FlowMaps: Modeling Long-Term Multimodal Object Dynamics with Flow Matching

FlowMaps: Modeling Long-Term Multimodal Object Dynamics with Flow Matching Francesco Argenziano1 Miguel Saavedra-Ruiz2,3 Sacha Morin2,3 Charlie Gauthier2,3 Daniele Nardi1 Liam Paull2,3 1

Sapienza University of Rome, Rome, Italy Université de Montréal, Montréal, QC, Canada 3 Mila - Quebec AI Institute, Montréal, QC, Canada 2

arXiv:2606.20209v1 [cs.RO] 18 Jun 2026

{argenziano,nardi}@diag.uniroma1.it {miguel-angel.saavedra-ruiz,sacha.morin}@mila.quebec {charlie.gauthier,paulll}@mila.quebec

Abstract: Joint spatial and temporal understanding of 3D scenes is a crucial requirement for robots deployed in everyday household environments. Such agents must not only comprehend and navigate spatial layouts, but also reason about how these spaces evolve over time. In particular, humans interact with objects daily, causing them to change position throughout the environment and making it difficult for robots to reliably associate current observations with previously seen objects. However, these interactions are not random: human habits and routines induce spatio-temporally consistent patterns in object locations, which robotic agents can potentially learn and then exploit for downstream tasks such as navigation. To this end, we introduce FlowMaps, a latent flow matching model for estimating multimodal distributions over the future locations of dynamic objects in a continuous 3D space. By learning the implicit dependencies among objects and their temporal evolution, FlowMaps predicts likely changes in object locations conditioned on past human interactions, while supporting generalization across previously unseen environments that share similar object routines. To demonstrate the utility of this method, we deploy FlowMaps in a downstream dynamic Object Navigation task in both simulated and real-world environments. Across more than 600 episodes, FlowMaps outperforms state-of-the-art approaches, showing that modeling object dynamics through continuous, multimodal spatio-temporal distributions improves robotic search and navigation in changing household environments. Code and additional material is available at https://fra-tsuna.github.io/flowmaps/. Keywords: Flow matching, Dynamic environments, Object navigation

1

Introduction

Environments inhabited by humans are intrinsically dynamic. This dynamism should not be attributed solely to the people living in them, but also to the everyday objects they contain. Through repeated human interactions, objects are frequently moved, displaced, and relocated over time, posing major challenges for robots expected to deliberate and act in such environments [1, 2]. Under these conditions, even a seemingly simple task (e.g., “help me find my glasses” [3]) becomes highly challenging: although the robot may have observed an object in the past, there is no guarantee that it will still be found at its previous location, and reasoning about its possible displacement requires considering where it may have moved in physical space. Humans constantly move objects within and across rooms, creating significant difficulties for navigation and retrieval. Nevertheless, humans tend to follow repetitive behavioral patterns [4]. For instance, a pair of glasses may be placed on

the nightstand before sleep, moved to the bathroom sink before a shower, and later returned to the nightstand. These routines are referred to as semantically consistent patterns [5, 6], and can potentially be leveraged to recover from failed retrieval attempts by directing the robot toward likely human-induced object placements. We hypothesize that, given sufficient observations of how objects move through environments over time, a generative model can learn latent regularities induced by human routines and exploit them to predict likely future object placements. Crucially, our key insight is that predicting where objects will be found does not require explicitly identifying the human activity that caused their displacement; instead, such structure can emerge directly from data. To be useful beyond a single observed home, these predictions should generalize to environments with different layouts and object arrangements, while remaining grounded in actual object locations rather than fixed, predefined alternatives such as receptacle labels. Since object displacements are inherently multimodal (i.e., they admit multiple modes, each corresponding to a distinct plausible placement), modeling them requires a distributional approach over continuous space. To this end, we present FlowMaps, a latent flow matching (FM) model that recovers multimodal spatio-temporal distributions of common household objects directly in continuous 3D space. Unlike prior dynamic object location models that primarily operate over discrete receptacle-level relations, FlowMaps models human-induced household object relocalization as a multimodal distribution in continuous 3D space. FlowMaps is composed of two main modules: (i) a Variational Autoencoder (VAE) that learns latent representations of object geometry and semantics, and (ii) a latent Diffusion Transformer (DiT) [7] that predicts likely object locations over time. To obtain training data at scale, we employ ProcTHOR [8] to generate meaningful dynamic object trajectories across procedurally generated household environments. We then demonstrate the usefulness of the learned prior on Object Navigation (ObjNav), using it as a representative downstream robotic task among the broader set of applications that can benefit from predictions of likely object placements. In this setting, FlowMaps is trained on dynamic trajectories from a set of training environments, while ObjNav performance is evaluated in disjoint, previously unseen homes, explicitly testing whether the learned prior transfers beyond the scenes observed during training. Across more than 600 ObjNav episodes in ProcTHOR, we compare FlowMaps against state-of-the-art approaches and show superior performance in retrieving target objects. We further validate our approach on a real robotic platform, showing how it can be deployed in a real-world setting. Contributions statement. Our contributions are three-fold: (i) FlowMaps, a FM-based architecture for representing and predicting continuous, multimodal spatio-temporal object distributions in dynamic environments; (ii) as an exemplary downstream robotic application, we demonstrate the use of FlowMaps for ObjNav in dynamic scenes, enabling a robot to reason about where target objects are likely to be found over time, while outperforming both zero-shot baselines and trained expert policies; (iii) an extensive quantitative and qualitative evaluation of both the distributional properties and practical applicability of FlowMaps, covering more than 600 simulated ObjNav episodes in disjoint environments together with real-world deployments, and demonstrating the soundness, effectiveness, cross-environment generalization, and practical viability of the proposed approach.

2

Related Work

Learning human habits and patterns. Learning human habits enables robots to move from reactive command execution to proactive assistance [9, 10]. Prior work models human activities for anticipation [11, 12], uses interaction histories for personalized collaboration [13], and studies anticipatory robot assistance [14, 15, 16]. The closest work to ours is HOMER [17], which uses a Graph Neural Network (GNN) to predict object displacements from recurring household activities. Unlike HOMER, which trains a separate model per household and mainly evaluates within-environment generalization, we train and test on disjoint environments sharing semantically consistent patterns, assessing generalization to unseen household layouts. 2

Object navigation in dynamic environments. ObjNav aims to locate target objects in known or unknown environments. Existing methods often rely on Large Language Models (LLMs) or Visual Language Models (VLMs) for zero-shot reasoning [18, 19], or learn end-to-end policies [20], but typically assume static scenes. Recent dynamic approaches incorporate object trajectories [21, 18], probabilistic object-location estimates [3, 22], or dynamic scene graph memories [23], yet they depend on costly VLM reasoning, hand-designed priors, online estimation, LLM-interpreted activity hints, or explicit graph structures, and have often been evaluated in limited settings. In contrast, FlowMaps models object dynamics as continuous distributions over future object placements in 3D, rather than discrete temporal link prediction over object-location relations [17, 23]. It requires no explicit scene graph structure at training or inference time, remains efficient through its FM formulation, and outperforms prior dynamic ObjNav methods. Flow matching in robotics. Generative models are increasingly used in robotics, supported by large cross-embodiment datasets such as Open-X Embodiment [24]. Recent work uses diffusion models [25, 26], FM [27, 28], and vision-language-action models [29, 30] mainly for policy learning. In contrast, we use FM for posterior inference, recovering spatio-temporal and multimodal distributions over plausible object placements rather than generating actions. To the best of our knowledge, this is the first use of FM for posterior inference in this setting.

3

Background and problem formulation

Vector fields and flows. An Ordinary Differential Equation (ODE) is defined by a time-dependent vector field u : Rd × [0, 1] → Rd , which assigns a velocity ut (x) to every position x ∈ Rd at each time t ∈ [0, 1]. A solution of the ODE is a trajectory X : [0, 1] → Rd , where Xt denotes the position of the system at time t. In this work, we are particularly interested in flows as solutions of the ODE. A flow is a time-dependent map ψ : Rd × [0, 1] → Rd , written ψt (x) = ψ(x, t), whose evolution is governed by d ψt (x) = ut (ψt (x)), dt

ψ0 (x0 ) = x0 .

