Context-Informed Ship Trajectory Prediction via Conditional Attention

# Context-Informed Ship Trajectory Prediction via Conditional Attention
Source: [https://arxiv.org/html/2607.27418](https://arxiv.org/html/2607.27418)
Chandler SquiresMachine Learning Department, Carnegie Mellon UniversityTimothy HuAdvanced Technology Laboratories Lockheed MartinPradeep RavikumarMachine Learning Department, Carnegie Mellon University

###### Abstract

Long\-term ship trajectory prediction is a fundamental capability for maritime safety and autonomous navigation\. While recent Transformer\-based architectures have improved forecasting horizons, they predominantly rely on historical kinematic states, treating vessel motion as an isolated system\. In reality, maritime navigation is profoundly modulated by extrinsic factors like weather and constrained by static vessel characteristics\. Existing multimodal approaches fundamentally model the joint distribution over states and contexts, treating environmental variables as peer features rather than encoding the directional physical dependence of vessel dynamics on environmental conditions\.

In this work, we propose the Conditional Informer, a novel encoder\-decoder architecture that formulates trajectory prediction as a conditional generation task\. We employ a dedicated Conditional Attention mechanism where the vessel state explicitly queries environmental contexts through cross\-attention, encoding the physical prior that weather modulates—but is not generated by—vessel dynamics\. Furthermore, to address the intermittency of real\-world data, we introduce a Modality Masking training strategy to prevent catastrophic degradation during sensor fallback\. Extensive experiments on AIS and ERA5 data demonstrate that our approach outperforms kinematic and concatenation\-based baselines by 15\.4% in prediction accuracy when context is available\. Crucially, Modality Masking prevents shortcut learning, reducing fallback error by nearly an order of magnitude compared to unconstrained models\.

††footnotetext:© 2026 IEEE\. Personal use of this material is permitted\. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works\.## 1Introduction

As autonomous vessel technologies advance, the ability to forecast a vessel’s future state—position, speed, and course—over long horizons becomes increasingly critical\. While the Automatic Identification System \(AIS\) is the primary source of real\-time vessel telemetry, its signals suffer dropouts from transmission failures, antenna obstructions, satellite coverage gaps, or deliberate transponder deactivation\. In such scenarios, accurate trajectory prediction models are essential for maintaining maritime domain awareness, enabling operators to track vessels through AIS gaps and to deconflict shipping lanes proactively\. Beyond AIS, similar kinematic features can be derived through satellite imagery and radar\-based trackingMilios et al\. \([2019](https://arxiv.org/html/2607.27418#bib.bib12)\), though at lower temporal resolution and higher computational cost\. Deep learning models, particularly Transformers, have largely superseded RNNsCapobianco et al\. \([2021](https://arxiv.org/html/2607.27418#bib.bib3)\)and kinematic models \(e\.g\., Kalman Filters\) for AIS trajectory forecasting by effectively capturing long\-range dependencies\. Notably, Informer\-TPXiong et al\. \([2024](https://arxiv.org/html/2607.27418#bib.bib17)\)utilizes ProbSparse Attention to significantly reduce long\-term error accumulation compared to standard Transformers\.

Despite these advances, a critical limitation remains: state\-of\-the\-art models treat vessel motion as an isolated kinematic process, relying exclusively on historical trajectory states\. In reality, a ship is an open physical system whose trajectory is profoundly modulated by extrinsic environmental forces \(such as winds, waves, and currents\) and constrained by static vessel characteristics\. While a broad range of exogenous factors—including seasonal shipping patterns, port congestion, and geopolitical eventsMarineTraffic \([2025](https://arxiv.org/html/2607.27418#bib.bib11)\)—may influence routing decisions, we focus on the physically measurable environmental forces \(wind speed and significant wave height\) that exert direct and continuous mechanical influence on vessel dynamics and are available globally through reanalysis products such as ERA5\. The conditional architecture we propose is, however, extensible to additional exogenous signals as they become available\.

In this work, we propose the Conditional Informer, a novel architecture that extends the Informer\-TP framework to explicitly model the conditional dependence of trajectory states on environmental context and static metadata\. Our approach utilizes separate embedding pathways and a specialized Conditional Attention mechanism to capture the effect of these other variables on the evolution of ship states\.

\(a\)State Trajectory𝐗k\\mathbf\{X\}^\{k\}
\(b\)Context Trajectory𝐖k\\mathbf\{W\}^\{k\}
\(c\)Timestamps

Figure 1:Example state and context trajectories for context\-informed ship trajectory prediction\.In \(a\), LAT, LON, SOG, and COG are short for*latitude*,*longitude*,*speed over ground*, and*course over ground*, respectively\. In \(b\), SWH and WS are short for*significant wave height*and*wind speed*, respectively\.### 1\.1Multimodal Data Fusion

Recognizing the importance of external factors, recent research has begun to explore Multimodal Trajectory Prediction\. The core challenge lies in effectively fusing high\-frequency, sparse kinematic data \(AIS\) with low\-frequency, dense environmental fields \(weather\) and static attributes \(metadata\)\. Existing approaches generally fall into two categories:

1. 1\.Parallel Feature Extraction:Zhang et al\. \([2024](https://arxiv.org/html/2607.27418#bib.bib18)\)proposed the Adaptive Multimodal Data \(AMD\) model, which employs separate feature extraction branches using GRUs for AIS and MLPs for environmental data before merging them in a central fusion block\.
2. 2\.Gated Fusion:Chen et al\. \([2023](https://arxiv.org/html/2607.27418#bib.bib4)\)introduced the Dual\-Path Spatial\-Temporal Attention Network \(DualSTMA\), which injects static vessel attributes into the dynamic stream of the Transform via a gating mechanism that modulates the learned features\.

