From Fixed Grids to Moving Particles:A Transferable Latent Operator for Fluid Dynamics

# From Fixed Grids to Moving Particles:A Transferable Latent Operator for Fluid Dynamics
Source: [https://arxiv.org/html/2608.14120](https://arxiv.org/html/2608.14120)
Chuqi ChenZhengqing GaoXi ZhouXiao SunYang Xiang\\correspondingHuaxi Huang\\corresponding

###### Abstract

Lagrangian modeling is vital to fluid dynamics, as it characterizes particle transport and complements the Eulerian description\. However, Lagrangian trajectories are less commonly available than Eulerian fields, while most neural operators are trained and evaluated primarily in the Eulerian representation\. This mismatch motivates a new learning problem: can a model trained solely on Eulerian observations generalize zero\-shot from Eulerian field prediction to Lagrangian particle rollout, without Lagrangian supervision or task\-specific adaptation? To address this problem, we propose the*Transferable Latent Operator*\(TLO\), which learns a unified flow representation shared by Eulerian field prediction and Lagrangian particle rollout\.TLOdecouples latent flow evolution from coordinate\-dependent decoding: querying the evolving latent representation at fixed spatial coordinates yields Eulerian fields, whereas querying velocities at particle positions and recursively updating these positions enables Lagrangian rollout\. Across five fluid\-dynamics benchmarks,TLOconsistently outperforms existing neural operators in both Eulerian field prediction and zero\-shot Lagrangian rollout, with further gains from limited Lagrangian fine\-tuning\.

## Introduction

Lagrangian particle trajectories provide a direct description of transport, mixing, and dispersion in fluid flows\. However, most fluid\-learning datasets are available as time\-dependent fields on fixed Eulerian grids\([11](https://arxiv.org/html/2608.14120#bib.bib1);[9](https://arxiv.org/html/2608.14120#bib.bib2);[18](https://arxiv.org/html/2608.14120#bib.bib9);[13](https://arxiv.org/html/2608.14120#bib.bib3)\)\. Native Lagrangian data are less commonly available because they require specifying and tracking particle populations\. At high particle counts, trajectory generation, storage, and particle\-based training can also become expensive in time and memory\([16](https://arxiv.org/html/2608.14120#bib.bib4);[20](https://arxiv.org/html/2608.14120#bib.bib5);[15](https://arxiv.org/html/2608.14120#bib.bib20)\)\. This mismatch motivates a practical question: can a model trained only on fixed\-grid Eulerian data also support Lagrangian particle rollout of the same flow?

We formulate this problem as*zero\-shot Eulerian\-to\-Lagrangian generalization*\. A model is trained exclusively with Eulerian field supervision, without particle coordinates, particle velocities, or trajectory labels\. At test time, the same model is queried at moving particle positions, and its predicted velocities are recursively integrated without parameter adaptation\. This is more than ordinary arbitrary\-coordinate evaluation: each velocity prediction changes the next particle position and therefore determines the next query location\. Errors can consequently induce a history\-dependent query shift and accumulate into trajectory drift\.

Although existing methods have explored Eulerian–Lagrangian modeling, they do not directly address zero\-shot Eulerian\-to\-Lagrangian rollout\. Approaches based on latent particles, joint field–trajectory learning, or trajectory reconstruction\([22](https://arxiv.org/html/2608.14120#bib.bib18);[19](https://arxiv.org/html/2608.14120#bib.bib19);[1](https://arxiv.org/html/2608.14120#bib.bib14);[12](https://arxiv.org/html/2608.14120#bib.bib17)\)typically require additional Lagrangian supervision or particle\-specific modeling, preventing zero\-shot transfer\. Alternatively, Eulerian neural operators\([9](https://arxiv.org/html/2608.14120#bib.bib2);[21](https://arxiv.org/html/2608.14120#bib.bib25);[14](https://arxiv.org/html/2608.14120#bib.bib26)\)can be combined with numerical interpolation to obtain particle velocities from predicted grid fields\. However, this strategy remains tied to the resolution and finite support of the Eulerian discretization\. Once particles move beyond the grid support, standard interpolation cannot continue without additional extrapolation rules\.

These limitations motivate a neural operator that learns flow dynamics from Eulerian observations while supporting direct evaluation at recursively evolving particle locations\. We therefore introduce the*Transferable Latent Operator*\(TLO\), which uses one latent flow evolution with two query modes\. An encoder maps fixed\-grid Eulerian observations into spatially localized tokens augmented with global flow context, and a shared latent processor advances this representation independently of the output query set\. A coordinate\-conditioned decoder then evaluates the predicted field at requested locations\. Querying on the fixed Eulerian grid produces an Eulerian field forecast, whereas querying the velocity channels at recursively advected particle positions produces a Lagrangian rollout\. Particle coordinates are used only as decoder queries and are not required by the Eulerian training objective\.

We evaluateTLOon five fluid\-dynamics benchmarks for Eulerian forecasting and on three benchmarks for zero\-shot Lagrangian particle rollout\.TLOachieves strong Eulerian accuracy while transferring directly to reference\-path velocity prediction and closed\-loop trajectories\. We additionally compare with a native particle\-based model and grid interpolation\. Direct decoding is competitive with interpolation within the Eulerian support and remains evaluable after particles leave that support, where standard interpolation becomes undefined\. We separately study decoder\-only adaptation using sparse particle\-velocity supervision\.

Our main contributions are summarized as follows:

- •We formulate and evaluate*zero\-shot Eulerian\-to\-Lagrangian generalization*, in which a model trained only with fixed\-grid Eulerian supervision performs closed\-loop particle rollout without particle supervision or parameter adaptation\.
- •We propose the*Transferable Latent Operator*, whose output\-query\-independent latent evolution and coordinate\-conditioned decoder allow the same trained model to support both fixed\-grid forecasting and moving particle queries\.
- •Across five Eulerian benchmarks and three particle\-rollout benchmarks,TLOachieves strong performance, compares favorably with neural\-operator and native particle\-based baselines, and remains evaluable beyond the fixed Eulerian grid support\.