The vector field ut is said to generate a probability path (pt )0≤t≤1 if the corresponding flow transports samples from the initial distribution p0 to the distribution pt at each time t. Equivalently, for X0 ∼ p0 , we have Xt = ψt (X0 ) ∼ pt . Conditional flow matching. Conditional flow matching (CFM) learns such a vector field by regressing onto tractable conditional velocities. The goal is to learn uθt whose flow transports a simple base distribution pinit to the data distribution pdata . Instead of directly constructing the marginal vector field, namely the vector field that generates the full probability path from pinit to pdata , CFM defines conditional paths pt (· | y) indexed by data samples y ∼ pdata , together with tractable target velocities utarget (· | y). The training objective is t h i 2 LCF M (θ) = Et,x,y uθt (x) − utarget (x | y) 2 , t ∼ U [0, 1], y ∼ pdata , x ∼ pt (· | y). (1) t Under standard assumptions, this objective has the same gradient as the corresponding marginal FM objective, so minimizing (1) learns the marginal vector field without requiring its explicit evaluation. For a more complete explanation and formalization, we refer the reader to [31, 32]. Latent flow matching. In latent flow matching, the same learning objective is applied in a latent representation space. Given an encoder E and decoder D, data samples x ∼ pdata are mapped to latents z = E(x), inducing a latent data distribution pzdata . A vector field uθt is then trained to transport samples from a simple latent prior pzinit to pzdata by minimizing the CFM objective in d Zt = uθt (Zt ), and decodes the terminal latent space. Sampling draws Z0 ∼ pzinit , integrates dt latent variable as x̂ = D(Z1 ). This reduces the dimensionality of the transport problem and lets the flow operate on compact, semantically structured representations, while the decoder maps generated latents back to the data domain. 3

Problem formulation. At time τ ≥ 0, we represent the map as Mτ = (Oτ , OBG ), where O Oτ = {Oi,τ }N i=1 denotes the set of dynamic objects observed in the scene, and OBG denotes the static background, which is assumed to remain unchanged over time. Each dynamic object is represented as Oi,τ = (bi,τ , li ), where bi,τ ∈ R6 denotes the object’s 3D axis-aligned bounding box in center-size format, bi,τ = (cxi,τ , cyi,τ , czi,τ , sxi , syi , szi ), with (cxi,τ , cyi,τ , czi,τ ) denoting the box center at time τ , and (sxi , syi , szi ) denoting its spatial extent. The label li ∈ L denotes the semantic class of the object. In each scene, there is at most one dynamic object per semantic class, so no two dynamic objects share the same label. Background elements in OBG are represented analogously, but their spatial states are fixed across time. Given a prediction horizon ∆τ ≥ 0 and an object query label lq identifying a dynamic object in Oτ , our goal is to infer the distribution over future bounding boxes of the queried object at time τf = τ + ∆τ . Namely, we want to infer the distribution
p bq,τf | Mτ , lq , τf .

4

FlowMaps

Directly computing this posterior is intractable, as the queried object’s future state depends on rich scene context and may admit multiple plausible outcomes. We approximate it with FM, learning a conditional transport from a Gaussian base distribution p0 = N (0, I) to the target distribution
p1 = p bq,τf | Mτ , lq , τf . Starting from zinit ∼ p0 , integrating the learned flow produces a sample zfinal ∼ p1 , corresponding to a plausible future bounding box of the queried object at time τf . We instantiate this conditional flow in latent space with FlowMaps, a Transformer-based FM model built on the Conditional Diffusion Transformer (CDiT) of Bar et al. [33]. A VAE first encodes object bounding boxes and semantic labels into a latent target space (Section 4.2). A CFM network then learns the flow in this space, predicting future object placements conditioned on the current scene, queried label, and prediction horizon (Section 4.3). Dynamic scene generation and data collection are described in Section 4.1. Architectural, hyperparameter, solver, and interpolation details are provided in the supplementary material. 4.1

Dynamic Scenes Generation and Data Collection

We used ProcTHOR [8] to generate dynamic indoor environments for training both components of FlowMaps. Object movements are driven by predefined human-like routines that produce semantically consistent patterns. We instantiated three representative routines, each capturing a different type of plausible indoor behavior. In Habit #1, the simulated human exhibits location preferences, repeatedly returning to a small set of favored places and therefore spending more time there. In Habit #2, the simulated human follows a balanced routine, distributing time approximately uniformly across the relevant locations in the environment. In Habit #3, the simulated human follows a highly dynamic routine, frequently transitioning between locations and spending only short intervals at each one. For each routine, we generate 2706 training and 918 validation environments. Each scene contains up to 15 dynamic objects moving between semantically compatible receptacles, e.g., a “Fork” can appear on a “Sink” or “DiningTable”, but not on a “ShelvingUnit”. Each environment is simulated for 4 weeks with hourly resolution dτ , yielding 672 timestamps per scene s, with sτ = [τ, Oτ, OBG]. This results in over 1.8M training samples per habit. For each habit, we train and evaluate a separate FlowMaps model. More information on habits can be found in the appendix. 4.2

Variational Autoencoder

We train a VAE to map individual object tokens into the latent space. Each token consists of a normalized 3D bounding box b and a semantic label l, and is encoded into a latent code z ∈ Rdz . Once trained, the VAE is frozen: its encoder eϕ provides the target latents for the CDiT, while its decoder dξ maps generated latents back to object predictions. The decoder is trained to predict both geometry and semantics: its geometry head reconstructs the normalized bounding box parameters b̂, 4

while its semantic head predicts logits over object classes. The network is trained with the standard VAE objective LVAE = Lrec + βt LKL , where βt is linearly annealed during training to prevent posterior collapse [34]. The reconstruction term combines geometric and semantic supervision: Lrec = λCIoU LCIoU +λL1 LL1 +λCE LCE . Here, LCIoU and LL1 supervise the reconstructed bounding box, while LCE supervises the semantic class prediction. 4.3

Latent Flow Matching network

The FlowMaps network parametrizes the velocity field uθt that transports a Gaussian latent z0 to the latent encoding z1 = eϕ (bq , lq ) of the queried object’s future bounding box. It is composed of: (i) a map encoder that aggregates the scene context into a sequence of contextualized tokens, and (ii) a stack of CDiT blocks that iteratively refine the noisy query latent by cross-attending to that context.

Scale

Pointwise Feedforward

Layer Norm

Scale, Shift

Scale

Multi-Head Cross-Attention

Scale, Shift

+

+

AdaLN-Zero

Layer Norm

(a) Map encoder.

Scale, Shift

K, V Layer Norm

Map encoder. At each timestep, the scene is a padded set Mτ of at most NO + NBG object tokens, with maximum length S. Each token represents either a dynamic object or a static background object through a 3D axisaligned bounding box b, semantic label l, and object-type flag fobj . Following Wald et al. [35], this flag acts as a set-membership indicator, where fobj = 1 for dynamic objects and fobj = 0 for furniture/background objects. The map encoder (Fig. 1a) maps Mτ to context tokens Hτ ∈ RS×dh by summing bounding box, class, type, and learned time embeddings, then processing them with NT pre-norm Transformer encoder layers. Self-attention lets each token incorporate scene-level context and relations to other objects. No positional encoding is used across tokens, making the encoder permutation-invariant: spatial structure is provided only by the bounding box embedding.

CDiT Block

(b) CDiT block.

CDiT block. Given Hτ , the flow is implemented by NC CDiT blocks (Fig. 1b) that update the noisy query latent Figure 1: Overview of the latent FM zt ∈ Rdz by cross-attending to the encoded scene. This network components: (a) the map enCDiT-style design, rather than full-attention DiT [7], re- coder, and (b) the CDiT block. flects the asymmetry of the task: a single query object is transported while the scene provides a variable-length context, avoiding repeated quadratic attention over the full scene at each FM step. Cross-attention and feed-forward layers are modulated with adaLN-Zero [7] using a conditioning vector c built from the flow time t ∈ [0, 1], final timestamp τf , and queried label lq . Since the residual stream contains only one query token, we remove the per-block self-attention of Bar et al. [33]; scene-level reasoning is handled by the map encoder, while the CDiT blocks condition the query trajectory through cross-attention. A final adaLN-modulated linear head maps the refined token to the VAE latent space, producing uθt (zt | Hτ , lq , τf ). Training and inference. We train the network with the CFM loss of Eq. 1 on a conditional optimaltransport path [36]. Target latents z1 are obtained by encoding ground-truth future bounding boxes with the frozen VAE encoder eϕ and standardizing them with training-set statistics, while source latents z0 ∼ N (0, Idz ) are sampled independently. For t ∼ T on [0, 1], we set zt = at z1 + bt z0 and regress uθt to żt = ȧt z1 + ḃt z0 with an L2 loss. Source-target pairs are matched by an exact minibatch optimal-transport plan [37, 38], yielding straighter trajectories and fewer integration steps. At inference, given Mτ and (lq , τf ), one map-encoder pass computes the query-independent Hτ . Since Hτ is query-independent, the same encoder pass is amortized across all dynamic objects and ˆt = uθt (ẑt | Hτ , lq , τf ) from z0 to ẑ1 in K fixed steps, posterior samples. We then integrate ż de-standardize ẑ1 , and decode it with dξ into b̂q,τf . Independent z0 draws produce samples from p(bq,τf | Mτ , lq , τf ). The full pipelines are shown in Fig. 2. 5

Interpolant

Robot POV

Inference Vase

Vase

+

Old location

New location

Map Enc

CDiT Blocks

Map Encoder

ODE Solver

Training

Figure 2: Training (left) and inference (right) pipeline for FlowMaps. We train the stack of CDiT blocks to learn the vector field uθt , which is used at inference time to predict b̂q,τf .