While these architectures move beyond naïve input concatenation, they model the joint distributionP​\(X,W\)P\(X,W\)over states \(XX\) and contexts \(WW\), treating environmental variables as peer features that help explain variance in the output\. Critically, this symmetric treatment does not encode the directional physical relationship in the maritime system: environmental forces influence vessel dynamics, but the vessel’s motion does not alter the weather\. As we demonstrate empirically in Section V\-D, this omission has practical consequences—without structural constraints enforcing the correct directionality, multimodal models are susceptible to shortcut learning\(Geirhos et al\.,[2020](https://arxiv.org/html/2607.27418#bib.bib6)\), collapsing onto the strongest available signal while suppressing essential but weaker modalities\.

### 1\.2Causal Perspective

Environmental conditions and vessel dynamics exhibit a strict physical asymmetry: weather perturbs vessel motion, but not vice versa\.

FollowingSchölkopf et al\. \([2021](https://arxiv.org/html/2607.27418#bib.bib13)\), encoding this known causal structure as an architectural inductive bias—separating the environment encoder from the state pathway—produces models that generalize more robustly than joint representations\.

The practical risk of ignoring this asymmetry isshortcut learning\(Geirhos et al\.,[2020](https://arxiv.org/html/2607.27418#bib.bib6)\): when multiple input modalities carry different levels of predictive signal, unconstrained models can exploit the strongest signal while suppressing weaker but essential features\. In multimodal trajectory prediction, this manifests asfeature collapse—the model learns to rely entirely on environmental context and disregards kinematic state, leading to catastrophic failure when context is unavailable at inference\.

We therefore adopt the design principle that the known physical hierarchy should be reflected in the model architecture\. In our Conditional Attention mechanism, the vessel state generates queries while the environment provides keys and values, forming directional attention matrices where the state attends to the context but not vice versa\.

## 2Problem Formulation

In general, we introduce the task of*context\-informed trajectory prediction*\. Letkkbe an index for the targets whose trajectories we wish to predict\. Each indexkkis associated with a*trajectory length*Tk∈ℕT\_\{k\}\\in\\mathbb\{N\}and*state trajectory*𝐗k\\mathbf\{X\}^\{k\}\. Further, eachkkmay optionally be associated with a*context trajectory*𝐖k\\mathbf\{W\}^\{k\}, and a*metadata vector*𝐚k\\mathbf\{a\}^\{k\}, as follows\.

### 2\.1States, Contexts, and Metadata

Letting𝒳=ℝdx\\mathcal\{X\}=\\mathbb\{R\}^\{d\_\{x\}\}denote the*state space*of our objects, a state trajectory is a sequence of states, i\.e\.,𝐗k=\{𝐱tk\}t=1Tk\\mathbf\{X\}^\{k\}=\\\{\\mathbf\{x\}^\{k\}\_\{t\}\\\}\_\{t=1\}^\{T\_\{k\}\}, where𝐱tk∈𝒳\\mathbf\{x\}^\{k\}\_\{t\}\\in\\mathcal\{X\}fort=1,…,Tkt=1,\\ldots,T\_\{k\}\. Similarly, letting𝒲=ℝdw\\mathcal\{W\}=\\mathbb\{R\}^\{d\_\{w\}\}denote our*context space*, a context trajectory is a sequence of contexts, i\.e\.,𝐖k=\{𝐰tk\}t=1Tk\\mathbf\{W\}^\{k\}=\\\{\\mathbf\{w\}^\{k\}\_\{t\}\\\}\_\{t=1\}^\{T\_\{k\}\}, where𝐰tk∈𝒲\\mathbf\{w\}^\{k\}\_\{t\}\\in\\mathcal\{W\}fort=1,…,Tkt=1,\\ldots,T\_\{k\}\. Finally, letting𝒜=ℝda\\mathcal\{A\}=\\mathbb\{R\}^\{d\_\{a\}\}denote the*metadata space*for our objects, the metadata vector𝐚k∈𝒜\\mathbf\{a\}^\{k\}\\in\\mathcal\{A\}is a static vector associated with each object\.

In the ship trajectory prediction task that we consider, the state𝐱tk\\mathbf\{x\}^\{k\}\_\{t\}includes elapsed time since trajectory start in minutes, the ship’s latitude, longitude, speed over ground \(SOG\), and course over ground \(COG\) at timett\. Then, the context𝐰tk\\mathbf\{w\}^\{k\}\_\{t\}corresponds to the significant wave height and wind speed at the same latitude and longitude as the ship, and at the same timett\. Finally, the metadata𝐚k\\mathbf\{a\}^\{k\}includes vessel hull characteristics \(draft, length, and width\) and cargo type\. In practice, both context and metadata features may be partially unobserved; we augment each with binary missingness indicators, yieldingdw=3d\_\{w\}=3\(wind speed, significant wave height, and a wave\-height missingness indicator\) andda=9d\_\{a\}=9\(four physical attributes and their associated indicators, plus MMSI\)\. An example trajectory is given in Fig\.[1](https://arxiv.org/html/2607.27418#S1.F1)\.

### 2\.2Context\-Informed Trajectory Prediction

For our prediction task, we consider a fixed*history length*LLand*prediction horizon*PP, such thatL\+P≤Tk∗L\+P\\leq T\_\{k^\{\*\}\}, wherek∗\{k^\{\*\}\}is the index of our target object\. At test time, we are given:

1. 1\.The historical state trajectory𝐗pastk∗:=\{𝐱tk∗\}t=1L\\mathbf\{X\}^\{k^\{\*\}\}\_\{\\text\{past\}\}\\mathrel\{:=\}\\\{\\mathbf\{x\}^\{k^\{\*\}\}\_\{t\}\\\}\_\{t=1\}^\{L\},
2. 2\.The historical context trajectory𝐖pastk∗:=\{𝐰tk∗\}t=1L\\mathbf\{W\}^\{k^\{\*\}\}\_\{\\text\{past\}\}\\mathrel\{:=\}\\\{\\mathbf\{w\}^\{k^\{\*\}\}\_\{t\}\\\}\_\{t=1\}^\{L\},*or*an indicator \(e\.g\.,None\) that it is unavailable, and
3. 3\.A metadata vector𝐚k∗∈𝒜\\mathbf\{a\}^\{k^\{\*\}\}\\in\\mathcal\{A\},*or*an indicator \(e\.g\.,None\) that the metadata is unavailable\.