## Related Work

#### Neural operators\.

Neural operators learn mappings between function spaces for PDE modeling\([11](https://arxiv.org/html/2608.14120#bib.bib1);[8](https://arxiv.org/html/2608.14120#bib.bib27);[24](https://arxiv.org/html/2608.14120#bib.bib22)\)\. Representative architectures include spectral operators, coordinate\-based models, and attention\- or token\-based operators\([9](https://arxiv.org/html/2608.14120#bib.bib2);[5](https://arxiv.org/html/2608.14120#bib.bib12);[10](https://arxiv.org/html/2608.14120#bib.bib16);[26](https://arxiv.org/html/2608.14120#bib.bib15);[1](https://arxiv.org/html/2608.14120#bib.bib14);[3](https://arxiv.org/html/2608.14120#bib.bib11)\)\. Although several of these models support arbitrary\-coordinate evaluation, they are generally evaluated on fixed or externally specified query locations\. We instead consider moving queries whose positions are recursively generated by the model’s predicted velocities\.

#### Eulerian–Lagrangian modeling\.

Classical particle\-grid methods couple Eulerian fields and Lagrangian particles through transfers between the two representations\([2](https://arxiv.org/html/2608.14120#bib.bib23);[17](https://arxiv.org/html/2608.14120#bib.bib24)\)\. Recent neural methods use latent particles to improve Eulerian forecasting\([12](https://arxiv.org/html/2608.14120#bib.bib17)\), infer fields or trajectories from particle observations\([22](https://arxiv.org/html/2608.14120#bib.bib18);[19](https://arxiv.org/html/2608.14120#bib.bib19)\), or learn particle\-based simulators\([15](https://arxiv.org/html/2608.14120#bib.bib20);[27](https://arxiv.org/html/2608.14120#bib.bib21)\)\. In contrast, the zero\-shot version ofTLOis trained only on fixed\-grid Eulerian fields, with particle positions introduced at inference time solely as decoder queries\. The same learned flow representation therefore supports both Eulerian field prediction and closed\-loop Lagrangian rollout\. Sparse decoder adaptation is studied separately as a non\-zero\-shot extension\.

## Problem Setting

LetΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}be the spatial domain\. At timett, the state of a flow is represented by a vector\-valued field𝒖t:Ω→ℝdu\\bm\{u\}\_\{t\}:\\Omega\\rightarrow\\mathbb\{R\}^\{d\_\{u\}\}, where theccchannels may contain velocity, pressure, density, or other physical variables\. We denote the velocity by𝒗t​\(𝒙\):=Πv​𝒖t​\(𝒙\)∈ℝd\\bm\{v\}\_\{t\}\(\\bm\{x\}\):=\\Pi\_\{v\}\\bm\{u\}\_\{t\}\(\\bm\{x\}\)\\in\\mathbb\{R\}^\{d\}, whereΠv\\Pi\_\{v\}is the projection operator extracting the velocity channels\.

The available data do not provide the continuous function𝒖t\\bm\{u\}\_\{t\}directly\. Instead, each state is observed at a fixed set of Eulerian coordinates

XE:=\{𝒙i\}i=1NE,𝐔t:=\[𝒖t\(𝒙i\)\]i=1NE∈ℝNE×du\.\\displaystyle X\_\{E\}:=\\\{\\bm\{x\}\_\{i\}\\\}\_\{i=1\}^\{N\_\{E\}\},\\,\\mathbf\{U\}\_\{t\}:=\[\\bm\{u\}\_\{t\}\(\\bm\{x\}\_\{i\}\)\]\_\{i=1\}^\{N\_\{E\}\}\\in\\mathbb\{R\}^\{N\_\{E\}\\times d\_\{u\}\}\.\(1\)The coordinates inXEX\_\{E\}remain fixed over time, while the field values𝐔t\\mathbf\{U\}\_\{t\}evolve\. The dataset therefore consists of temporal sequences\{𝐔0,𝐔1,…\}\\\{\\mathbf\{U\}\_\{0\},\\mathbf\{U\}\_\{1\},\\ldots\\\}observed on the same Eulerian coordinates\.

Our goal is to learn the one\-step evolution from the current Eulerian observation𝐔t\\mathbf\{U\}\_\{t\}to the next physical field𝒖t\+1\\bm\{u\}\_\{t\+1\}\. In particular, the learned operator should not be restricted to returning values only onXEX\_\{E\}\. Given any finite set of output coordinatesQ=\{𝒒j\}j=1NQ⊂ΩQ=\\\{\\bm\{q\}\_\{j\}\\\}\_\{j=1\}^\{N\_\{Q\}\}\\subset\\Omega, it predicts

𝐔^t\+1​\(Q\):=𝒢θ​\(XE,𝐔t,Q\)\\displaystyle\\widehat\{\\mathbf\{U\}\}\_\{t\+1\}\(Q\):=\\mathcal\{G\}\_\{\\theta\}\(X\_\{E\},\\mathbf\{U\}\_\{t\};Q\)=\[𝒖^t\+1​\(𝒒1\),…,𝒖^t\+1​\(𝒒NQ\)\]⊤\.\\displaystyle=\[\\widehat\{\\bm\{u\}\}\_\{t\+1\}\(\\bm\{q\}\_\{1\}\),\\ldots,\\widehat\{\\bm\{u\}\}\_\{t\+1\}\(\\bm\{q\}\_\{N\_\{Q\}\}\)\]^\{\\top\}\.Here, a*query*is simply a coordinate𝒒j\\bm\{q\}\_\{j\}at which the next field is requested\. SettingQ=XEQ=X\_\{E\}produces the next Eulerian grid field\. SettingQQto particle coordinates provides the local velocity values needed to evolve Lagrangian particles\.

![Refer to caption](https://arxiv.org/html/2608.14120v1/Figures/rollout_workflow.png)Figure 1:Workflow ofTLOfor unified Eulerian field forecasting and Lagrangian particle rollout from a shared latent flow representation\. NotationsXEX\_\{E\},U^t\\hat\{U\}\_\{t\}, andy^t\\hat\{y\}\_\{t\}follow Section Problem Setting\.#### Learning from Eulerian fields\.