5

Experimental Results

Distributional Evaluation. We evaluate FlowMaps using the joint accuracy and distributional metrics reported in Table 1. All entries are averaged over the three considered habits. We report two best-of-K accuracy metrics, following the protocol introduced for multimodal trajectory prediction [39]. For each query, we draw K=50 samples. minFDE@K measures the 3D distance between the ground-truth bounding box center and the closest predicted sample, while Rec@1 reports the corresponding best-of-K recall within δ=1m on the floor plane. To assess the predicted distribution beyond point accuracy, we also report coverage and density following Naeem et al. [40]. Coverage is the fraction of ground-truth positions whose k-Nearest Neighbor ball contains at least one generated sample, whereas density is the average number of such balls occupied by each sample. These metrics separate mode coverage from over-concentration, without penalizing valid multimodal predictions. We further compute TV and JS, respectively the total-variation and Jensen-Shannon divergences between predicted and ground-truth samples discretized as histograms on an xz floor grid with 0.5 m resolution. These divergences measure how spatial mass is allocated across the scene, independently of best-of-K accuracy. We compare FlowMaps against four baselines. FreqPrior estimates a training-set distribution P (r | lq ) over receptacle categories r. At test time, given (Mτ , lq ), it samples a receptacle category, selects an instance of that category in the scene, and samples a point uniformly from the footprint of its top face. This captures scene-conditional object co-occurrence but ignores temporal structure. LLMPrior replaces this empirical distribution with a ranked top-T list of receptacles suggested by an LLM. EmpiricalMean is an oracle point predictor that returns the mean of all ground-truth positions of objects with label lq in the environment, providing a strong upper bound for regression methods with the same (Mτ , lq ) conditioning. MeanFlowMaps collapses FlowMaps samples to their centroid, isolating the effect of multimodality from the conditioning signal. More details on the distributional baselines are provided in the appendix. FlowMaps achieves the best minFDE (0.342 m, 7.6% below the strongest baseline), while nearly saturating mode coverage. It also raises the Naeem density from 0.647 (FreqPrior) to 0.886, a 37% gain, indicating that its samples concentrate more tightly around ground-truth modes. The two single-point baselines are consistently worse on accuracy (minFDE > 2 m, Rec@1 < 0.42); MeanFlowMaps, despite using the same conditioning, is over 7× worse in minFDE, confirming the need for a multimodal generator. FreqPrior matches FlowMaps only in best-of-K recall, whereas FlowMaps also reduces the error tail, increases density, and lowers both TV and JS, showing that it captures where on the plausible surfaces mass concentrates, not just which surfaces are plausible. Object Navigation. We further evaluate FlowMaps on ObjNav, our primary downstream application, using a minival split of 25 random validation environments. For each environment, we select 5 query objects and sample 5 (τ, τf ) pairs per object, obtaining more than 600 ObjNav episodes. An episode is successful if the agent reaches the target within Nsteps steps, is within distance δdist of the object’s ground-truth position, and observes the object from its point of view. All simulated experiments are conducted in AI2-THOR [41] with ProcTHOR [8] environments, where visibility is 6

Table 1: Distributional evaluation averaged across the three habits. Density, coverage, TV and JS are distribution-shape metrics and are reported only for methods that emit a non-degenerate sample distribution. Method FreqPrior LLMPrior EmpiricalMean MeanFlowMaps FlowMaps (ours)

minFDE(m)↓

Rec@1↑

Density↑

Coverage↑

TV↓

JS↓

0.370 0.907 2.040 2.712 0.342

0.958 0.780 0.414 0.260 0.942

0.647 0.773 — — 0.886

0.991 0.967 — — 0.999

0.848 0.796 — — 0.829

0.514 0.471 — — 0.482

assessed using ground-truth information. In the real-world lab demo, a VLM is used only to trigger the candidate “found” signal, while success is still evaluated using the same three criteria, so false positives do not count as successful episodes. We compare FlowMaps against representative dynamic ObjNav baselines: TAP-LGX [21], a zeroshot VLM-based extension of LGX [18]; OSG [3] and CEG [22], which implement the CP-SAT planning approximation of Rudra et al. [3]; HOMER [17] and SGM [23], which model dynamic ObjNav as temporal link prediction with GNNs, with SGM additionally exploiting the scene graph hierarchy; and a Naive LLM baseline directly prompted to rank future object locations. Since OSG and CEG require object location likelihoods, we follow Wang et al. [22] and evaluate scene-prior (SP) variants based on object-receptacle co-occurrences, as well as LLM-based variants that parse human habit hints into relevant events for posterior estimation. All methods return a ranked list of K candidate future locations; for FlowMaps, we obtain this list by running inference Npreds times, clustering predictions with DBSCAN [42], and ranking clusters by mass. Additional baseline details are provided in the appendix. We report Success Rate (SR) and Success weighted by Path Length (SPL) [43], including SR@K, SPL@K, mean SR (mSR), and mean SPL (mSPL) averaged over K. Success at smaller K indicates that the model localizes the object’s future position with fewer trials. We also report the mean path length and mean number of steps required to reach the target. Table 2 shows that FlowMaps consistently achieves the best mSR and mSPL across all settings, while remaining optimal or competitive on the other metrics. Notably, FlowMaps’ SR@1 exceeds the SR@5 of some baselines, indicating that its first prediction can be more accurate than several competing top-5 proposal sets. Among the baselines, GNN-based methods perform best on the easier Habit #1 and Habit #2 settings, but are surpassed by zero-shot LLM-based methods on the harder Habit #3. This suggests that harder behavioral patterns benefit more from transferable dynamic exploration than from directly generalizing learned patterns. Overall, FlowMaps performs best on all three habits, showing stronger dynamic-pattern learning and cross-environment generalization. Real-world deployment. We further showcase our proposed approach in a real-world demonstration with a TIAGo robot1 . The experiment is depicted in Fig. 3: it consists in finding a “CellPhone” that belongs to a person operating in this environment and alternates between the different desks, spending an equal amount of time in each of them (Habit #2). The scene is observed when the phone is at the right desk. A couple of hours pass, and the robot is fetched to find again the phone. Predictions of FlowMaps are clustered together and become ranked navigation targets for the agent. Eventually, the phone gets found at second targeted position.

