people
shoae often
thrrugh
trigg Cadded
10m
Figure 7: Layout of the ATC dataset environment BBášči et al. 2013), showing the main east corridor and open areas annotated with semantic information such as entry and exit points, shops, seating areas, and stairs.
C IMPLEMENTATION DETAILS
The output of the network & parameterizes an SWGMM over speed and orientation. For mix- ture components, the network predicts 6J raw values per query coordinate. Each component defined by: a mixture weight Wy. obtained by applying a softmax over the raw weights; mean speed M4ss = max(0, PJ.s) and mean orientation μJa = Íija mod 2T;; variances 03.- exp(clamp(0)... - 10, 10)) and 07,a = exp(clamp(@r.. -10, 10)); and a correlation coefficient Pj = 0.99 tanh Altogether, the network defines a valid SWGMM with parameters as in Eq. (3), where μ4 = (J. Mz.a) and Σj is the covariance matrix with diagonal entries 07... o3,, and corre- lation In the experiments, is set to and coordinates are normalized to [-1, Spatial input processed by MIP with hidden sizes [128, 64) and ReL.U activations. In the ATC dataset, the environment exhibits strong daily regularity and the timestamps are provided in epoch format. We therefore convert them time-of-day to capture long-term temporal patterns. In the ETH/UCY dataset where no such periodic structure is present, we instead use the frame as the temporal vari- able. Temporal input is processed by two-layer SIREN (sine activations with w 30 in the first
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1.4
1.4
Average NLL
Average NLL
0.8
0.6
0.6
2
8
10
12
14
8
16
32
64
128
256
SWGMM components
Feature grid resolution
Figure 8: Hyperparameter ablations the SWGMM component number J and the spatial feature
grid resolution x W.
layer and wp = in the hidden layer). The two streams are fused via FiLM modulation (Perez et al.) 2018). The fused representation is passed to linear head producing 6./ outputs. Models are trained using the Adam optimizer with learning rate 10-3 for 100 epochs. An ablation of alternative temporal encodings is provided in Sec. 4.5
D DOWNSTREAM TASK VALIDATION
In the experiments of long-term human motion prediction, for observations, trajectory and use the remaining (up to the maximum prediction horizon) as the prediction ground truth. We compare our method against MoD-based predictors, Trajectory Unified TRansformer (TUTR, Shi et al. (2023) and Motion Indeterminacy Diffusion (MID, Gu et al. (2022)).
3s of each
TUTR unifies social interaction modeling and multimodal trajectory prediction components in transformer mecoerr architecture. It predicts multiple trajectories with corresponding prob- abilities, and we evaluate using the trajectory with the highest probability. MID formulates trajectory prediction as reverse process of motion indeterminacy diffusion, which gradually discards the in- determinacy to obtain desired trajectory from ambiguous walkable areas, Both MID and TUTR are trained for 100 epochs on both datasets.
For the evaluation metrics, we use Averuge and Final Displacement Errors (ADE and FDE), ADE describes the mean 1 distance between predicted trajectories and the ground truth. FDE describes 2 distance between the predicted and ground truth positions at the last prediction time step. For each ground truth trajectory generate l = 5 prediction trajectories. Based on practical applications for autonomous robots, we evaluate using the most likely predicted trajectory. When probability information for the predicted trajectories is not available, such in the of the baseline MID, we report the mean ADE and FDE values over the predicted trajectories.
E HYPERPARAMETER ABLATION STUDY
For hyperparameter ablations, we study the effect of the SWGMM component number and the spatial feature grid resolution H/ W. Increasing J allows the model to represent more complex human motion behaviors. However, a larger number of components also increases the output di- mension of the MoD representation and leads to higher computational cost during both training and inference. In the left figure of Fig. g performance stabilizes when / 2 3, showing that it is suffi- cient to capture the dominant multimodal motion patterns in human environments. Therefore, recommend starting with J = 3 in practical applications.
For the spatial feature grid resolution x W, finer grid enables better representation of local motion variations, but also increases memory consumption and training time. As shown in the right figure of Fig. 8 the performance remains stable the range from 16 16 to x 64. We therefore recommend starting in this range to have good trade-off between performance and efficiency.