Fluid\-dynamics datasets are commonly available as time\-dependent fields on fixed spatial grids\. We therefore use consecutive Eulerian fields as the primary source of supervision\. Although the operator accepts a general query setQQ, its training targets are provided only onXEX\_\{E\}:

ℒEul=𝔼t​\[1NE​du​‖𝒢θ​\(XE,𝐔t,XE\)−𝐔t\+1‖F2\]\.\\mathcal\{L\}\_\{\\mathrm\{Eul\}\}=\\mathbb\{E\}\_\{t\}\\\!\\left\[\\frac\{1\}\{N\_\{E\}d\_\{u\}\}\\left\\\|\\mathcal\{G\}\_\{\\theta\}\(X\_\{E\},\\mathbf\{U\}\_\{t\};X\_\{E\}\)\-\\mathbf\{U\}\_\{t\+1\}\\right\\\|\_\{F\}^\{2\}\\right\]\.\(2\)Thus, neither particle coordinates nor trajectory labels are required to train the field\-evolution model\. Starting from an observed initial state𝐔0\\mathbf\{U\}\_\{0\}, choosingQ=XEQ=X\_\{E\}at every step gives the autoregressive Eulerian rollout

𝐔^t\+1=𝒢θ​\(XE,𝐔^t,XE\),𝐔^0=𝐔0\.\\widehat\{\\mathbf\{U\}\}\_\{t\+1\}=\\mathcal\{G\}\_\{\\theta\}\(X\_\{E\},\\widehat\{\\mathbf\{U\}\}\_\{t\};X\_\{E\}\),\\qquad\\widehat\{\\mathbf\{U\}\}\_\{0\}=\\mathbf\{U\}\_\{0\}\.\(3\)This produces the predicted field sequence\{𝐔^1,…,𝐔^H\}\\\{\\widehat\{\\mathbf\{U\}\}\_\{1\},\\ldots,\\widehat\{\\mathbf\{U\}\}\_\{H\}\\\}on the fixed Eulerian grid support\. Besides providing a field forecast, this rollout supplies the evolving flow state from which velocities at moving coordinates can be evaluated\.

![Refer to caption](https://arxiv.org/html/2608.14120v1/Figures/unfied_frame.png)Figure 2:Overview of the Transferable Latent Operator \(TLO\)\. TLO learns coordinate\-independent latent dynamics from Eulerian observations\. The hierarchical encoder extracts local and global flow representations, while the latent processor evolves the fused tokens without spatial coordinates\. A coordinate\-conditioned decoder then queries the latent state at arbitrary locations, enabling both fixed\-grid Eulerian prediction and Lagrangian rollout\. Particle positions are used only as inference\-time queries and are not required for Eulerian training\.
#### Recovering Lagrangian trajectories\.

Fixed\-grid forecasts describe how the flow evolves at stationary spatial locations\. Many transport problems, however, concern material carried by the flow and therefore require following moving particles\. Particle trajectories provide a direct description of transport and expose accumulated drift that may not be evident from an average fixed\-grid field error\. To obtain this Lagrangian view, letY0=\{𝒚i,0\}i=1NPY\_\{0\}=\\\{\\bm\{y\}\_\{i,0\}\\\}\_\{i=1\}^\{N\_\{P\}\}denote a set of initial particle positions\. Once𝒖^t\\widehat\{\\bm\{u\}\}\_\{t\}has been predicted, its velocity channels are evaluated at the current particle positions and integrated through

𝒚^i,t\+1=ℬ⁡\(𝒚^i,t\+Δ​t​𝒗^t​\(𝒚^i,t\)\),𝒚^i,0=𝒚i,0,\\widehat\{\\bm\{y\}\}\_\{i,t\+1\}=\\mathcal\{B\}\\\!\\left\(\\widehat\{\\bm\{y\}\}\_\{i,t\}\+\\Delta t\\,\\widehat\{\\bm\{v\}\}\_\{t\}\(\\widehat\{\\bm\{y\}\}\_\{i,t\}\)\\right\),\\qquad\\widehat\{\\bm\{y\}\}\_\{i,0\}=\\bm\{y\}\_\{i,0\},\(4\)whereℬ\\mathcal\{B\}enforces the physical boundary condition during particle integration, ensuring that the updated particle positions remain within the spatial domainΩ\\Omega\. The Eulerian state continues to evolve through Equation \([3](https://arxiv.org/html/2608.14120#Sx3.E3)\); the particles only determine where the predicted velocity field is evaluated\. Thus, in the Eulerian\-only setting, the same trained operator predicts fields on the fixed grid and provides velocities at moving particle positions\. The particle trajectories are then obtained by numerical integration, without training a separate particle\-dynamics model\.

#### Field and trajectory errors\.

Since Lagrangian rollout is driven by velocities queried from the predicted flow, fixed\-grid field error alone does not fully characterize model performance\. We therefore distinguish errors in Eulerian field prediction, velocity evaluation at particle positions, and the resulting closed\-loop trajectories\. We refer to these quantities as*Eul*,*Ref*, and*Path*, respectively, and report them throughout the experiments\.

LetHHdenote the rollout horizon,NEN\_\{E\}the number of Eulerian grid points, andNpN\_\{p\}the number of particles\. We usedud\_\{u\},dvd\_\{v\}, anddxd\_\{x\}for the dimensions of the predicted state, velocity, and particle position, respectively\.

First, the fixed\-grid Eulerian error is

ℰEul:=1H​NE​du​∑t=1H∑𝒈∈XE‖𝒖^t​\(𝒈\)−𝒖t​\(𝒈\)‖22\.\\mathcal\{E\}\_\{\\mathrm\{Eul\}\}:=\\frac\{1\}\{HN\_\{E\}d\_\{u\}\}\\sum\_\{t=1\}^\{H\}\\sum\_\{\\bm\{g\}\\in X\_\{E\}\}\\left\\\|\\widehat\{\\bm\{u\}\}\_\{t\}\(\\bm\{g\}\)\-\\bm\{u\}\_\{t\}\(\\bm\{g\}\)\\right\\\|\_\{2\}^\{2\}\.\(5\)This measures the autoregressive field prediction on the Eulerian supportXEX\_\{E\}\.