We train the model by minimizing the mean squared error \(MSE\) over the full predicted state vector\. For model selection and evaluation, we use the average Haversine distance between predicted and true geographic coordinates,

m​\(Yk∗,Y^k∗\)=1P​∑t=L\+1L\+Phav​\(xtk∗,x^tk∗\),m\\\!\\left\(Y^\{k^\{\*\}\},\\,\\hat\{Y\}^\{k^\{\*\}\}\\right\)=\\frac\{1\}\{P\}\\sum\_\{t=L\+1\}^\{L\+P\}\\mathrm\{hav\}\\\!\\left\(x\_\{t\}^\{k^\{\*\}\},\\,\\hat\{x\}\_\{t\}^\{k^\{\*\}\}\\right\),\(1\)wherehav​\(x,x′\)\\mathrm\{hav\}\(x,x^\{\\prime\}\)is the great\-circle distance computed from the latitude and longitude coordinates ofxxandx′x^\{\\prime\}\. We select this metric for checkpointing because accurate geographic position prediction is the primary operational objective; ancillary state variables \(speed, course\) are of secondary concern for maritime domain awareness\. We prefer the mean over the full horizon to the terminal\-point metrichav​\(x^L\+Pk∗,xL\+Pk∗\)\\text\{hav\}\(\\hat\{x\}^\{k^\{\*\}\}\_\{L\+P\},x^\{k^\{\*\}\}\_\{L\+P\}\)because maritime applications such as collision avoidance and route de\-confliction require accurate intermediate waypoint predictions throughout the horizon, not only the final position\.

In our application to ship trajectory prediction, the potential unavailability of the context and metadata aligns with common real\-world situations, e\.g\. weather data may be unavailable due to coverage gaps or the latency of weather feeds, while ship metadata may be unavailable or unreliable due to receiver outages or strategic manipulation\.

### 2\.3Training Regimes and Related Learning Setups

To learn a trajectory predictor, we will consider several different training regimes, with training datasets wherek≠k∗k\\neq k^\{\*\}:

- •In the*state\-only*\(S\) regime, we have a training dataset consisting only of state trajectories𝐗k\\mathbf\{X\}^\{k\}\.
- •In the*state\+metadata*\(S\+M\) regime, we have a training dataset of tuples\(𝐗k,𝐚k\)\(\\mathbf\{X\}^\{k\},\\mathbf\{a\}^\{k\}\)\.
- •Similarly, in the*state\+context*\(S\+C\) regime, we have a training dataset of tuples\(𝐗k,𝐖k\)\(\\mathbf\{X\}^\{k\},\\mathbf\{W\}^\{k\}\)\.
- •Finally, in the*state\+context\+metadata*\(S\+C\+M\) regime, we have a training dataset of tuples\(𝐗k,𝐖k,𝐚k\)\(\\mathbf\{X\}^\{k\},\\mathbf\{W\}^\{k\},\\mathbf\{a\}^\{k\}\)\.

Notably, the data modalities available during training may differ from those available during inference time, e\.g\., if we consider predicting when the context trajectory and metadata are unavailable \(i\.e\.,𝐖pastk∗=None\\mathbf\{W\}^\{k^\{\*\}\}\_\{\\text\{past\}\}=\\textsc\{None\}and𝐚k∗=None\\mathbf\{a\}^\{k^\{\*\}\}=\\textsc\{None\}\) after training in the S\+C\+M regime\. We refer to this possibility as the*deployment modality mismatch \(DMM\)*problem, which is a key design consideration for our proposed architecture and training strategy\.

The DMM problem is closely related to two existing learning setups\. First, the DMM setup generalizes*learning using privileged information \(LUPI\)*\(Vapnik and Vashist,[2009](https://arxiv.org/html/2607.27418#bib.bib14); Vapnik et al\.,[2015](https://arxiv.org/html/2607.27418#bib.bib15); Lopez\-Paz et al\.,[2016](https://arxiv.org/html/2607.27418#bib.bib10)\), where the additional modalities are guaranteed to be absent during inference time \(or must simply be “thrown out” if they are present\)\. Second, the DMM setup is an important special case of the more general idea of*learning with side information \(LWSI\)*\(Jonschkowski et al\.,[2015](https://arxiv.org/html/2607.27418#bib.bib9); Adel and Levene,[2023](https://arxiv.org/html/2607.27418#bib.bib1)\), which considers a wider variety of inputs, e\.g\. facts from a knowledge base\. By concentrating on the DMM setup, we can benefit from the additional modalities when they are available at inference time \(unlike LUPI\), while focusing on the specific problem structure \(unlike LWSI in its more general form\)\.

## 3Methods

We propose the Conditional InformerTP, an encoder\-decoder Transformer architecture designed to efficiently fuse heterogeneous maritime data\. The architecture consists of three key components: a Multimodal Embedding Layer, a Conditional Encoder, and a Conditional Decoder\.

### 3\.1Base Architecture

Our proposed model builds upon the Informer architecture\(Zhou et al\.,[2021](https://arxiv.org/html/2607.27418#bib.bib19)\), and its extension Informer\-TP\(Xiong et al\.,[2024](https://arxiv.org/html/2607.27418#bib.bib17)\), a specialized Transformer designed for long\-term ship trajectory prediction\. We leverage its ProbSparse Attention to reduce complexity toO​\(L​log⁡L\)O\(L\\log L\), and Self\-Attention Distilling \(1\-D convolution and MaxPool\) to downsample redundant features in long sequences\. We also adopt Informer\-TP’s domain\-specific embedding layers for kinematic states\.