15


Reviews:
[{"review_id": "FFRiXWsjwv", "reviewer": "Reviewer_9dui", "summary": "This paper proposes a continuous representation for Maps of Dynamics (MoDs) that models statistical human motion patterns using implicit neural functions rather than traditional discrete spatial grids.\n\nThe authors replace grid-based discretization with a neural implicit function that maps continuous spatio-temporal coordinates ((x, t)) directly to the parameters of a Semi-Wrapped Gaussian Mixture Model (SWGMM), capturing both linear (speed) and circular (orientation) components of human motion. This approach allows smooth generalization across space and time and removes the need for manual resolution tuning or interpolation in sparsely sampled regions.\n\nThe model architecture combines:\n\n* A learnable spatial feature grid queried via bilinear interpolation.\n* A temporal encoder using SIREN (sinusoidal representation networks) to model daily periodicity.\n* A fully connected MLP that outputs SWGMM parameters for multimodal, continuous motion distributions.\n\nExperiments on the ATC pedestrian dataset* demonstrate that NeMo-map achieves:\n\n* The lowest negative log-likelihood (NLL = 0.775 ± 2.052), outperforming established baselines such as CLiFF-map, Online CLiFF-map, and STeF-map.\n* Smooth temporal adaptation of motion fields without explicit geometric maps.\n\nIn summary, the paper’s key contribution is the NeMo-map, the first continuous spatio-temporal MoD representation using implicit neural fields.", "questions": "1. Could the authors provide qualitative visual comparisons between NeMo-map and the baselines (e.g., CLiFF-map, Online CLiFF-map, STeF-map)?\n2. How well does NeMo-map generalize to motion domains with very different spatial and temporal dynamics (e.g., outdoor environments, vehicle flows, or multi-agent interactions)?\n3. Since one of the motivations is to support robot navigation and trajectory prediction, could the authors show how NeMo-map impacts these tasks compared to prior MoD-based representations?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper addresses the limitation of existing Maps of Dynamics (MoDs)—their reliance on discrete spatial grids—by introducing a continuous implicit neural representation. While this idea of applying neural implicit fields to motion modeling is an incremental extension of trends in continuous scene representations (e.g., NeRF, SIREN), it is relatively new within the specific context of MoD construction for human motion modeling.\n\nThe technical formulation is mathematically sound and builds on established probabilistic foundations (Semi-Wrapped Gaussian Mixture Models). The experimental setup uses a large-scale and reputable dataset (ATC) and provides clear quantitative comparisons with several reasonable baselines (CLiFF-map, Online CLiFF-map, and STeF-map).\n\nThe paper is generally well-written and structured, with clear motivations, illustrative figures (e.g., flow-field visualizations in Fig. 5). The proposed NeMo-map provides a modest but meaningful step forward for motion-aware mapping, particularly for robotics tasks requiring real-time adaptation to human flow patterns. Its computational efficiency and smoothness could make it a practical alternative to existing grid-based approaches in large or dynamic environments.", "weaknesses": "1. Limited novelty and conceptual contribution: While the paper presents a “continuous” MoD via implicit neural functions, the underlying methodological idea—using an implicit neural field to map coordinates to local probabilistic parameters—is conceptually similar to existing implicit representation frameworks (e.g., NeRF, SIREN, and Neural Fields for flow estimation). The novelty primarily lies in adapting these tools to an existing MoD formulation rather than introducing fundamentally new modeling principles. In this sense, NeMo-map reads as an engineering refinement of the CLiFF-map rather than a conceptual breakthrough.\n\n2. Narrow experimental validation and lack of domain diversity: The evaluation is conducted entirely on a single dataset (ATC), which—while large—is limited to indoor pedestrian motion in a controlled shopping mall environment. The paper’s claims about generalization in spatio-temporal motion fields would be more convincing if validated across domains (e.g., outdoor crowd datasets, vehicle or mixed-agent motion). Testing only on ATC raises concerns that the model’s performance gains might be dataset-specific, particularly since temporal periodicity (daily cycles) is a strong prior in ATC but not in all motion environments. Expanding experiments to additional datasets (e.g., ETH/UCY or MOTChallenge) would considerably strengthen the generality claim.\n\n3. Lack of downstream evaluation or task relevance: Although the paper emphasizes applications such as socially aware navigation and long-term prediction, the evaluation is limited to negative log-likelihood (NLL) metrics. These metrics assess representational fit but not utility. It remains unclear whether the proposed NeMo-map actually improves downstream performance in motion planning, collision avoidance, or human trajectory forecasting compared with prior MoD-based methods. Including an evaluation in a planning or prediction pipeline—e.g., comparing robot navigation success rates or forecast accuracy using NeMo vs. CLiFF—would provide stronger evidence of practical benefit.