To evaluate velocity without position feedback, we query at the reference positions𝒚i,t\\bm\{y\}\_\{i,t\}:

ℰRef:=1H​Np​dv​∑t=1H∑i=1Np‖𝒗^t​\(𝒚i,t\)−𝒗t​\(𝒚i,t\)‖22\.\\mathcal\{E\}\_\{\\mathrm\{Ref\}\}:=\\frac\{1\}\{HN\_\{p\}d\_\{v\}\}\\sum\_\{t=1\}^\{H\}\\sum\_\{i=1\}^\{N\_\{p\}\}\\left\\\|\\widehat\{\\bm\{v\}\}\_\{t\}\(\\bm\{y\}\_\{i,t\}\)\-\\bm\{v\}\_\{t\}\(\\bm\{y\}\_\{i,t\}\)\\right\\\|\_\{2\}^\{2\}\.\(6\)The Eulerian field is still predicted autoregressively, but the query coordinates are reset to the reference pathline at every step\. Hence,ℰRef\\mathcal\{E\}\_\{\\mathrm\{Ref\}\}measures velocity accuracy along the correct trajectory without propagating particle\-position errors\.

Finally, in the closed\-loop rollout, the particle positions𝒚^i,t\\widehat\{\\bm\{y\}\}\_\{i,t\}are recursively generated using the model’s queried velocities\. The resulting trajectory error is

ℰPath:=1H​Np​dx​∑t=1H∑i=1NpdΩ2​\(𝒚^i,t,𝒚i,t\),\\mathcal\{E\}\_\{\\mathrm\{Path\}\}:=\\frac\{1\}\{HN\_\{p\}d\_\{x\}\}\\sum\_\{t=1\}^\{H\}\\sum\_\{i=1\}^\{N\_\{p\}\}d\_\{\\Omega\}^\{2\}\\\!\\left\(\\widehat\{\\bm\{y\}\}\_\{i,t\},\\bm\{y\}\_\{i,t\}\\right\),\(7\)wheredΩd\_\{\\Omega\}accounts for the domain geometry and uses the minimum\-image distance on periodic domains\.

## Transferable Latent Operator

TLOfollows an encode–process–decode architecture\([16](https://arxiv.org/html/2608.14120#bib.bib4)\)that separates latent flow modeling from coordinate\-dependent field evaluation\. Given an Eulerian observation onXE=\{𝒙i\}i=1NEX\_\{E\}=\\\{\\bm\{x\}\_\{i\}\\\}\_\{i=1\}^\{N\_\{E\}\}, we denote its sampled state by𝐔t=\[𝒖t​\(𝒙i\)\]i=1NE\\mathbf\{U\}\_\{t\}=\[\\bm\{u\}\_\{t\}\(\\bm\{x\}\_\{i\}\)\]\_\{i=1\}^\{N\_\{E\}\}\. The model advances this state by one time step and evaluates the predicted field at a specified query setQ=\{𝒒j\}j=1NQQ=\\\{\\bm\{q\}\_\{j\}\\\}\_\{j=1\}^\{N\_\{Q\}\}\.

Formally, the one\-step operator𝒢θ\\mathcal\{G\}\_\{\\theta\}is decomposed into an encoderℰθ\\mathcal\{E\}\_\{\\theta\}, a latent processor𝒯θ\\mathcal\{T\}\_\{\\theta\}, and a coordinate\-conditioned decoder𝒟θ\\mathcal\{D\}\_\{\\theta\}:

𝒁t0\\displaystyle\\bm\{Z\}\_\{t\}^\{0\}=ℰθ​\(XE,𝐔t\),\\displaystyle=\\mathcal\{E\}\_\{\\theta\}\(X\_\{E\},\\mathbf\{U\}\_\{t\}\),\(8\)𝒁tL\\displaystyle\\bm\{Z\}\_\{t\}^\{L\}=𝒯θ​\(𝒁t0\),\\displaystyle=\\mathcal\{T\}\_\{\\theta\}\(\\bm\{Z\}\_\{t\}^\{0\}\),𝐔^t\+1\|Q\\displaystyle\\widehat\{\\mathbf\{U\}\}\_\{t\+1\}\\big\|\_\{Q\}=𝒟θ​\(𝒁tL,Q\)\.\\displaystyle=\\mathcal\{D\}\_\{\\theta\}\(\\bm\{Z\}\_\{t\}^\{L\},Q\)\.The encoder constructs a latent representation of the observed flow, the processor advances this representation, and the decoder evaluates the predicted state at the requested coordinates\. Importantly, the encoder and latent processor are independent ofQQ; only the final readout depends on the output coordinates\. SettingQ=XEQ=X\_\{E\}produces the next Eulerian field, whereas settingQQto the current particle positions provides the velocities used for Lagrangian rollout\. The following sections describe each component in detail\.

Patch Encoderemploys cross\-attention as a permutation\-invariant aggregation mechanism to map local and global observations into a fixed\-size latent representation\.