### 3\.2Motivation

Unlike standard models that treat vessel motion as an isolated system, maritime navigation is physically modulated by extrinsic forces like wind and waves\. We employ an encoder\-decoder structure to model the conditional distributionP​\(Y\|X,W\)P\(Y\|X,W\)rather than a symmetric joint distributionP​\(X,W\)P\(X,W\)\. This ensures a directional information flow where the Encoder acts as a Context Modeler \(aggregatingXXandWW\) and the Decoder acts as a State Generator \(querying the context via cross\-attention\)\.

### 3\.3Conditional InformerTP

#### 3\.3\.1Embedding

To handle the heterogeneous nature of the inputs, we employ three distinct embedding streams\. Each temporal stream \(state and context\) produces a token embedding that is the sum of three components: a*value projection*, a*positional encoding*, and a*temporal embedding*\.

##### Value Projection\.

The normalized state vector𝐱tk∈𝒳\\mathbf\{x\}^\{k\}\_\{t\}\\in\\mathcal\{X\}\(wheredx=d​i​m​\(𝒳\)=5d\_\{x\}=dim\(\\mathcal\{X\}\)=5: elapsed time, latitude, longitude, SOG, COG\) is mapped to adXd\_\{X\}\-dimensional representation by a learnable linear projection \(bias\-free\):

ut=𝐱tk​Wval,Wval∈ℝdx×dX\.u\_\{t\}=\\mathbf\{x\}^\{k\}\_\{t\}\\,W\_\{\\text\{val\}\},\\quad W\_\{\\text\{val\}\}\\in\\mathbb\{R\}^\{d\_\{x\}\\times d\_\{X\}\}\.\(2\)The context vector𝐰tk∈𝒲\\mathbf\{w\}^\{k\}\_\{t\}\\in\\mathcal\{W\}undergoes an analogous projection toℝdW\\mathbb\{R\}^\{d\_\{W\}\}\.

##### Positional Encoding\.

To preserve sequence ordering—particularly important given that ProbSparse attention selects a subset of query positions—we add a fixed sinusoidal positional encodingPEt∈ℝdX\\mathrm\{PE\}\_\{t\}\\in\\mathbb\{R\}^\{d\_\{X\}\}followingVaswani et al\. \([2017](https://arxiv.org/html/2607.27418#bib.bib16)\)\.

##### Temporal Embedding\.

To capture absolute temporal patterns \(e\.g\., time\-of\-day effects on shipping traffic, seasonal weather variation\), we extract five calendar features from each timestamp: minute, hour, ISO weekday, month, and year\. Each is normalized to approximately\[−0\.5,0\.5\]\[\-0\.5,\\,0\.5\]and the five scalars are jointly projected todXd\_\{X\}via a pointwise convolution \(Conv1dwith kernel size 1\):

TEt=Conv1d5→dX​\(\[c~1,c~2,c~3,c~4,c~5\]t\),\\mathrm\{TE\}\_\{t\}=\\texttt\{Conv1d\}\_\{5\\to d\_\{X\}\}\\\!\\bigl\(\[\\tilde\{c\}\_\{1\},\\,\\tilde\{c\}\_\{2\},\\,\\tilde\{c\}\_\{3\},\\,\\tilde\{c\}\_\{4\},\\,\\tilde\{c\}\_\{5\}\]\_\{t\}\\bigr\),\(3\)wherec~i\\tilde\{c\}\_\{i\}denotes the normalizedii\-th calendar scalar\.

##### State and Context Embeddings\.

The final embedding for the state stream is the summation of the three components:

HX\(t\)=ut\+PEt\+TEt\.H\_\{X\}^\{\(t\)\}=u\_\{t\}\+\\mathrm\{PE\}\_\{t\}\+\\mathrm\{TE\}\_\{t\}\.\(4\)The context stream follows an identical embedding procedure \(with its own projectionWval\(W\)∈ℝdw×dWW\_\{\\text\{val\}\}^\{\(W\)\}\\in\\mathbb\{R\}^\{d\_\{w\}\\times d\_\{W\}\}and temporal embeddingConv1d5→dW\\texttt\{Conv1d\}\_\{5\\to d\_\{W\}\}\), sharing the same calendar features to ensure temporal alignment between the vessel state and its environmental conditions\.

##### Metadata Embedding\.

Static vessel attributes \(draft, length, width, cargo type, associated missing indicators, and MMSI\) are min–max normalized and concatenated into a single vector𝐚k∈ℝda\\mathbf\{a\}^\{k\}\\in\\mathbb\{R\}^\{d\_\{a\}\}, which is then projected through a linear layer:

Ha=𝐚k​Wmeta,Wmeta∈ℝda×dM\.H\_\{a\}=\\mathbf\{a\}^\{k\}\\,W\_\{\\text\{meta\}\},\\quad W\_\{\\text\{meta\}\}\\in\\mathbb\{R\}^\{d\_\{a\}\\times d\_\{M\}\}\.\(5\)SinceHaH\_\{a\}is static \(identical across time steps for a given trajectory\), it is broadcast to the temporal dimension during conditioning \(Section[3\.3\.2](https://arxiv.org/html/2607.27418#S3.SS3.SSS2)\)\.

#### 3\.3\.2Conditional Attention

The Conditional Attention layer enables the state stream to selectively incorporate information from auxiliary modalities while preserving modality\-specific structure\. It consists of three parallel branches—state self\-attention, context cross\-attention, and metadata conditioning—whose outputs are summed and passed through a shared output projection\.

LetHX∈ℝB×L×dXH\_\{X\}\\in\\mathbb\{R\}^\{B\\times L\\times d\_\{X\}\}denote the state embedding,HW∈ℝB×L×dWH\_\{W\}\\in\\mathbb\{R\}^\{B\\times L\\times d\_\{W\}\}the context embedding, andHa∈ℝB×dMH\_\{a\}\\in\\mathbb\{R\}^\{B\\times d\_\{M\}\}the metadata embedding\. We writenhn\_\{h\}for the number of attention heads and define the per\-head dimensionsdk=dv=dX/nhd\_\{k\}=d\_\{v\}=d\_\{X\}/n\_\{h\}\.