6

Limitations and Future Directions

The current formulation leaves room for further extensions. First, we consider a closed set of 41 object classes and 17 predefined receptacles, which makes FlowMaps not directly applicable to open-vocabulary settings. This choice was motivated by our focus on indoor household environments, where relevant object classes and receptacles are relatively constrained. Extending the method to open-vocabulary labels could broaden its applicability to more diverse indoor scenarios, such as offices, laboratories, or factories. 1

https://pal-robotics.com/robot/tiago/

7

Habit #1

42.13 35.25 18.69 42.79 22.30 42.95 48.36 46.72 57.70

59.34 57.05 56.89 62.46 58.03 62.62 69.18 67.54 71.15

63.77 62.62 64.10 63.44 64.75 63.61 71.80 69.84 72.62

57.44 54.03 51.85 60.13 52.66 60.29 67.26 64.46 69.26

33.54 28.88 15.34 34.65 18.16 34.93 38.74 37.31 45.31

38.40 40.03 31.46 44.03 33.32 44.57 46.52 44.59 49.52

39.03 44.00 33.21 44.27 34.82 44.81 46.84 44.98 49.72

37.80 43.62 29.29 42.90 30.98 43.40 45.64 43.62 48.95

12.01 10.62 12.99 8.38 12.42 8.00 8.82 9.29 9.01

195.1 193.6 224.3 131.9 212.7 126.3 137.4 145.9 141.1

Habit #2

SR@1 (%)↑ SR@5 (%)↑ SR@10 (%)↑ mSR (%)↑ SPL@1 (%)↑ SPL@5 (%)↑ SPL@10 (%)↑ mSPL (%)↑ Path (m)↓ Steps↓

Naive LLM TAP-LGX [21] OSG+SP [3] OSG+LLM [3] CEG+SP [22] CEG+LLM [22] SGM [23] HOMER [17] FlowMaps (ours) Naive LLM TAP-LGX [21] OSG+SP [3] OSG+LLM [3] CEG+SP [22] CEG+LLM [22] SGM [23] HOMER [17] FlowMaps (ours)

39.02 34.26 19.18 36.89 22.46 37.38 47.54 41.48 50.49

58.52 54.92 57.38 47.38 58.36 48.20 66.23 67.38 66.56

60.66 61.64 66.07 50.16 65.74 50.98 68.52 69.18 67.21

55.46 53.57 52.69 46.43 53.08 47.25 63.85 62.90 63.92

31.30 28.12 15.74 29.85 18.31 30.11 38.07 33.59 41.18

37.95 40.73 32.03 34.97 33.39 35.47 44.83 41.65 46.51

38.21 43.18 34.49 35.69 35.27 36.17 45.14 41.85 46.59

36.94 39.47 29.99 34.40 31.17 34.90 43.94 40.37 45.72

8.72 10.98 13.37 8.14 12.29 8.02 7.76 8.79 7.55

135.2 199.6 233.3 129.5 212.6 127.5 117.6 135.4 113.5

Habit #3

Method

Naive LLM TAP-LGX [21] OSG+SP [3] OSG+LLM [3] CEG+SP [22] CEG+LLM [22] SGM [23] HOMER [17] FlowMaps (ours)

41.15 36.72 15.90 38.52 18.52 38.03 27.21 38.03 50.98

56.23 57.05 56.23 54.26 57.21 54.10 62.62 62.95 66.72

61.48 64.10 65.25 55.74 64.43 55.74 65.41 65.74 68.20

55.46 55.61 51.21 52.33 52.21 52.00 56.84 58.93 64.39

33.52 29.82 12.95 31.72 14.85 31.23 22.19 30.29 41.07

38.31 42.30 31.00 38.97 31.48 38.59 33.64 38.55 45.99

39.21 44.86 33.31 39.33 33.30 38.93 34.02 38.94 46.17

37.89 40.97 28.55 38.04 29.37 37.62 31.84 37.25 45.30

9.45 9.37 11.92 7.68 11.54 7.91 12.24 11.51 7.90

145.1 163.0 207.0 115.7 199.2 119.9 198.1 186.9 117.4

Table 2: Results of FlowMaps and the baselines for the dynamic ObjNav episodes. We report metrics grouped by the three distinct habits that induce the objects displacement.

A second limitation concerns the amount of data needed to train the models. In our experiments, ProcTHOR enabled us to generate sufficient training data for the considered setup. However, extending the approach to more general environments would likely require comparable data, possibly without access to the same simulation tools. One direction to mitigate this issue is to reduce reliance on fully offline training. For examFigure 3: FlowMaps deployed in a real-world environment. ple, after an appropriate sampleefficiency analysis, a compact offline backbone could be trained and then fine-tuned online using data collected during deployment. Finally, the current evaluation is limited to scenarios with at most three habits. While sufficient to assess the proposed formulation, further analysis is needed to understand how the method scales with the number and diversity of habits. These limitations will be investigated and addressed in future work.

7

Conclusions

In this paper, we introduced FlowMaps, a latent FM framework for modeling multimodal spatiotemporal distributions of dynamic objects in human-inhabited environments. Rather than predicting discrete receptacle-level states, FlowMaps models object relocalization as a distribution over future bounding boxes in continuous 3D space. Conditioned on the current scene, queried object, and prediction horizon, it learns recurring human-driven motion patterns directly from data, without explicit habit or activity labels. By combining a VAE-based latent object representation with a CDiT flow model, FlowMaps remains grounded in object geometry while generalizing to unseen household 8

layouts. We evaluated both its distributional properties and practical utility in dynamic ObjNav, where predicted future-location distributions generate ranked navigation targets. Across ProcTHOR simulations and real-world robotic deployments, FlowMaps captures continuous, multimodal, and cross-environment spatio-temporal structure, improving object search and providing an effective prior for robotic reasoning beyond static scene assumptions. Acknowledgments This work has been carried out while Francesco Argenziano was enrolled in the Italian National Doctorate on Artificial Intelligence run by Sapienza University of Rome. This research was conducted while Francesco Argenziano was enrolled as a visiting researcher at Mila - Quebec AI Institute. The work conducted at the Université de Montréal was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through an NSERC Discovery Grant and by the CIFAR AI Chair program (Liam Paull). Individual support was provided through NSERC Postgraduate Scholarships-Doctoral (PGS D) (Sacha Morin, Charlie Gauthier, Miguel Saavedra-Ruiz). This research was also enabled in part by computational resources provided by Mila (mila.quebec).

9

Appendix Table 3: Closed-set semantic vocabularies used for dataset generation. Pickupable objects Oobj

Receptacles OBG

ID

Class

ID

Class

ID

Class

ID

Class

ID

Class

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Apple Egg Fork Kettle Ladle Lettuce Pencil Potato SaltShaker SoapBottle SprayBottle Tomato WineBottle Book

14 15 16 17 18 19 20 21 22 23 24 25 26 27

Bottle Knife Spatula Bread ButterKnife Candle CellPhone Newspaper Vase AlarmClock KeyChain Laptop Pen RemoteControl

28 29 30 31 32 33 34 35 36 37 38 39 40

Spoon Watch BaseballBat BasketBall DishSponge ToiletPaper TissueBox TeddyBear SoapBar PaperTowelRoll DeskLamp TennisRacket Cloth

0 1 2 3 4 5 6 7 8

CounterTop Plate Pot ShelvingUnit Chair DiningTable Dresser ArmChair Bed

9 10 11 12 13 14 15 16

Sink Sofa TVStand GarbageCan SideTable Desk CoffeeTable Box

A

Dataset and environments

Environment classes. We generated the environments using ProcTHOR and adopted the closed set of semantic labels available in the simulator for dynamic pickupable objects and receptacles. This resulted in a vocabulary Oobj of 41 pickupable object classes and a vocabulary of OBG 17 receptacle classes. The former contains the dynamic objects whose locations may change over time, while the latter contains the static receptacles and furniture elements that define the nvironment layout and provide support surfaces for object placement. The complete list can be observed in Table 3. Scene layouts. The generated dataset contains indoor ProcTHOR scenes with diverse spatial layouts and room configurations. Scenes range from compact single-room environments to larger multiroom households, with different floor plans, furniture arrangements, and numbers of receptacle instances. This variability affects both the geometric structure of the environment and the set of valid support surfaces available for object placement. A sample of 30 validation environments can be observed in Fig. 5 Object-receptacle com<MJWJNXYMJ'TTP$9MWJJMFGNYUFYYJWSXFHWTXX\JJPX -FGNY patibility and habitSNY conditioned schedules. To 8MJQ[NSL: QJ G F avoid arbitrary or seman- )NSNSL9 TU YJW9 S Z tically invalid placements, (T -FGNY we use ProcTHOR’s SNY : L S pickupable and receptacle 8MJQ[N GQJ weighting files to deter- )NSNSL9F U T 9 mine which pickupable (TZSYJW -FGNY object classes can be asSNY : L signed to which receptacle 8MJQ[NS QJ classes. These metadata )NSNSL9FG U T define a class-level com- (TZSYJW92TS 9ZJ <JI 9MZ +WN 8FY 8ZS 2TS 9ZJ <JI 9MZ +WN 8FY 8ZS 2TS 9ZJ <JI 9MZ +WN 8FY 8ZS 2TS 9ZJ <JI 9MZ +WN 8FY 8ZS 2TS 9ZJ patibility prior between Figure 4: Different spatio-temporal behaviors of the same object objects and support sur(“Book”) under different human habits in a ProcTHOR scene. faces, ensuring that each object is placed only on semantically plausible receptacles. For example, kitchen objects such as Fork or Tomato may be assigned to counters, sinks, plates, or dining tables, while unrelated receptacles are excluded from their candidate sets. For each object class o ∈ Oobj , we define the compatible receptacle set as R(o) ⊆ OBG . Then, for each object instance in a scene, we select candidate receptacle instances whose class belongs to R(o).