To capture both localized transport dynamics and long\-range flow dependencies, we construct a hierarchical domain representation consisting of local patchesΩp\\Omega\_\{p\}and the global domainΩg\\Omega\_\{g\}:

𝒟=\{Ω1,…,ΩP⏟local patches,Ωg⏟global context\},Ωg=⋃p=1PΩp\.\\mathcal\{D\}=\\\{\\underbrace\{\\Omega\_\{1\},\\dots,\\Omega\_\{P\}\}\_\{\\text\{local patches\}\},\\underbrace\{\\Omega\_\{g\}\}\_\{\\text\{global context\}\}\\\},\\qquad\\Omega\_\{g\}=\\bigcup\_\{p=1\}^\{P\}\\Omega\_\{p\}\.
For each regionΩ∗∈𝒟\\Omega\_\{\*\}\\in\\mathcal\{D\}, we first lift coordinate\-attached observations into position\-physical features\. Specifically,

𝒓i\\displaystyle\\bm\{r\}\_\{i\}=ϕ⁡\[PE⁡\(𝒙i\),PE⁡\(𝒙ip\),𝒖i,ep\],\\displaystyle=\\phi\\left\[\\mathrm\{PE\}\(\\bm\{x\}\_\{i\}\),\\mathrm\{PE\}\(\\bm\{x\}\_\{i\}^\{\\,p\}\),\\bm\{u\}\_\{i\},e\_\{p\}\\right\],R∗\\displaystyle R\_\{\*\}=\{𝒓i∣𝒙i∈Ω∗\},\\displaystyle=\\\{\\bm\{r\}\_\{i\}\\mid\\bm\{x\}\_\{i\}\\in\\Omega\_\{\*\}\\\},where𝒙i\\bm\{x\}\_\{i\}denotes the global coordinate inΩ\\Omega,𝒙ip\\bm\{x\}\_\{i\}^\{\\,p\}represents the local coordinate relative to its patch, and𝒖i\\bm\{u\}\_\{i\}is the associated physical state\. The fixed Fourier feature encodingPE⁡\(⋅\)\\mathrm\{PE\}\(\\cdot\)provides continuous coordinate information, while the learnable patch embeddingepe\_\{p\}identifies local spatial regions\. The lifting functionϕ\\phifollows the standard formulation used in neural operator learning\([9](https://arxiv.org/html/2608.14120#bib.bib2)\)\.

For each local patch and the global domain, learnable latent queries𝑸l\\bm\{Q\}^\{l\}and𝑸g\\bm\{Q\}^\{g\}aggregate observations through cross\-attention:

𝒁∗=Attn⁡\(𝑸∗,K⁡\(R∗\),V⁡\(R∗\)\),\\bm\{Z\}\_\{\*\}=\\mathrm\{Attn\}\(\\bm\{Q\}^\{\*\},K\(R\_\{\*\}\),V\(R\_\{\*\}\)\),where∗\*indicates either a local patch or the global domain\. The resulting latent tokens from different spatial regions are fused to form the initial latent state:

𝒁0=concat​\(\{𝒁∗\}\)\\bm\{Z\}^\{0\}=\\text\{concat\}\(\\\{\\bm\{Z\}\_\{\*\}\\\}\)By integrating local coordinate encoding with global context aggregation, the hierarchical Patch Encoder converts coordinate\-attached observations into a discretization\-independent latent representation\. This latent state preserves both local transport structures and global flow dependencies, providing a unified representation that can be queried on fixed Eulerian grids or evolving Lagrangian particle trajectories\.

The Latent Processorlearns the temporal evolution of underlying dynamics and separats physical evolution from spatial discretization\. We implement the Latent Processor withLLattention\-based blocks inspired by the slice\-based aggregation mechanism in Transolver\([26](https://arxiv.org/html/2608.14120#bib.bib15)\)\. Different from coordinate\-conditioned neural operators, the proposed processor operates purely on latent tokens\.

At layerℓ\\ell, the latent tokens are flattened as𝒁ℓ∈ℝN×h\\bm\{Z\}^\{\\ell\}\\in\\mathbb\{R\}^\{N\\times h\}, whereNNdenotes the total number of latent tokens\. The processor first assigns tokens intoMMadaptive latent slices:

𝑾ℓ=softmax⁡\(ψ⁡\(𝒁ℓ\)\)∈ℝN×M,\\bm\{W\}^\{\\ell\}=\\operatorname\{softmax\}\(\\psi\(\\bm\{Z\}^\{\\ell\}\)\)\\in\\mathbb\{R\}^\{N\\times M\},where each row of𝑾ℓ\\bm\{W\}^\{\\ell\}represents the contribution of a token to the latent slices\. The slice representations are obtained by weighted and are then updated through self\-attention:

𝑺ℓ=Attn⁡\(𝑺^ℓ\),𝑺^ℓ=𝑾~ℓ,T​𝒁ℓ,\\bm\{S\}^\{\\ell\}=\\operatorname\{Attn\}\(\\hat\{\\bm\{S\}\}^\{\\ell\}\),\\,\\hat\{\\bm\{S\}\}^\{\\ell\}=\\widetilde\{\\bm\{W\}\}^\{\\ell,T\}\\bm\{Z\}^\{\\ell\},whereW~i,jℓ=Wi,jℓ/∑iWi,jℓ\\widetilde\{W\}^\{\\ell\}\_\{i,j\}=W^\{\\ell\}\_\{i,j\}/\\sum\_\{i\}W^\{\\ell\}\_\{i,j\}\. The updated slice information is distributed back to the latent tokens:

𝒁ℓ\+1=𝒁ℓ\+Proj⁡\(𝑾ℓ​𝑺ℓ\)\+α​FFN⁡\(𝒁ℓ\)\.\\bm\{Z\}^\{\\ell\+1\}=\\bm\{Z\}^\{\\ell\}\+\\operatorname\{Proj\}\(\\bm\{W\}^\{\\ell\}\\bm\{S\}^\{\\ell\}\)\+\\alpha\\operatorname\{FFN\}\(\\bm\{Z\}^\{\\ell\}\)\.
Through latent\-space evolution independent of spatial coordinates, the processor provides a reusable dynamic representationZLZ^\{L\}that can be decoded at arbitrary Eulerian locations or evolving Lagrangian particle positions\.