##### State Self\-Attention \(ProbSparse\)\.

Queries, keys, and values are obtained from independent linear projections of the state embedding:

QX=HX​WXQ,KX=HX​WXK,VX=HX​WXV,Q\_\{X\}=H\_\{X\}W\_\{X\}^\{Q\},\\quad K\_\{X\}=H\_\{X\}W\_\{X\}^\{K\},\\quad V\_\{X\}=H\_\{X\}W\_\{X\}^\{V\},\(6\)whereWXQ,WXK∈ℝdX×dk​nhW\_\{X\}^\{Q\},W\_\{X\}^\{K\}\\in\\mathbb\{R\}^\{d\_\{X\}\\times d\_\{k\}n\_\{h\}\}andWXV∈ℝdX×dv​nhW\_\{X\}^\{V\}\\in\\mathbb\{R\}^\{d\_\{X\}\\times d\_\{v\}n\_\{h\}\}\. The self\-attention output is computed via ProbSparse attention\(Zhou et al\.,[2021](https://arxiv.org/html/2607.27418#bib.bib19)\):

OX=ProbSparseAttn​\(QX,KX,VX\),O\_\{X\}=\\mathrm\{ProbSparseAttn\}\(Q\_\{X\},\\,K\_\{X\},\\,V\_\{X\}\),\(7\)which selects the top\-uudominant queries \(u=c​⌈ln⁡L⌉u=c\\,\\lceil\\ln L\\rceilfor sparsity factorcc\) and computes scaled dot\-product attention only at those positions, reducing complexity fromO​\(L2\)O\(L^\{2\}\)toO​\(L​log⁡L\)O\(L\\log L\)\.

##### Context Cross\-Attention \(Full\)\.

To condition the state on the environmental context, we compute a cross\-attention in which queries are derived from the state embedding and keys/values from the context embedding\. Crucially, this branch uses a*separate*query projection from the self\-attention branch, allowing the model to learn distinct “questions to ask” of itself versus of the environment:

QX→W=HX​WX→WQ,KW=HW​WWK,VW=HW​WWV,\\displaystyle Q\_\{X\\to W\}=H\_\{X\}\\,W\_\{X\\to W\}^\{Q\},K\_\{W\}=H\_\{W\}\\,W\_\{W\}^\{K\},V\_\{W\}=H\_\{W\}\\,W\_\{W\}^\{V\},\(8\)whereWX→WQ∈ℝdX×dk​nhW\_\{X\\to W\}^\{Q\}\\in\\mathbb\{R\}^\{d\_\{X\}\\times d\_\{k\}n\_\{h\}\},WWK∈ℝdW×dk​nhW\_\{W\}^\{K\}\\in\\mathbb\{R\}^\{d\_\{W\}\\times d\_\{k\}n\_\{h\}\}, andWWV∈ℝdW×dv​nhW\_\{W\}^\{V\}\\in\\mathbb\{R\}^\{d\_\{W\}\\times d\_\{v\}n\_\{h\}\}\. Note that the query and key projections map from different input dimensions \(dXd\_\{X\}anddWd\_\{W\}, respectively\) into the commondkd\_\{k\}\-dimensional head space, enabling dot\-product compatibility scores to be computed across modalities\. The cross\-attention output is:

OW=softmax​\(QX→W​KW⊤dk\)​VW\.O\_\{W\}=\\mathrm\{softmax\}\\\!\\left\(\\frac\{Q\_\{X\\to W\}\\,K\_\{W\}^\{\\top\}\}\{\\sqrt\{d\_\{k\}\}\}\\right\)V\_\{W\}\.\(9\)This directional design ensures that the state queries the context but not vice versa, encoding the physical asymmetry that the environment modulates—but is not generated by—vessel dynamics\.

##### Metadata Conditioning \(Project and Broadcast\)\.

Static vessel attributesaka^\{k\}\(e\.g\., draft, length, cargo type\) are projected toHa∈ℝB×dMH\_\{a\}\\in\\mathbb\{R\}^\{B\\times d\_\{M\}\}\. SinceHaH\_\{a\}is constant across all time stepsLL, applying standard dot\-product attention over this single key\-value pair results in a softmax weight of exactly11at every query position \. Consequently, the attention mechanism reduces to a linear projection followed by a temporal broadcast:Oa=1L⊗\(Ha​WaV\)O\_\{a\}=1\_\{L\}\\otimes\(H\_\{a\}W\_\{a\}^\{V\}\)\. This formulation avoids redundant query\-key projections while maintaining mathematical equivalence to full cross\-attention, allowing us to bypass the mechanism entirely:

Oa=𝟏L⊗\(Ha​WaV\),O\_\{a\}=\\mathbf\{1\}\_\{L\}\\otimes\\bigl\(H\_\{a\}\\,W\_\{a\}^\{V\}\\bigr\),\(10\)whereWaV∈ℝdM×dv​nhW\_\{a\}^\{V\}\\in\\mathbb\{R\}^\{d\_\{M\}\\times d\_\{v\}n\_\{h\}\}and𝟏L⊗\\mathbf\{1\}\_\{L\}\\otimesdenotes broadcasting the projected vector to allLLtime steps\. The metadata thus acts as a*global additive bias*that uniformly conditions every position in the trajectory\.††Some of our experiments retain the full query\-key\-value formulation for metadata\. Since the softmax over a single key collapses to unity, the two implementations are mathematically equivalent; we present the simplified form here for clarity\.