10

Figure 5: An overview of 30 ProcTHOR environments coming from the validation split. Each dynamic scene is generated by assigning every pickupable object a habit-conditioned trajectory over this candidate set. The compatibility prior constrains which receptacles are valid, while the habit determines the temporal structure of the object’s receptacle sequence. We consider the three habit families described in the main text. Habit#1 captures a preference-driven routine, where the simulated human spends more time in a few favored locations. Habit#2 represents a balanced rou11

Table 4: Tokenization and bounding box encoder hyperparameters. Parameter

Value

Sequence length S Hidden width d Center normalization range [cmin , cmax ] Log-size offset εs Bounding box branches Bounding box MLP depth Label/flag MLP depth Linear init, bounding box branches and label MLPs LayerNorm init

32 256 [−0.5, 34.0] m 10−4 4 translation/size × direction/magnitude branches 4 layers, GELU 2 layers, SiLU N (0, 0.022 ), bias 0 gain 1, bias 0

tine, with time distributed roughly evenly across relevant locations. Habit#3 models a more dynamic routine, characterized by frequent transitions and short stays. An example of such rouines can be observed in Fig. 4. Continuous placement by potential-field sampling. The habit schedule specifies the target receptacle for each object at each timestamp, but not a single deterministic 3D placement. To obtain continuous object locations, we sample placements from a potential field defined over the support surface of the selected receptacle. The sampled surface point determines the continuous position of the object on the receptacle, and the object is then teleported to the corresponding pose in the simulator. This separates the discrete semantic component of the data generation process from the continuous geometric one: the habit determines which receptacle an object moves to, while the potential field determines where on that receptacle surface the object is placed. This procedure produces trajectories in which objects follow semantically consistent, habit-conditioned patterns while still exhibiting continuous spatial variation. As a result, the model is not trained to predict only discrete object-receptacle relations, but rather full future bounding boxes in continuous 3D space.

B

Implementation Details

B.1

Tokenization and encoding strategies

Bounding box normalization. Because ProcTHOR generates scenes with varying layouts, we normalize all bounding box center coordinates to [0, 1] using fixed global bounds, cmin and cmax , computed from the extrema observed in the training set. Since object sizes vary substantially across semantic categories, for example between an “Egg” and a “Bed”, we map bounding box extents to log-space and normalize them using the observed log-size range in the training data. Scenes are then tokenized to a fixed sequence length S = 32, as no environment exceeded this number of receptacles plus dynamic objects. Embeddings. Each normalized bounding box is embedded through four separate branches, corresponding to translation magnitude, translation direction, size magnitude, and size direction. Each branch is a 4-layer Multi Layer Perceptron (MLP), implemented as two stacked Linear → GELU → Linear blocks with hidden width d = 256. Translation- and size-group sums are individually layernormalized, summed, and passed through a final layer norm to produce the per-token geometric embedding. Semantic labels and the dynamic/background flag are embedded using learned tables of width d, each followed by a 2-layer Linear → SiLU → Linear MLP. Receptacle and dynamic-object classes use separate embedding tables to reflect their different cardinalities and priors. Hyperparameters are summarized in Table 4. B.2

Variational Autoencoder

A general overview of the VAE can be observed in Fig. 6. The encoder eϕ and decoder dξ are residual MLPs with hidden width dvae = 256, depth Lvae = 3, and an MLP expansion ratio of 4. Each residual block is a Linear → LayerNorm → SiLU → Dropout → Linear → LayerNorm → SiLU → Dropout 12

Table 5: VAE hyperparameters. Parameter

Value

Latent dimension dz Hidden width dvae Depth Lvae MLP expansion ratio Dropout Linear init

32 256 3 4 0.1 Xavier-uniform, bias 0

Iterations Batch size Optimizer Learning rate (β1 , β2 ) ε Weight decay LR schedule

30,000 256 AdamW 10−4 (0.95, 0.999) 10−8 10−6 cosine, 500 linear-warmup steps

λCIoU λL1 λCE Final KL weight β KL warm-up steps Tkl

1.0 5.0 1.0 10−2 10,000, linear from 0 to β

stack with a skip connection. The encoder input is the bounding box embedding summed with the class-label embedding described in Section B.1; its final linear layer projects to 2dz and is split into (µ, log σ 2 ). The decoder mirrors the encoder and terminates in two heads: a 6-dimensional geometry head with sigmoid activation, so outputs stay in [0, 1]6 , and a |Oobj |-way class-logit head. The CIoU term is comVAE pretraining puted in metric space after inverting the log-size mapping, so that the area/volEncoder Decoder ume penalty reflects actual r overlap rather than overlap in the normalised representation. After training, per-dimension latent statistics (µD , σ D ) are computed once on the training Bread ButterKnife Egg Kettle Potato Tomato Vase split and cached on disk. The FM stage operates en- Figure 6: VAE used to encode and decode object tokens in a ProcTHOR environment. Dotted bounding boxes mean decoded objects. tirely on standardized latents (z − µD )/σ D , with the inverse transform applied to predicted latents before decoding. All other architectural and optimization hyperparameters are reported in Table 5. B.3

Latent Flow Matching Network

Map encoder. The map-encoder layers are pre-norm Transformer Encoder modules with GELU activations. The dynamic/background flag is implemented as a 2-entry learned parameter, with one d-dimensional vector per type, added on top of the bounding box, class, and observation time embeddings before the first layer. CDiT block. The adaLN-Zero projection is a single SiLU → Linear(d, 8d) MLP whose output is split into eight chunks (γ1 , β1 , α1 , γ2 , β2 , α2 , γ3 , β3 ). The first triple modulates the query token and gates the cross-attention residual. The second triple modulates and gates the feed-forward sublayer. The last pair modulates the context tokens, namely keys and values, before the cross-attention sublayer. The feed-forward sublayer is a two-layer Linear → GELU → Linear MLP with the same 13

Table 6: Latent flow-matching network hyperparameters. Parameter

Value

Hidden width d Attention heads H MLP expansion ratio Map encoder depth NT CDiT block depth NC Dropout

256 8 4 8 8 0.1

Latent input embedder Flow time embedder τ , τf embedders Query label embedder Conditioning vector c

2-layer MLP, dz → d, SiLU sinusoidal, 256, plus 2-layer MLP separate Tmax × d tables 2-layer MLP, separate table emb(t) + emb(τf ) + emb(lq )

Init: adaLN-Zero and final gate Init: other linear layers Init: embedding tables, class/time MLPs

zero Xavier-uniform N (0, 0.022 )

expansion ratio 4 as the map encoder. Cross-attention uses H heads with biases on the key/value projections. Conditioning embeddings. The flow time t ∈ [0, 1] is encoded with a 256-dimensional sinusoidal frequency embedding followed by a 2-layer Linear → SiLU → Linear MLP. The final timestamp τf and the observation time τ use separate learned tables of size Tmax × d. The queried object label lq uses its own 2-layer-MLP class embedder, distinct from the map-encoder object table, so that the query and map tokens are not tied. The conditioning vector fed to each CDiT block is the sum of the embeddings of t, τf , and lq . The noisy latent zt is lifted from dz = 32 to d = 256 by a 2-layer Linear → SiLU → Linear input embedder. Initialization. All adaLN-Zero projections and the gate projection of the final output head are zeroinitialized so that the network starts as an identity map on the noisy latent [7]. Other linear layers are initialized with Xavier-uniform. Embedding tables and the projection MLPs of the class and timestep embedders use N (0, 0.022 ). Architectural hyperparameters are reported in Table 6.