Query Decoderprovides a coordinate\-conditioned readout from the coordinate\-independent latent state, decoupling latent dynamics evolution from spatial discretization\. For each query point, we first identify its corresponding patchΩp⁡\(j\)\\Omega\_\{p\(j\)\}and construct a query representation

𝒈j=ψ⁡\[PE⁡\(𝒒j\),PE⁡\(𝒒jp\)\],\\bm\{g\}\_\{j\}=\\psi\[\\mathrm\{PE\}\(\\bm\{q\}\_\{j\}\),\\mathrm\{PE\}\(\\bm\{q\}\_\{j\}^\{\\,p\}\)\],using the previous embedding strategy\. The physical quantity at the query location is recovered through cross\-attention between the query feature and latent tokens from neighboring patches:

𝒖^​\(𝒒j\)=∑pk∈𝒩⁡\(𝒒j\)αk​Attn⁡\(Q⁡\(𝒈j\),K⁡\(𝒁pk\),V⁡\(𝒁pk\)\),\\hat\{\\bm\{u\}\}\(\\bm\{q\}\_\{j\}\)=\\sum\_\{p\_\{k\}\\in\\mathcal\{N\}\(\\bm\{q\}\_\{j\}\)\}\\alpha\_\{k\}\\operatorname\{Attn\}\\left\(Q\(\\bm\{g\}\_\{j\}\),K\(\\bm\{Z\}\_\{p\_\{k\}\}\),V\(\\bm\{Z\}\_\{p\_\{k\}\}\)\\right\),where𝒩⁡\(𝒒j\)\\mathcal\{N\}\(\\bm\{q\}\_\{j\}\)denotes the set of neighboring patches around the patch containing𝒒j\\bm\{q\}\_\{j\}, andαk\\alpha\_\{k\}represents the aggregation weight of each neighboring patch\.

## Error Analysis

TLOpredicts velocity, whereas a Lagrangian rollout is ultimately evaluated through particle position\. Consider particleiiover a rollout ofTTsteps, indexed byt=0,…,Tt=0,\\ldots,T\. LetΔ​t\\Delta tbe the integration step, and let𝒗t\\bm\{v\}\_\{t\}and𝒗^t\\widehat\{\\bm\{v\}\}\_\{t\}denote the reference and predicted velocity fields at steptt, respectively\. The corresponding reference and predicted particle positions are denoted by𝒚i,t\\bm\{y\}\_\{i,t\}and𝒚^i,t\\widehat\{\\bm\{y\}\}\_\{i,t\}\.

Using forward Euler, their positions evolve as

𝒚i,t\+1\\displaystyle\\bm\{y\}\_\{i,t\+1\}=ℬ⁡\(𝒚i,t\+Δ​t​𝒗t​\(𝒚i,t\)\),\\displaystyle=\\mathcal\{B\}\\\!\\left\(\\bm\{y\}\_\{i,t\}\+\\Delta t\\,\\bm\{v\}\_\{t\}\(\\bm\{y\}\_\{i,t\}\)\\right\),\(9\)𝒚^i,t\+1\\displaystyle\\widehat\{\\bm\{y\}\}\_\{i,t\+1\}=ℬ\(𝒚^i,t\+Δt𝒗^t\(𝒚^i,t\)\),t=0,…,T−1\.\\displaystyle=\\mathcal\{B\}\\\!\\left\(\\widehat\{\\bm\{y\}\}\_\{i,t\}\+\\Delta t\\,\\widehat\{\\bm\{v\}\}\_\{t\}\(\\widehat\{\\bm\{y\}\}\_\{i,t\}\)\\right\),\\,t=0,\\ldots,T\-1\.whereℬ\\mathcal\{B\}is the bounce operator that handles particles that reach the boundary of the physical\-domain during rollout\. Define the position error and the velocity error evaluated at the reference particle position as

δi,t:=dΩ​\(𝒚^i,t,𝒚i,t\),ri,t:=‖𝒗^t​\(𝒚i,t\)−𝒗t​\(𝒚i,t\)‖2\.\\delta\_\{i,t\}:=d\_\{\\Omega\}\(\\widehat\{\\bm\{y\}\}\_\{i,t\},\\bm\{y\}\_\{i,t\}\),\\,r\_\{i,t\}:=\\left\\\|\\widehat\{\\bm\{v\}\}\_\{t\}\(\\bm\{y\}\_\{i,t\}\)\-\\bm\{v\}\_\{t\}\(\\bm\{y\}\_\{i,t\}\)\\right\\\|\_\{2\}\.\(10\)
###### Proposition 1\(Velocity error to trajectory error\)\.

Assume that𝐲^i,0=𝐲i,0\\widehat\{\\bm\{y\}\}\_\{i,0\}=\\bm\{y\}\_\{i,0\}, thatℬ\\mathcal\{B\}is non\-expansive underdΩd\_\{\\Omega\}, and that𝐯^t\\widehat\{\\bm\{v\}\}\_\{t\}isL^t\\widehat\{L\}\_\{t\}\-Lipschitz over the region visited by the two particles\. Then, for everyt=0,…,T−1t=0,\\ldots,T\-1,

δi,t\+1≤\(1\+Δ​t​L^t\)​δi,t\+Δ​t​ri,t\.\\delta\_\{i,t\+1\}\\leq\(1\+\\Delta t\\,\\widehat\{L\}\_\{t\}\)\\delta\_\{i,t\}\+\\Delta t\\,r\_\{i,t\}\.\(11\)Consequently, the position error at the rollout horizonTTsatisfies

δi,T≤Δ​t​∑s=0T−1ri,s​∏k=s\+1T−1\(1\+Δ​t​L^k\),\\delta\_\{i,T\}\\leq\\Delta t\\sum\_\{s=0\}^\{T\-1\}r\_\{i,s\}\\prod\_\{k=s\+1\}^\{T\-1\}\(1\+\\Delta t\\,\\widehat\{L\}\_\{k\}\),\(12\)with the convention that an empty product equals one\.

Proposition[1](https://arxiv.org/html/2608.14120#Thmproposition1)connects the velocity predicted by the model to the particle position evaluated in a Lagrangian rollout\. At each step, a velocity errorri,tr\_\{i,t\}causes an immediate position error of approximatelyΔ​t​ri,t\\Delta t\\,r\_\{i,t\}\. The updated position is then used as the query coordinate at the next step\. Therefore, once the predicted particle deviates from the reference path, subsequent velocities are evaluated at a different location, which may introduce further error\. The factor1\+Δ​t​L^t1\+\\Delta t\\,\\widehat\{L\}\_\{t\}describes the strength of this feedback\. Over a long rollout, these errors accumulate, and errors made earlier can affect more subsequent steps\. Thus, a small*Ref vel\.*error is important but does not necessarily imply a small*Path*error, which motivates reporting both metrics\.