##### Summation\.

The final output of the Conditional Attention layer is the element\-wise sum of the three branches, followed by an output projection:

Ofinal=\(OX\+OW\+Oa\)​WO,O\_\{\\text\{final\}\}=\\bigl\(O\_\{X\}\+O\_\{W\}\+O\_\{a\}\\bigr\)\\,W^\{O\},\(11\)whereWO∈ℝdv​nh×dXW^\{O\}\\in\\mathbb\{R\}^\{d\_\{v\}n\_\{h\}\\times d\_\{X\}\}\. This additive fusion ensures that when an auxiliary modality is masked—either via Modality Masking during training \(Section[5\.4](https://arxiv.org/html/2607.27418#S5.SS4)\) or due to unavailability at inference—its contribution is zeroed out without disrupting the tensor shapes of the primary state stream\.

#### 3\.3\.3Conditional Encoder

Figure[2\(a\)](https://arxiv.org/html/2607.27418#S3.F2.sf1)illustrates our encoder architecture\. The encoder consists of a stack of Conditional Encoder Layers\. In each layer, the vessel trajectory𝐇X\\mathbf\{H\}\_\{X\}attends to itself via self\-attention while simultaneously attending to the context𝐇W\\mathbf\{H\}\_\{W\}and metadata𝐇a\\mathbf\{H\}\_\{a\}\. The output of theii\-th layer is defined as:

𝐇X\(i\+1\)=LayerNorm​\(𝐇X\(i\)\+CondAttn​\(𝐇X\(i\),𝐇W,𝐇a\)\)\\mathbf\{H\}\_\{X\}^\{\(i\+1\)\}=\\text\{LayerNorm\}\\left\(\\mathbf\{H\}\_\{X\}^\{\(i\)\}\+\\text\{CondAttn\}\(\\mathbf\{H\}\_\{X\}^\{\(i\)\},\\mathbf\{H\}\_\{W\},\\mathbf\{H\}\_\{a\}\)\\right\)
Following the standard Informer design, we also employ distilling operations \(Conv1d \+ MaxPool\) between layers to reduce the sequence length and highlight dominant features\.

#### 3\.3\.4Conditional Decoder

Figure[2\(b\)](https://arxiv.org/html/2607.27418#S3.F2.sf2)illustrates our decoder architecture\. The decoder aims to generate the future trajectory𝐇p​r​e​d\\mathbf\{H\}\_\{pred\}\. It utilizes a standard masked self\-attention layer to model the temporal dependencies of the prediction sequence itself\. However, for the cross\-attention step—where the decoder attends to the encoder’s output—we replace the standard mechanism with a Conditional Cross\-Attention Layer\. In this layer, the query comes from the decoder’s current state \(the generated future trajectory so far\)\. This query attends to three separate sets of keys/values provided by the encoder: the encoded past trajectory states, the encoded context, and the encoded metadata\. This architecture ensures that the generation of future positions is directly conditioned on both the historical motion pattern and the prevailing weather conditions\.

![Refer to caption](https://arxiv.org/html/2607.27418v1/x1.png)\(a\)Encoder
![Refer to caption](https://arxiv.org/html/2607.27418v1/x2.png)\(b\)Decoder

Figure 2:Architecture of the Conditional Informer\.\(a\) Encoder with state self\-attention, context cross\-attention, and metadata bias\. \(b\) Decoder with masked self\-attention and conditional cross\-attention over encoder representations\.

### 3\.4Training Strategy

#### 3\.4\.1Causal Masking

To ensure the validity of prediction, we apply a regular Triangular Causal Mask in the decoder’s self\-attention layers\. This maskM∈\{0,−∞\}P×PM\\in\\\{0,\-\\infty\\\}^\{P\\times P\}sets the attention score to−∞\-\\inftyfor all upper\-triangular elements, preventing the model from attending to future positionsj\>tj\>twhen predicting the state at timett\.

#### 3\.4\.2Modality Masking

A model trained on complete data triplets\{𝐗,𝐖,𝐚\}\\\{\\mathbf\{X\},\\mathbf\{W\},\\mathbf\{a\}\\\}often learns to over\-rely on the auxiliary modalities, leading to catastrophic performance degradation when they are missing at inference time\. To enforce robustness, we introduce a Modality Masking strategy during training\. For each training batch, we independently mask the context𝐖\\mathbf\{W\}and the metadata𝐚\\mathbf\{a\}with probabilitiespWp\_\{W\}andpap\_\{a\}respectively\.

Formally, letbW,ba∼Bernoulli​\(1−p\)b\_\{W\},b\_\{a\}\\sim\\text\{Bernoulli\}\(1\-p\)be binary indicator variables where11indicates presence and0indicates absence\. The input to the model during a training step becomes:

𝐖′=bW⋅𝐖,𝐚′=ba⋅𝐚\\mathbf\{W\}^\{\\prime\}=b\_\{W\}\\cdot\\mathbf\{W\},\\quad\\mathbf\{a\}^\{\\prime\}=b\_\{a\}\\cdot\\mathbf\{a\}In our implementation, masking is applied batch\-wise: when a modality is dropped, its corresponding tensor is replaced withNone\. Due to the summation computation of our Conditional Attention mechanism, this zeroes out the contribution of that branch\.

## 4Experiments

In this section, we evaluate the proposed Conditional Informer against varied baselines to validate the effectiveness of incorporating environmental context and the robustness of our conditional architecture\.