\n\n4. Inadequate comparative visualization of results: The qualitative results (Fig. 5) present visualizations only for the proposed NeMo-map, without showing side-by-side comparisons with baseline methods such as CLiFF-map, STeF-map, or Online CLiFF-map. This omission makes it difficult for readers to visually assess the claimed advantages in smoothness, continuity, or multimodality. Including qualitative comparisons—e.g., overlaying the predicted flow fields of different models in the same regions—would provide a much clearer and more compelling demonstration of NeMo-map’s improvements.", "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": "eKWSUV8XPs", "reviewer": "Reviewer_g9DM", "summary": "This paper introduces a novel approach called Neural Motion Map (NeMo-map) for modeling human motion patterns in complex environments. Unlike traditional Maps of Dynamics (MoDs) that rely on discrete spatial grids, NeMo-map uses a continuous spatio-temporal implicit neural representation. It maps any given (x, y, t) coordinate to the parameters of a Semi-Wrapped Gaussian Mixture Model (SWGMM), enabling smooth and multimodal modeling of velocity distributions. The method is evaluated on the large-scale ATC pedestrian dataset, demonstrating superior accuracy, smoother flow fields, and significantly faster map construction compared to existing baselines such as CLiFF-map and STeF-map.", "questions": "This paper presents an effective approach to modeling human motion dynamics using continuous neural representations. The method significantly outperforms existing baselines in both accuracy and efficiency. While there are areas for improvement—particularly in online adaptability, structural integration. Please see the weakness.\n\nAnd I would like to clarify that I am not a specialist in this particular field. Therefore, my review reflects my understanding and interpretation as a general reviewer, and I hope my comments are still helpful.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This work to apply implicit neural representations to the MoD problem, offering a continuous and differentiable alternative to grid-based methods.\n- The use of SWGMM allows joint modeling of speed and orientation, capturing multimodal and correlated motion patterns effectively.\n- The method achieves the lowest NLL on the ATC dataset and is orders of magnitude faster in training than CLiFF-map.", "weaknesses": "- While the model implicitly learns spatial constraints, it does not explicitly incorporate geometric or semantic map information, which could improve robustness in complex environments.\n- The model assumes daily periodicity in motion patterns, which may not hold in environments with irregular or event-driven dynamics.\n- The proposed method is trained offline and does not support incremental updates or online learning, limiting its applicability in non-stationary environments", "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": "usvBgoxL2a", "reviewer": "Reviewer_xjtj", "summary": "This paper proposes a new way to represent how humans move in an environment. It introduces a continuous map of dynamics called NeMo-map, which models motion as a smooth function across space and time instead of using fixed grids. The method uses an implicit neural network to predict local motion distributions based on spatial and temporal inputs. This continuous approach preserves detailed motion patterns, enables flexible querying at any point in time or location, and avoids the data loss caused by discretization", "questions": "- In Line 146, the authors explicitly state that the proposed method is fundamentally different from trajectory prediction models. However, in Figure 1, to illustrate the application of MoD, the authors present results alongside existing human trajectory prediction models such as Social LSTM and MID. This raises the question of whether such comparisons should be included as part of actual experimental validation, rather than being presented merely as conceptual illustrations. Furthermore, as mentioned in Line 036, “a planner informed by MoDs can exploit prior knowledge of human motion patterns to generate a trajectory that aligns with the expected flow, allowing the robot to reach the goal safely and efficiently. MoDs can also be applied to long-term human motion prediction (Zhu et al., 2023). As shown in the right of Fig. 1, MoDs help predict realistic trajectories that implicitly respect the complex topology of the environment, such as navigating around corners or avoiding obstacles.” Given this description, it would be valuable to verify whether, in the field of human motion prediction or human trajectory prediction, the proposed approach indeed provides better priors that contribute to the performance improvements observed in “With MoD guidance” in Figure 1. Such experiments would substantially reinforce the credibility of the claim that MoD guidance effectively aids motion planning and human motion prediction.\n\n- The authors conducted experiments on the ATC dataset and included the analysis in Section 4.5 (Qualitative Results). However, the explanation could be more detailed. Figure 5 merely shows the dataset distribution across different time periods, but it would be insightful to elaborate on what specific human lifestyle patterns lead to certain behaviors or activities at particular locations and times. Moreover, it would be helpful to clarify under which scenarios(situations) or environmental conditions the model’s predictions are accurate and where they fail. A deeper discussion on these aspects would strengthen the interpretability and practical relevance of the results.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The proposed NeMo-map, unlike existing methods such as CLiFF-map and STeF-map, continuously models space and time, enabling smooth and generalizable representations of human mobility patterns.