B.4

FlowMaps Training and Sampling

We adopt the CondOT path [36] with σt = 0, which reduces the conditional probability path to the straight-line interpolation zt = (1−t)z0 +tz1 with constant target velocity utarget = z1 −z0 . Within t each mini-batch, source and target samples are paired by an exact Hungarian solver on the squaredEuclidean cost ∥z0 − z1 ∥22 . Only z0 is permuted, leaving z1 and every conditioning variable in their original order so that semantic alignment with (Mτ , lq , τf ) is preserved. We sample t = sigmoid(s) with s ∼ N (0, 1), giving a logit-normal flow-time distribution. This concentrates supervision on the middle of the trajectory, where the target velocity has its largest variance. We also experimented with sampling t from a uniform distribution U [0, 1] and from a Beta distribution, but empirically found the logit-normal schedule to yield the best results. During inference, we integrate with a fixed-step Euler solver on a uniform grid over [0, 1], with a step of 0.05. Smaller steps and different intregration methods (e.g., midpoint) produced no measurable improvement. The map encoder is run once per scene and its output reused across all dynamic objects in the scene and across the Npred posterior samples drawn for each query. Training and sampling hyperparameters are summarized in Table 7. 14

Table 7: Flow-matching training and sampling hyperparameters. Parameter

Value

Iterations Batch size Optimizer Learning rate (β1 , β2 ) Weight decay LR schedule Validation interval

30,000 1024 AdamW 10−4 (0.95, 0.999) 10−6 cosine, 500 linear-warmup steps every 2,500 iterations

Path Coupling Flow-time sampler Loss

CondOT [36], σ = 0 exact mini-batch OT, Hungarian logit-normal, s ∼ N (0, 1) squared-L2 CFM regression

EMA decay EMA inverse-gamma / power EMA warm-up

0.9999 1.0 / 0.75 none

ODE solver, inference Step size

euler 0.05

C

Experiments

C.1

Distributional Evaluation: Experimental Details

We evaluate on a shared minival split of 25 validation environments, identical across the three habits described in the main paper. For each scene with T hourly timestamps we select nτ evenly-spaced observation times τ ∈ [0, T −2] and, for each τ , up to 10 evenly-spaced query times τf ∈ (τ, T −1], giving at most 30 (τ, τf ) pairs per (env, lq ). For each pair we draw K=50 samples from the predictor. Per-pair, per-instance minFDE@K and Rec@1 are then averaged over all (τ, τf ) pairs and all instances of class lq in the environment. In contrast, the distributional shape metrics are computed once per (env, lq ) using pooled point clouds. The predicted cloud is obtained by concatenating the K sampled centers across all pairs, yielding M ≈ 1500 points. The ground-truth cloud is the union of the bounding box centers of every instance of class lq at every evaluated τf . Regarding the metrics, minFDE uses the full 3D Euclidean distance to the ground-truth bounding box center, while Rec@1 uses xz-only distance with δ=1m. Density and Coverage follow Naeem et al. [40] with kNN =5, augmented with a 0.5m radius floor that prevents degenerate k-NN radii on classes whose ground-truth positions coincide across timestamps (e.g., static objects on the same receptacle). TV and JS are computed on an xz floor grid of resolution 0.5 m, with JS reported in nats (bounded by ln 2 ≈ 0.693). We rely on sample-based histogram divergences rather than the change-of-variables likelihood obtainable from the flow itself for two distinct reasons. First, the inverse-ODE likelihood is defined only for FlowMaps, and even there only in the VAE latent space the flow operates on; the VAE decoder is not invertible, so a likelihood in 3D position units would require an importance-weighted estimate that introduces its own variance and bias. The baselines fare no better: FreqPrior and LLMPrior have closed-form densities, but those are two-dimensional delta measures concentrated on receptacle top surfaces, while EmpiricalMean and MeanFlowMaps are point masses. In every, case the pointwise log-likelihood at a generic 3D ground-truth coordinate is either −∞ or ill-defined, and the only way to make the numbers comparable is to fit a separate density estimator (e.g., kernel density estimation) on each method’s samples, which introduces additional hyperparameters and confounds the comparison. Second, even within FlowMaps the inverse-ODE likelihood scores how much density the model assigns at the ground-truth coordinates, whereas the downstream use of FlowMaps consumes samples drawn from the model rather than its evaluated density: a metric defined on the predicted samples is therefore more directly aligned with the actual object of comparison. TV and JS satisfy both requirements simultaneously: 15

they are computable from any baseline’s samples without further fitting, and they measure how the predicted mass is spatially allocated across the scene rather than how dense it is at the reference point. FreqPrior. This baseline represents a class-conditional categorical over receptacle categories. From the training split of the current habit we estimate, for each pickupable class lq and receptacle category r, the empirical frequency of lq resting on instances of r, and store the resulting table P (r | lq ) alongside the data. At test time, for each of the K samples we (i) restrict P (· | lq ) to receptacle categories present in Mτ and renormalize, (ii) sample a category r, emph(iii) sample a concrete receptacle of that category uniformly, and (iv) sample a point uniformly inside its footprint polygon. The prior is time-invariant and the scene receptacles are read at τ and reused for any τf , so this baseline has no notion of time. LLMPrior. LLMPrior uses the same point-sampling pipeline as FreqPrior, but the distribution over receptacle instances is replaced by a ranked top-T list suggested from an LLM. We use llama3.1:8b as our LLM, with T =5, sampling temperature 0, and a rank-softmax temperature of 1.0, so the rank-i receptacle receives mass proportional to exp(−i). One LLM call is issued per (env, lq , τ, τf ): the returned indeces induce the receptacle distribution, and the K predicted points are obtained by drawing receptacles from it and sampling uniformly inside the chosen footprint at its top face. The system prompt remains unchanged across queries. For each query, we generate the user prompt by filling the bold-bracketed placeholders in the box below. These placeholders specify the observation time τ , the scene state Mτ , the queried class lq , the receptacle that currently contains the target, the elapsed time τf − τ , the query time τf , and the top-T size. The scene state is written as a bullet list over receptacles, with each line reporting the pickupable categories currently on that receptacle, or “empty” when none are present. Timestamps are formatted as Week W, Weekday HH:00 starting from Week 1, Monday 00:00, and elapsed time as X day(s) and Y hour(s).

System Prompt: You are an agent that has to find an object that may have moved from its previous location. You will be given the layout of receptacles in a scene and the object’s previous location. Reason about where the object is most likely to be now. User Prompt: You are an agent that has to find an object that could have moved from its previous location. You know that on [τ ] the environment has the following layout: [Mτ receptacle list]. Also, on [τ ], the [lq ] is on [current receptacle of lq ]. [τf − τ ] later, it is now [τf ]. From the list of receptacles above, enumerate the top [T ] most likely receptacles where the [lq ] can be found on [τf ], ordered from most to least likely. Reply with one receptacle id per line, exactly as written in the list above. No commentary, no numbering, no extra text.

When the queried object is not assigned to any receptacle at τ , the “[lq ] is on [current receptacle of lq ]” clause is replaced by “the [lq ] is not on any of the listed receptacles”. EmpiricalMean. This baseline represents an oracle point predictor: for each (env, lq ), it returns the centroid of every ground-truth bounding-box center of class lq pooled across the scan’s timestamps, replicated K times. The predictor consults the evaluation set and is independent of τ and τf ; since this centroid minimizes L2 error against the pooled ground-truth cloud, it upper-bounds any deterministic regression baseline conditioned on at least (Mτ , lq ). Its distribution-shape entries in the main results table are not reported since these metrics are not informative for a predictor that emits K identical samples. MeanFlowMaps. This baseline is a wrapper around the trained FlowMaps predictor that, per query, draws Kint =50 samples internally, returns their arithmetic mean, and replicates that point K times in the output. The conditioning (Mτ , lq , τf ) is inherited from FlowMaps; the only operation removed is multimodality, which isolates its contribution to the metric gains. Its distribution-shape entries are not reported for the same reason as EmpiricalMean. 16