### 4\.1Datasets and Preprocessing

- •Vessel Trajectories \(AIS\):Raw Automatic Identification System \(AIS\) data was sourced from the Marine Cadastre AIS vessel traffic archive\(Bureau of Ocean Energy Management and National Oceanic and Atmospheric Administration \(NOAA\)\(2024\),[BOEM](https://arxiv.org/html/2607.27418#bib.bib2)\), which provides over 100,000 open\-source trajectories spanning 2009–2025 across diverse vessel types\. Our study focuses on the period from June 2023 to September 2023 within the Gulf of Mexico region \(Lat: 23–27∘N, Lon: 84–77∘W\)\. We focus on cargo vessels \(IMO type 70\) to maintain a homogeneous vessel population and to remain methodologically consistent with Informer\-TP\(Xiong et al\.,[2024](https://arxiv.org/html/2607.27418#bib.bib17)\), which evaluates on the same vessel class and region\. The conditional architecture is vessel\-type agnostic; extension to heterogeneous fleets is a natural direction for future work \(Section[6](https://arxiv.org/html/2607.27418#S6)\)\. Following the pipeline ofXiong et al\. \([2024](https://arxiv.org/html/2607.27418#bib.bib17)\), data cleaning consisted of four steps: \(1\) DBSCAN outlier removal withε\\varepsilonestimated as the 95th percentile of the 5\-nearest\-neighbor distance distribution; \(2\) gap\-based segmentation, splitting trajectories at intervals exceeding 60 minutes; \(3\) linear interpolation \(for continuous variables\) and forward/backward filling \(for categorical attributes\) to bridge minor dropouts under11minute; and \(4\) uniform resampling to55\-minute intervals\.
- •Environmental Context \(ERA5\):To enrich the kinematic trajectories, we integrated environmental features from the ERA5 reanalysis dataset\(Hersbach et al\.,[2020](https://arxiv.org/html/2607.27418#bib.bib8)\), obtained from the Climate Data Store\(Copernicus Climate Change Service, Climate Data Store,[2026](https://arxiv.org/html/2607.27418#bib.bib5)\)\. The dataset provides a global latitude\-longitude grid of atmospheric and oceanic state variables at hourly resolution\. We extracted two features: 10\-meter wind speed \(m/s\) and significant wave height \(m\)\. Given the resolution mismatch between the high\-frequency AIS stream and the hourly, discrete ERA5 grid \(0\.25∘0\.25^\{\\circ\}\), we employed a nearest\-neighbor lookup strategy in both space and time\. Each AIS point was assigned environmental variables from the spatially nearest ERA5 grid coordinate and the preceding full hour\. Missing significant wave height values \(undefined over certain regions and time periods\) were zero\-imputed after normalization\.
- •Static Metadata:We included vessel dimension features—draft, length, and width—along with the cargo type attribute\. These physical characteristics capture the vessel’s inertia and hydrodynamic profile, which directly influence its responsiveness to environmental forces\. Missing dimension values were zero\-imputed with corresponding binary missing indicators appended to the feature vector\.

Trajectory segments with non\-recoverable missing data or fewer than 48 resampled points \(4 hours\) were discarded, ensuring all tracks comfortably exceeded theL\+P=36L\+P=36step model requirement\. The remaining 1,571 valid trajectory segments were split strictly at the trajectory level \(80/10/10\) to prevent data leakage across sets, yielding 1,256 training, 157 validation, and 158 test tracks\. Finally, a sliding window with a history length ofL=24L=24\(2 hours\) and prediction horizonP=12P=12\(1 hour\) was applied to these segments\. This produced a final dataset of 123,771 samples \(98,944 training / 12,785 validation / 12,042 test\)\.

Table 1:Main Results: Haversine distance \(km\) and MSE \(×10−3\\times 10^\{\-3\}\) comparison\. Each cell reports mean±\\pmstd across 5 random seeds\.Boldmarks the best result per column\. Metadata \(MM\) shows limited independent impact due to vessel population homogeneity, while Context \(CC\) drives a∼\\sim15\.4% accuracy gain over the baseline\.
### 4\.2Baselines and Experimental Setup

We compare the following configurations:

1. 1\.InformerTP \(State\-Only\):The baseline Informer model that utilizes only kinematic features \(Latitude, Longitude, SOG, COG\)\.
2. 2\.InformerTP \(Concatenated\):A fusion baseline where environmental context and metadata embeddings are concatenated directly with the state embeddings at the input level, treating context simply as additional feature channels\.
3. 3\.Conditional Informer \(Ours\):Our proposed model using Conditional Attention to fuse state, context, and metadata via separate pathways\.

All models were trained with a model dimensiondm​o​d​e​l=512d\_\{model\}=512,nh​e​a​d​s=8n\_\{heads\}=8, and a 2\-layer encoder/1\-layer decoder architecture\. We utilized the Adam optimizer with initial learning rate of5×10−65\\times 10^\{\-6\}and betas of\(0\.9,0\.98\)\(0\.9,0\.98\)\. To prevent overreliance on other modalities, we applied a masking rate of 0\.4 for context and 0\.3 for metadata—training employed mixed\-precision \(BF16\) to optimize memory usage on 2x NVIDIA A6000 GPUs\. The model was trained for a maximum of 100 epochs with an early stopping patience of 20 epochs, monitoring the validation Haversine distance to save the best\-performing checkpoint\. Training took about 2 hours per model, and inference took approximately7\.57\.5ms for a single\-vessel forward pass\.

## 5Results and Analysis

### 5\.1Comparison to Baselines

Table[1](https://arxiv.org/html/2607.27418#S4.T1)summarizes the performance of all model configurations\. We compare the proposed Conditional Informer against the InformerTP baseline \(state\-only\) and InformerTP\-Concat \(concatenated multimodal fusion\)\. All results report mean and standard deviation over 5 random seeds\.

When all modalities are available at test time, the Conditional Informer consistently outperforms both baselines\. The best overall result is achieved by the Conditional \(S\+C\+M\) model with full inputs, yielding a Haversine error of5\.105\.10km—a15\.4%15\.4\\%reduction compared to the InformerTP baseline \(6\.036\.03km\) and a16\.7%16\.7\\%improvement over the Concat model’s full\-modality result \(6\.126\.12km\)\. These gains confirm that our conditional attention mechanism more effectively leverages environmental context than naïve concatenation, by enforcing the causal direction from weather conditions to vessel dynamics rather than treating all modalities symmetrically\.

The Conditional \(S\) model, which uses the same architecture but with auxiliary branches disabled, achieves5\.975\.97km—on par with the InformerTP baseline \(6\.036\.03km\) given overlapping standard deviations\. This parity is expected by construction: when auxiliary modalities are absent, the architecture naturally reduces to the baseline’s computation, confirming that the additional branches introduce no interference\.