## Experiments

#### Benchmarks\.

We evaluateTLOon five fluid benchmarks covering particle\-based simulations, grid\-based PDEs, and real\-world SEA \(ocean current\)\. DAM2D and TGV3D are SPH datasets from LagrangeBench\([20](https://arxiv.org/html/2608.14120#bib.bib5)\)and are converted to Eulerian fields by kernel splatting; see supplementary material for details\. NS2D, Burgers3D, and SEA provide grid\-based velocity fields from, respectively, incompressible Navier–Stokes dynamics, nonlinear advection–diffusion, and ocean reanalysis data\([7](https://arxiv.org/html/2608.14120#bib.bib6);[6](https://arxiv.org/html/2608.14120#bib.bib7);[12](https://arxiv.org/html/2608.14120#bib.bib17)\)\. DAM2D uses its native particle trajectories, while virtual tracers for Burgers3D and SEA are generated from the reference velocity fields using Parcels\([4](https://arxiv.org/html/2608.14120#bib.bib8)\)\. These particles are passive tracers and do not influence the Eulerian dynamics\.

Table 1:Benchmark summary\. Eul, Ref, and Path denote fixed\-grid field, reference\-query velocity, and closed\-loop trajectory evaluation\.
#### Baselines\.

We compare with LSM\([25](https://arxiv.org/html/2608.14120#bib.bib10)\), GINO\([10](https://arxiv.org/html/2608.14120#bib.bib16)\), GNOT\([5](https://arxiv.org/html/2608.14120#bib.bib12)\), Transolver\([26](https://arxiv.org/html/2608.14120#bib.bib15)\), UPT\([1](https://arxiv.org/html/2608.14120#bib.bib14)\), LNO\([23](https://arxiv.org/html/2608.14120#bib.bib13)\), and DeepLag\([12](https://arxiv.org/html/2608.14120#bib.bib17)\), covering latent\-space, geometry\-aware, attention\-based, and Eulerian–Lagrangian operator architectures\. For particle evaluation, query\-native models are evaluated directly at particle coordinates, whereas grid\-output models use bilinear or trilinear interpolation of their predicted fields\. We additionally evaluate both direct decoding and grid interpolation from the sameTLOcheckpoint\.

#### Training and evaluation\.

All models are trained for one\-step Eulerian field prediction and evaluated through autoregressive rollout\. We report fixed\-grid field error \(*Eul*\), velocity error along reference particle trajectories \(*Ref*\), and closed\-loop particle\-position error \(*Path*\), as defined in Eqs\. \(5\)–\(7\)\. For Ref, reference particle positions are supplied as queries at each step, while the predicted field continues to evolve autoregressively\. For Path, only the initial particle positions are given; subsequent positions are obtained by integrating the predicted velocities and are used as the next queries\. Thus, Path measures the accumulated effect of velocity errors on the particle trajectory\. The main experiments use only Eulerian training data, with the sparse decoder\-adaptation study reported separately\.

### Eulerian prediction

#### Autoregressive Eulerian field rollout\.

We first examine whetherTLOpreserves predictive accuracy in the standard Eulerian setting\. Starting from the ground\-truth initial field, each model recursively predicts the next state on the fixed Eulerian grid and feeds its previous prediction back as input\. We report the fixed\-grid field error \(*Eul*\) defined in Eq\. \(5\), averaged over all rollout stepsHH, spatial locations, state channels, and test trajectories\. This evaluation isolates temporal field\-forecasting error from spatial\-query error and particle\-position feedback\.

As shown in Table[2](https://arxiv.org/html/2608.14120#Sx6.T2),TLOachieves strong Eulerian rollout performance across all five benchmarks\. The improvements cover SPH\-derived and grid\-native data, two\- and three\-dimensional systems, and both simulated and reanalysis flows\. These results show thatTLOretains strong fixed\-grid Eulerian forecasting accuracy while supporting moving particle queries through the same learned representation\.

Table 2:Mean squared error \(MSE\) of autoregressive Eulerian rollout\. Lower is better\. The best result in each column is highlighted in bold\.
#### Model ablations analysis\.

Figure[3](https://arxiv.org/html/2608.14120#Sx6.F3)shows that the12×1212\\times 12patch lattice yields the lowest error, suggesting a trade\-off between finer spatial partitioning and the amount of information available within each patch\.

Under the same total token budget, Local\+Global consistently outperforms Local, showing that its gain cannot be explained by additional latent capacity alone\. And the Local\+Global error exhibits an overall downward trend and reaches its lowest value at 160 tokens\. These results support the complementary roles of local tokens for spatially resolved information and global tokens for domain\-level context\.

![Refer to caption](https://arxiv.org/html/2608.14120v1/Figures/ns2d_ablation.png)Figure 3:Patch and token ablations on NS2D\. Left: Eul MSE versus patch lattice size\. Right: two models use matched total tokens, with a fixed8×88\\times 8patch and global token equals 8 for Local\+Global\. Lower is better\.In Table[3](https://arxiv.org/html/2608.14120#Sx6.T3), all variants are controlled with comparable model sizes to ensure a fair comparison\. TLO has 150\.8K parameters versus 146\.7K without FFN, while Local\-only and Global\-only use the same number of tokens as TLO\. Thus, the gains cannot be explained solely by model capacity, but arise from the local–global representation and latent processing\.

Table 3:Ablation of TLO components on NS2D under comparable model sizes\. Token\-allocation variants share the same 64\-token budget\. Lower L2 error is better\.

### Transfer from Eulerian to Lagrangian rollout

Having established strong Eulerian forecasting accuracy, we next evaluate whether the learned dynamics can transfer from fixed\-grid field prediction to closed\-loop rollout at moving particle coordinates\. Because the latent processor inTLOevolves the flow representation independently of the output query set, the same Eulerian\-trained checkpoint can be used by changing the decoder queries from fixed\-grid locations to recursively advected particle positions\.