\n- By leveraging implicit neural representation, it achieves both high expressive power and computational efficiency, while reducing map construction time by more than an order of magnitude compared to previous approaches.", "weaknesses": "- From the perspective of generalization, there remains considerable room to assess the significance of the proposed methodology.\nThe study was validated only on the ATC dataset. Although, as shown in Figure 4, it includes various indoor scenarios and conditions, it still represents a single dataset distribution. Therefore, additional experiments and performance evaluations on other datasets are necessary.\n\n- Since the proposed NeMo-map heavily relies on neural implicit representation, its internal representations are limited in terms of interpretability. Consequently, it is challenging to analyze the model’s decision-making process or evaluate its reliability in real-world robotic operation environments.", "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": "P1HF2KA3GS", "reviewer": "Reviewer_ETtv", "summary": "This paper proposes NeMo-map, a continuous neural implicit representation for modeling spatio-temporal motion dynamics in human environments. Unlike prior discretized Maps of Dynamics (MoDs) such as CLiFF-map and STeF-map, NeMo-map directly maps spatio-temporal coordinates to the parameters of a Semi-Wrapped Gaussian Mixture Model (SWGMM), enabling smooth, multimodal velocity distributions without grid discretization. The model combines a learnable spatial feature grid with a SIREN-based temporal encoder that captures daily periodic variations in human motion, producing continuous flow fields across both space and time. Experiments on a large-scale pedestrian tracking dataset show that NeMo-map achieves higher accuracy and significantly faster training than baseline MoD approaches, while qualitatively generating smooth, topologically consistent flow fields aligned with real-world environment structures. The approach is presented as a scalable and efficient foundation for dynamic scene understanding, motion prediction, and socially aware navigation.", "questions": "please address the concerns above", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper makes a clear and technically meaningful contribution by replacing discrete MoD representations with a continuous implicit field, eliminating the need for grid-based interpolation and yielding smooth, queryable flow estimates across both space and time. The probabilistic backbone—the Semi-Wrapped Gaussian Mixture Model—retains interpretability while enabling multimodal velocity representation, capturing correlations between speed and direction (Eq. 1–2). The use of SIREN-based temporal encoding is a strong design choice, effectively modeling daily periodicities and outperforming Fourier and discrete temporal grids in ablations (Table 3). Experimental validation is solid: the authors benchmark against multiple strong baselines (CLiFF-map, Online CLiFF, and STeF-map) and demonstrate both quantitative accuracy gains and significant efficiency improvements in training time (Table 2). The qualitative results are well-presented (Fig. 5), showing realistic spatio-temporal evolution of human flow patterns that align with semantic structure (e.g., benches, exits, corridors). Overall, the paper is technically rigorous, well-motivated, and carefully evaluated, making a strong contribution to spatio-temporal modeling in robotics.", "weaknesses": "1. The proposed implicit mapping formulation assumes a globally smooth function Φθ(x, t), which may be overly restrictive for environments with sharp spatial or temporal discontinuities (e.g., doorways, temporary barriers, or sudden event-driven flow changes). The model’s reliance on a fixed neural parameterization prevents local adaptation to abrupt changes, and the paper does not provide any mechanism (e.g., hierarchical grids or local residuals) to handle non-smooth motion transitions (Sec. 3.2).\n\n2. The periodicity assumption in time modeling (Sec. 3.2, lines 215–218) simplifies human motion to a daily cycle, which is unrealistic for many public spaces with irregular or event-driven patterns. The evaluation dataset (ATC) naturally exhibits strong daily regularity, but no experiment tests NeMo-map on non-periodic or transient patterns, making the claimed generality questionable. The model’s performance could degrade if queried outside the trained temporal domain or under distribution shifts in time.\n\n3. The evaluation metric focuses solely on negative log-likelihood (Sec. 4.4, Table 1), which measures statistical fit but not flow-level accuracy or spatial consistency—key factors for downstream robotic use. The paper reports no task-based or trajectory-level validation (e.g., motion prediction or navigation performance), making it difficult to assess whether the smoother velocity fields actually improve practical planning or prediction outcomes.\n\n4. While the authors emphasize NeMo-map’s faster training (Table 2), the comparison fairness is limited: CLiFF-map is retrained per hour and cell, while NeMo-map jointly trains across space and time with GPU acceleration. The paper does not normalize for total computation per prediction or per environment, so the reported “two orders of magnitude speedup” may overstate the practical advantage. Moreover, the ablation study (Sec. 4.6) isolates temporal encodings but omits critical architecture factors such as mixture component count (J) or grid resolution (Gs), leaving the method’s scalability characteristics underexplored.", "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)