### 5\.2Effect of Environmental Context

Environmental context emerges as the primary driver of prediction improvement\. The Conditional \(S\+C\) model achieves5\.105\.10km when context is provided, compared to 8\.89 km without—a42\.6%42\.6\\%reduction in error attributable solely to wind and wave information\. Similarly, the Conditional \(S\+C\+M\) model improves from8\.768\.76km \(state\-only\) to5\.105\.10km \(full inputs\) when context is available\. This pattern is consistent with the physical reality that environmental forces directly modulate vessel ground speed and heading, providing predictive signal that cannot be recovered from kinematic history alone\.

In contrast, the Concat baseline derives comparatively less benefit from context: its full\-modality result \(6\.126\.12km\) improves only modestly over state\+metadata \(6\.476\.47km\), and its state\+context result \(9\.209\.20km\) barely improves over state\-only \(9\.289\.28km\)\. This suggests that the concatenation approach struggles to disentangle the causal influence of weather from the kinematic signal, whereas the conditional architecture’s separated attention pathways enable more effective use of environmental data\.

We additionally evaluated a shorter history length ofL=P=12L=P=12\. In this setting, context provides negligible benefit: Conditional\-\(S\+C\) achieves5\.975\.97km with context versus5\.995\.99km without, compared to the42\.6%42\.6\\%gain atL=24L=24\. We attribute this to insufficient temporal signal atL=12L=12for the model to identify weather\-driven deviations from the nominal kinematic trajectory, corroborating that a two\-hour context window is necessary to observe environmentally\-modulated behavioral patterns\.

### 5\.3Effect of Metadata

Vessel metadata shows limited independent impact on prediction accuracy\. The Conditional \(S\+M\) model achieves6\.026\.02km with metadata and6\.046\.04km without—a negligible difference that falls within the standard deviation\. Similarly, the full S\+C\+M model’s performance with all modalities \(5\.105\.10km\) is essentially identical to the S\+C model with context alone \(5\.105\.10km\), suggesting that metadata provides minimal additional signal beyond what context already captures\.

### 5\.4Modality Masking

The “Without Masking” ablation demonstrates the necessity of Modality Masking\. Without it, the Conditional \(S\+C\+M\) model catastrophically fails when environmental data is missing at inference, with state\-only error reaching67\.0567\.05km—an order of magnitude worse than the masked\-trained model\(8\.76\(8\.76km\) and the kinematic baseline \(6\.036\.03km\)\. This confirms the unmasked model treats weather as a non\-optional shortcut rather than learning a robust kinematic representation\. Furthermore, masking acts as a regularizer that improves full\-modality performance: the masked S\+C\+M model achieves5\.105\.10km compared to the unmasked variant’s6\.446\.44km \(a20\.8%20\.8\\%improvement\)\.

When context is withheld at test time, the context\-trained models naturally degrade, validating that they actively utilize wind and wave vectors as physical drivers of trajectory adjustment\. The state\-only fallback errors increase to8\.898\.89km for Conditional \(S\+C\) and8\.768\.76km for \(S\+C\+M\)\. Although slightly worse than the state\-only baseline \(6\.036\.03km\), this represents a manageable degradation compared to the unmasked model’s failure \(67\.0567\.05km\)\.

## 6Discussion

It is important to note that the proposed Conditional Attention mechanism is not architecturally bound to the Informer framework\. Because it relies on a fundamental query\-key\-value separation — where the state trajectory queries the auxiliary contexts — this module can be integrated into any attention\-based architecture, including standard Transformers, Performers, or State Space ModelsGu et al\. \([2022](https://arxiv.org/html/2607.27418#bib.bib7)\)\.

Limitations\.Although Modality Masking substantially mitigates feature collapse, it does not fully eliminate the DMM problem \(Section[2\.3](https://arxiv.org/html/2607.27418#S2.SS3)\): when context is absent at test time, Conditional\-\(S\+C\+M\) reaches8\.768\.76km versus6\.036\.03km for the kinematic baseline\. This residual degradation suggests the current masking still permits some shortcut reliance on auxiliary data; future work on adaptive masking schedules or auxiliary contrastive objectives may close this gap further\.

The present evaluation focuses on open\-water transit trajectories typical of cargo vessels\. Whether the model accurately captures non\-trivial behaviors—sudden course changes, speed modulations in heavy weather, or port approach maneuvers—remains an open question that requires targeted qualitative analysis\.

Finally, the study is limited to cargo vessels \(type7070\)\. Expanding to a heterogeneous fleet—tankers, container ships, fishing vessels—would test whether the metadata branch contributes more substantially under greater variation in hull characteristics, and is a priority for future work\.

Several additional directions remain\. Enriching the environmental context with ocean currents and bathymetry would provide stronger physical constraints\. Extending the framework to incorporate port schedules or geopolitical routing constraints could further improve prediction in operationally complex scenarios\.

Data Availability Statement\.The AIS data were obtained from[https://marinecadastre\.gov/ais/](https://marinecadastre.gov/ais/)and are publicly available; no restrictions on use apply beyond appropriate attribution\. The ERA5 reanalysis data were obtained from the Climate Data Store \([https://cds\.climate\.copernicus\.eu/datasets/reanalysis\-era5\-single\-levels](https://cds.climate.copernicus.eu/datasets/reanalysis-era5-single-levels)\), provided by the Copernicus Climate Change Service\. The data are distributed under the Creative Commons Attribution 4\.0 International \(CC\-BY 4\.0\) license; no restrictions on use apply by appropriate attribution\.

##### Acknowledgements

This research was developed with funding from the Defense Advanced Research Projects Agency \(DARPA\) via HR0011\-25\-3\-0239, FA8750\-23\-2\-1015, ONR via N00014\-23\-1\-2368, and NSF via IIS\-1909816\.

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