We first consider the*Eulerian\-only*\(EO\) setting\. The model is trained using only Eulerian field supervision, without particle coordinates, particle velocities, or trajectory labels during training\. At inference time, particle positions are introduced only as decoder queries, and the queried velocities are recursively integrated to form a closed\-loop rollout\. As shown in Table[4](https://arxiv.org/html/2608.14120#Sx6.T4),TLOachieves the lowest Ref and Path errors among the compared methods on all three Lagrangian benchmarks\. These results show that an Eulerian\-trainedTLOcheckpoint can transfer directly to moving\-particle queries without particle\-based training supervision or parameter adaptation\. They also motivate evaluating closed\-loop trajectories in addition to fixed\-grid forecasting accuracy\.

We further consider a*decoder\-adapted*\(DA\) setting\. Starting from the Eulerian\-trained checkpoint, we fine\-tune only the decoder using sparse velocity samples at particle coordinates, while keeping the encoder and latent processor fixed\. This adaptation provides additional, dataset\-dependent improvements in the coordinate readout\. The results are consistent with the Eulerian\-trained latent representation already containing transport\-relevant information, which can be further calibrated for moving\-coordinate evaluation through limited particle supervision\.

Figure[4](https://arxiv.org/html/2608.14120#Sx6.F4)provides a qualitative comparison of closed\-loop particle rollouts\. All methods start from the same initial particle positions and recursively update them using their predicted velocities\. The resulting trajectory deviations visualize the accumulated effects of velocity\-prediction error and feedback through recursively generated query locations\.

Table 4:Transfer from Eulerian training to Lagrangian rollout\. EO directly applies the Eulerian\-trained checkpoint without particle\-based training supervision\. ForTLO, DA fine\-tunes only the decoder using sparse velocity samples at particle coordinates\. Eul, Ref, and Path denote fixed\-grid field, reference\-path velocity, and closed\-loop particle\-position MSE, respectively\. Lower is better\.![Refer to caption](https://arxiv.org/html/2608.14120v1/Figures/sea_rollout_traj.png)Figure 4:Qualitative closed\-loop Lagrangian rollout on SEA\. Each column shows the Eulerian velocity field and the corresponding particle trajectories at different rollout steps \(T=2,6,10T=2,6,10\)\.#### Comparison with Lagrangian baseline\.

We compareTLO\-DA with GNS\([16](https://arxiv.org/html/2608.14120#bib.bib4)\), a native Lagrangian model trained from particle trajectories, in Table[5](https://arxiv.org/html/2608.14120#Sx6.T5)\. This comparison examines whether latent dynamics learned from Eulerian fields, together with lightweight decoder adaptation, can support competitive particle rollout\. GNS learns particle interactions directly through trajectory supervision, whereasTLO\-DA keeps its Eulerian\-trained encoder and latent processor fixed and fine\-tunes only the decoder using sparse velocity samples at particle coordinates, without a trajectory loss\.

On DAM2D,TLO\-DA achieves more accurate particle rollout with lower computational overhead than GNS\. This advantage comes from the different representations used for dynamics modeling: GNS explicitly propagates information among particles through graph message passing, whose cost grows with particle interactions, whileTLOevolves a compact latent flow representation and only decodes velocities at queried locations\. These results show that an Eulerian latent representation can recover accurate particle dynamics without the cost of explicit particle interaction modeling\.

Table 5:Comparison with a native Lagrangian model on DAM2D particle rollout\. TLO and GNS contain 144K and 146K parameters, respectively\. Memory and runtime are reported in MiB and ms/step, respectively\.

### Beyond\-Grid Lagrangian Rollout

We further evaluateTLOon Lagrangian rollouts beyond the fixed Eulerian grid support\. Starting from the same latent flow state, we compare two readout strategies:TLO\-Interp obtains particle velocities by interpolating the predicted Eulerian field, whereasTLO\-Direct evaluates the coordinate\-conditioned decoder at particle locations\.

Within the grid support, the two strategies achieve comparable accuracy\. Once particles leave the grid domain, however, standard interpolation is no longer defined without an additional extrapolation rule, whileTLO\-Direct can still be evaluated at the off\-grid particle coordinates\.TLO\-DA, which fine\-tunes only the decoder using sparse particle\-velocity supervision, further reduces the off\-grid rollout error\. These results highlight the advantage of decoupling latent flow evolution from coordinate\-dependent readout: Lagrangian rollout is not restricted to the spatial support of the Eulerian output grid\.

Table 6:Direct decoding and grid interpolation on DAM2D\. A dash indicates that interpolation cannot continue after grid exit\.![Refer to caption](https://arxiv.org/html/2608.14120v1/Figures/dam2d_beyond_grid_steps_updated_cropped.png)Figure 5:Closed\-loop rollout beyond the fixed Eulerian grid support on DAM2D\. Light\- and dark\-blue points show reference particles at Steps 15 and 20, respectively\. The inset follows a particle crossingΩgrid\\Omega\_\{\\mathrm\{grid\}\}: grid interpolation becomes undefined after the exit, whereas direct decoding continues the rollout\.

## Conclusion

To answer whether flow dynamics learned only from fixed\-grid Eulerian data can support closed\-loop Lagrangian particle rollout, we formulated zero\-shot Eulerian\-to\-Lagrangian generalization and developedTLO\. By separating query\-independent latent dynamics from coordinate\-conditioned decoding,TLOsupports both fixed\-grid field prediction and rollout at recursively evolving particle locations\. Across five field\-prediction and three particle\-rollout benchmarks,TLOachieves strong Eulerian accuracy and lower rollout errors than the evaluated baselines; sparse decoder adaptation further improves accuracy and outperforms GNS on DAM2D\. The decoder can also be queried after particles leave the fixed\-grid domain, whereas interpolation requires an additional extrapolation rule\. Overall, these results show that fixed\-grid supervision can learn transferable flow dynamics beyond fixed\-grid inference, while closed\-loop moving queries provide a stringent test of this transferability\.

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