C.2

Object navigation: experimental details

We evaluate on the same minival split of 25 validation environments used in the distributional protocol, with the agent embodied in AI2-THOR through ProcTHOR scene assets. For every environment we sample 5 target objects, and for each target we draw 5 (τ, τf ) pairs (the observation timestamp at which the agent perceives the scene, and the future timestamp at which we ask it to retrieve the object), yielding 600+ episodes per habit. Each method receives, at the start of an episode, the per-timestep data that backs the scene up to τ (the observation history of receptacle-object assignments) and is asked to produce a ranked list of K candidate future receptacles for the target at τf . The agent then navigates the AI2-THOR controller toward those candidates in order, querying GetShortestPathToPoint for the planner-relevant distances, with a per-episode budget of 1000 environment steps. An episode is successful at depth K if any of the first K candidates places the agent within δdist = 1.25ms of the target and the target is observed from the agent’s point of view. In simulator runs, observation is assessed using AI2-THOR ground-truth visibility. In the real-robot demonstration, the VLM is used only to trigger a candidate target observation; a trial is counted as successful only when this trigger occurs at a valid target location, so VLM false positives are not counted as successful detections. For FlowMaps, the ranked candidate list is built by drawing Npreds =25 samples per query from the trained flow, clustering them with DBSCAN (ε=1.0 m, min =2) and ranking clusters by mass. Ties are broken by the cluster centroid that minimises the distance from the current agent pose. Across all baselines we report Success Rate, SPL, and their per-K variants exactly as in the main results table. TAP-LGX. We follow the formulation of [21], which extends the zero-shot exploration of LGX [18] with a rolling transit memory of past observations. At each LGX step the agent performs a 360◦ scan, the visible objects are fed into a VLM together with the current memory, the VLM returns a single object name, and the agent advances a fixed distance toward it; failures and target sightings are appended to the memory. We reproduce the published exploration loop in full: the LGX stepdistance is 2m, the per-episode budget is 30 LGX steps, and the memory horizon is the last 30 entries; on lock-on the agent navigates directly to the detected target. We deviate from the original setup in three points. First, we substitute the VLM with the same llama3.1:8b used elsewhere in our evaluation, constraining its output to the visible-object set with a JSON schema so the language model behaves as a discrete chooser instead of a free-form generator.Second, we replace the RGBframe YOLO detections used in the original paper with ground-truth visibility queries from AI2THOR, filtered to the LGX vocabulary. This is a substantial strengthening of the perception input, since it removes detector noise and gives the system the same perception-noise floor used by the other baselines in the table. Third, we provide input-parity to the other basilines, we pre-seed the rolling memory with the full per-receptacle scene state at every past timestep t ≤ τ , and grows during exploration with one entry per LGX step recording the agent’s pose, the list of visible objects of the current scan, and the heading the LLM chose. The system and user prompts are shown in the box below: the system prompt renders the rolling memory verbatim, the user prompt is issued once per LGX step to elicit a single visible-object choice, and the bold-bracketed placeholders are filled per query with the target class lq , the current memory contents, and the set of objects detected at the current scan.

System Prompt: I am a smart robot trying to find a [lq ] in my house. The target sometimes moves between receptacles over time; the lines below record where I have seen it or the scene before, and what I observed as I walked around. Use this to reason about where it likely is now. Memory: [rolling memory: one line per past entry, e.g. “[step s] near (x, z) heading h deg observed [obj list] → went toward a” or “[prior t] target on r”; pre-seeded scene-state lines in TAP-LGX] User Prompt: I want to find a [lq ] in my house. Which object from [visible objects at the current scan] should I go towards? Reply in ONE word.

17

CEG. The Cost-Effective Greedy planner of Wang et al. [22] can be viewed as a multiplicative simplification of the original OSG additive blend described below. It ranks unvisited receptacles by the ratio pi /di , where pi is the estimated probability that receptacle i currently contains the target, and di is the navigation distance from the agent to that receptacle. At each step, the agent visits the highest-ranked receptacle. If the target is not found, the corresponding probability mass is set to zero, the remaining masses are renormalized, and the procedure continues until the target is found, all receptacles have been visited, or the per-episode receptacle budget is exhausted. We set this budget to 10, matching the typical scene size. Navigation distances are computed using AI2THOR’s GetShortestPathToPoint. To avoid numerical artifacts when receptacles are collocated or nearly collocated, we use a 0.25m epsilon in the denominator of pi /di , matching the ProcTHOR grid resolution. We evaluate two variants of the probability estimate pi . The scene-prior (SP) variant uses the per-class receptacle co-occurrence table P (r | lq ), which is the same table used by FreqPrior in the distributional evaluation. The LLM variant instead queries llama3.1:8b for an unnormalized probability for each receptacle. The LLM estimator prompt, shown in the box below, follows the structure of the paper, but replaces the “simulated resident” human-activity hint with one of three habit-specific lines. • Habit #1: ‘‘This [lq ] has a few favorite places. Most often on [r1 ] ([p1 ]%), sometimes [r2 ] ([p2 ]%), and occasionally [r3 ] ([p3 ]%).’’ • Habit #2: ‘‘This [lq ] doesn’t have a favorite spot. likely to be on any receptacle it can sit on.’’

It is equally

• Habit #3: ‘‘This [lq ] moves between receptacles quickly without dwelling. Past placements aren’t strong predictors of where it is now. Recently seen at [r1 , r2 , r3 ].’’ where ri and pi are filled from the target’s empirical placement history in the environment. This grounds the LLM estimator in the dataset’s residence model in the same way the original paper paper used the simulated resident to elicit human knowledge. System Prompt: You are an expert assistant that estimates where household objects are likely to be found. Given a scene’s receptacle list and the object’s previous location, you must output a probability for every receptacle, indicating how likely the object currently rests on it. Probabilities may be unnormalized but must be non-negative. User Prompt: Behavioral context: [habit-specific hint, see CEG description above]. You are an agent that has to find an object that could have moved from its previous location. On [τ ] the environment has the following layout: [Mτ receptacle list]. Also, on [τ ], the [lq ] is on [current receptacle of lq ]. [τf − τ ] later, it is now [τf ]. Estimate the probability that the [lq ] is on each of the receptacles listed above on [τf ]. Probabilities may be unnormalized but must be non-negative. Reply with one line per receptacle in the exact form: <receptacle id>\t<probability>. Use every receptacle id from the list, no extra commentary or numbering.

OSG. The One-Step Greedy planner of [3] uses the original additive-blend score from which CEG α can be seen as a simplified variant. It ranks receptacles according to dip + (1 − αp )pi , where di is the navigation distance to receptacle i and pi is the estimated probability that it contains the target. The coefficient αp controls the trade-off between cost and reward: αp = 1 yields purely distance-based selection, while αp = 0 yields purely probability-based selection. We set αp = 0.5, following the default used in [22]. All remaining components are kept identical to the CEG setting described above. In particular, we use the same two probability estimators, namely the scene-prior table P (r | lq ) and the LLM-based variant; the same online visit-fail-zero-renormalize procedure; the same AI2-THOR GetShortestPathToPoint distance computation; the same 0.25m denominator guard; and the same per-episode receptacle budget of 10. The LLM variant also uses the prompt shown in the previous box. HOMER. We adopt the spatio-temporal object-tracking model of [17]. The model represents each scene as a heterogeneous graph whose nodes correspond to rooms, furniture, and objects, and whose edges encode containment and spatial proximity. Given this graph and the observation history, a 18

Graph Translator Network predicts, for each pickupable object, a probability distribution over its possible receptacles at the next timestep. We train the network on our training split using the official implementation and training protocol. However, instead of fitting one model per household, we train a single cross-environment model. This setting is aligned with the generalization regime evaluated by FlowMaps: the same checkpoint is applied to all minival environments, without any per-house fine-tuning. At test time, we run the network on the environment scene graph and the observation history up to time τ . We then apply a softmax over destination receptacle nodes and extract the row corresponding to the queried target object. This row defines the predicted receptacle distribution at the future time τf . The top-K = 10 receptacles under this distribution are returned, in ranked order, as navigation goals, preserving the one-shot prediction semantics of the original method. SGM. We use the Scene Graph Memory (SGM) predictor of [23]. SGM represents the environment as a hierarchical scene graph, with nodes organized as HOUSE → ROOM → FURNITURE → OBJECT, and applies a HEAT neural edge predictor over this structure. The graph is augmented with temporal node and edge counters, which summarize the observation history up to time τ . Given a queried object, the predictor scores hypothetical edges between that object and each candidate receptacle. These scores are converted to sigmoid probabilities and used to rank receptacles directly. We take the top-K = 10 ranked receptacles as the navigation-goal sequence. Our implementation follows the published SGM architecture, featurizer, prior, and temporal-counter pipeline. The only departure from the original per-household training setup is that we train a single cross-environment checkpoint. This matches the generalization setting evaluated by FlowMaps: the same model is applied to every minival environment, with no per-house fine-tuning. Naive LLM. This approach consists in a direct prompting baseline that asks the LLM to rank the receptacles by likelihood of currently hosting the target. The model and the prompting template are identical to those of the LLMPrior baseline used in the distributional evaluation, with the single difference that the requested top-T is raised from 5 to 10 to match the ranked-list size of the other ObjNav baselines. We refer to the distributional appendix for the verbatim system and user prompts. C.3

Ablation Studies

We additionally report a full suite of ablation studies to display how each components of our architecture affect the final results. In particular, we consider the following ablation cases. A first family probes the role of semantic identity, that is, whether the model benefits from knowing what the entities in the problem are rather than only where they sit. In the no-query-class case we drop the class of the query object from the conditioning vector, forcing the model to place an object without knowing its category. In the no-scene-class case we remove the per-token class embeddings from the scene map, so that furniture and object tokens are described purely by their bounding boxes, with no semantic label. In the no-type-emb case we remove the coarser type embedding that distinguishes furniture tokens from object tokens, testing whether even this binary distinction within the scene carries useful signal. A second family examines two architectural and encoding choices. In the no-scene-encoder case we remove the Transformer encoder that performs self-attention over the scene tokens before they are consumed by the diffusion backbone, letting the cross-attention layers reason over the raw scene representation directly. In the linear-bbox-embedding case we replace our structured three-dimensional bounding-box embedding, which decomposes each box into separate direction and scale components processed by dedicated MLPs, with a single linear projection, in order to assess whether this structured decomposition is worth its additional complexity. The ablations are reported in Table 8 and Table 9. Overall, the ablations show that semantic information is the dominant factor. Removing the query class causes the largest degradation, especially in ObjNav metrics, indicating that the model needs the target object category to generate placements that are useful for downstream navigation. Removing scene-level class information also substantially hurts both distributional quality and navigation performance, while removing only the coarse object/furniture type embedding has a milder but still consistent effect. The architectural ablations 19

minFDE(m) ↓

Rec@1↑

Density↑

Coverage↑

TV↓

JS↓

FlowMaps (full)

0.342

0.942

0.886

0.999

0.829

0.482

no-query-class no-scene-class no-type-emb no-scene-encoder linear-bbox-emb

0.438 0.400 0.377 0.372 0.376

0.930 0.932 0.937 0.934 0.934

0.438 0.636 0.674 0.805 0.755

0.997 0.998 0.997 0.995 0.995

0.922 0.894 0.885 0.844 0.865

0.580 0.544 0.533 0.498 0.514

Method

Table 8: Distributional metrics ablations. Results are averaged over the three habits.

Method

SR@1↑

SR@5↑

SR@10↑

mSR↑

SPL@1↑

SPL@5↑

SPL@10↑

mSPL↑

FlowMaps (full)

53.06

68.14

69.34

65.86

42.52

47.34

47.50

46.66

no-query-class no-scene-class no-type-emb no-scene-encoder linear-bbox-emb

19.18 39.02 47.21 47.21 47.05

47.38 59.02 63.61 64.59 61.97

50.82 59.51 64.59 65.25 63.11

43.33 55.52 60.79 61.51 59.67

15.15 31.31 38.03 37.92 37.58

24.30 37.83 43.42 43.47 42.75

24.79 37.90 43.61 43.54 42.90

23.02 36.90 42.65 42.62 42.00

Table 9: ObjNav ablations. Results are averaged over the three habits.

are less severe, but still meaningful: removing the scene encoder or replacing the structured bounding box embedding with a linear projection reduces performance across most metrics. This suggests that both contextual reasoning over scene tokens and the structured geometric encoding contribute to the final model, although their impact is smaller than that of semantic conditioning. Failure analysis. We further analyze how navigation episodes terminate by decomposing each run into successful discoveries at different ranked modes and two failure cases: mode exhaustion, where all proposed targets are visited without observing the object, and step-budget failure, where the agent runs out of steps before completing the ranked search. Fig. 7 reports this decomposition for FlowMaps, HOMER, and SGM across the three habits. Across all habits, FlowMaps concentrates a larger fraction of successful episodes in the first predicted mode. In Habit #1, FlowMaps finds the target at the first mode in 57.7% of episodes, compared with 46.7% for HOMER and 48.4% for SGM. The same trend holds in Habit #2, where FlowMaps reaches 49.7% first-mode success, and in Habit #3, where the gap becomes largest: 51.0% for FlowMaps, compared with 38.0% for HOMER and 27.2% for SGM. This indicates that the continuous multimodal distribution learned by FlowMaps is not only useful for covering plausible future locations, but also for ranking them: when the method succeeds, it more often does so without requiring the agent to exhaust later proposals. The failure modes reveal a complementary trend. FlowMaps consistently produces fewer stepbudget failures than the graph-based baselines. For example, in Habit #1 only 10.5% of FlowMaps episodes exceed the step budget, compared with 20.5% for HOMER and 18.7% for SGM. In Habit #3, the same failure accounts for 17.4% of FlowMaps episodes, versus 24.1% for both HOMER and SGM. This suggests that FlowMaps proposes targets that are spatially more efficient for the downstream navigation policy, reducing failures caused by long or unproductive search trajectories. At the same time, FlowMaps exhibits a larger fraction of mode-exhaustion failures. This effect is visible in all three habits, with 16.9%, 16.4%, and 14.4% exhausted episodes for Habits #1, Habit #2, and Habit #3, respectively. By contrast, HOMER and SGM generally have lower exhaustion rates, but compensate with more late-mode successes and more step-budget failures. This points to a distinct error profile: FlowMaps tends to make sharper and more useful early predictions, but when its posterior misses the correct region, additional ranked modes are less likely to recover the object. In contrast, HOMER and SGM spread successful detections over later modes, which can improve recovery in some cases but increases the cost of search. Overall, the Sankey analysis shows that FlowMaps’ advantage is primarily driven by earlier target localization and fewer navigationinduced failures. The remaining errors are more often due to the predicted distribution missing the correct region than to inefficient traversal. This suggests that future improvements should focus on increasing tail-mode coverage while preserving the strong first-mode ranking behavior, for exam20

Figure 7: Success and failure decomposition for ObjNav episodes across habits. FlowMaps produces more first-mode successes and fewer step-budget failures, while its remaining failures are mostly due to exhausting the proposed modes. ple by encouraging more diverse samples before clustering or by explicitly calibrating cluster mass during target ranking. C.4

Additional Results

Object Navigation Examples. Figs. 8-15 collect representative successful episodes across different habits, environments and objects. Every row reads left to right as one episode. The first panel shows the raw predictive distribution: all Npreds =25 samples drawn from the flow for the queried object lq at τf , projected onto the top-down map. The second panel overlays the DBSCAN clusters ranked by mass (numbered #k) together with the ground-truth box of lq both at the observation time τ (dashed) and at the query time τf (solid): the offset between the two makes explicit that retrieval requires anticipating where the object moved, not recalling where it was last seen. The third panel draws the trajectory the agent actually executed, colored by attempt, as it visits the ranked modes in order from its start pose. The fourth panel is the egocentric view at the moment the target is confirmed. Two patterns recur. For low-depth successes the highest-mass mode lands directly on the τf ground truth and the agent reaches it in a near-straight path, confirming that FlowMaps concentrates probability on the correct future receptacle even when it is far from the last observed location. For higher-depth successes the trajectory visibly re-routes between modes: the agent rules out the leading clusters and falls back through the ranked list, which is exactly the behavior the mass-based ranking is meant to support when the future placement is genuinely multimodal.

21

Figure 8: Representative successful ObjNav episodes with FlowMaps, set 1. Each row shows the predicted samples, ranked clusters, executed trajectory, and final target confirmation view.

Figure 9: Representative successful ObjNav episodes with FlowMaps, set 2. Each row shows the predicted samples, ranked clusters, executed trajectory, and final target confirmation view.

22

Figure 10: Representative successful ObjNav episodes with FlowMaps, set 3. Each row shows the predicted samples, ranked clusters, executed trajectory, and final target confirmation view.

Figure 11: Representative successful ObjNav episodes with FlowMaps, set 4. Each row shows the predicted samples, ranked clusters, executed trajectory, and final target confirmation view.

23

Figure 12: Representative successful ObjNav episodes with FlowMaps, set 5. Each row shows the predicted samples, ranked clusters, executed trajectory, and final target confirmation view.

Figure 13: Representative successful ObjNav episodes with FlowMaps, set 6. Each row shows the predicted samples, ranked clusters, executed trajectory, and final target confirmation view.

24

Figure 14: Representative successful ObjNav episodes with FlowMaps, set 7. Each row shows the predicted samples, ranked clusters, executed trajectory, and final target confirmation view.

Figure 15: Representative successful ObjNav episodes with FlowMaps, set 8. Each row shows the predicted samples, ranked clusters, executed trajectory, and final target confirmation view.

25

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