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From Fixed Grids to Moving Particles:A Transferable Latent Operator for Fluid Dynamics
Meng Li, Chuqi Chen, Zhengqing Gao, Xi Zhou, Xiao Sun, Yang Xiang, Huaxi Huang
TL;DR
Most fluid-learning data are fixed-grid Eulerian fields, leaving open whether Eulerian-only training can support Lagrangian particle rollout. TLO learns a shared latent flow representation for both query modes and outperforms evaluated baselines across field prediction and particle-rollout benchmarks.
Problem
Because Lagrangian trajectories are less commonly available than fixed-grid Eulerian fields, the paper asks whether Eulerian-only training can support Lagrangian particle rollout.
Method
TLO separates query-independent latent flow evolution from coordinate-conditioned decoding, enabling fixed-grid field prediction and recursively queried particle rollout with one model.
Results
Across five field-prediction and three particle-rollout benchmarks, TLO achieves strong Eulerian accuracy and lower rollout errors than evaluated baselines; sparse decoder adaptation further improves accuracy and outperforms GNS on DAM2D.
Takeaways & Limitations
TLO remains queryable after particles leave the fixed-grid domain, whereas interpolation requires an additional extrapolation rule.
Takeaways & Limitations
Training supervision is provided only on consecutive Eulerian fields and targets on the fixed Eulerian grid.
Abstract
from arXiv · showhide
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. TLO decouples 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, TLO consistently outperforms existing neural operators in both Eulerian field prediction and zero-shot Lagrangian rollout, with further gains from limited Lagrangian fine-tuning.
Introduction
The paper formulates zero-shot Eulerian-to-Lagrangian generalization: models trained only on fixed-grid Eulerian fields must perform closed-loop particle rollout without particle supervision or parameter adaptation. TLO addresses this with shared latent flow evolution and coordinate-conditioned queries for both Eulerian forecasting and moving particles.
- Motivation: Lagrangian trajectories directly describe transport, mixing, and dispersion, but datasets more commonly provide time-dependent fields on fixed Eulerian grids.Native Lagrangian data are less common because particle populations must be specified and tracked.
- Problem formulation: Zero-shot Eulerian-to-Lagrangian generalization trains exclusively on Eulerian field supervision, then recursively integrates predicted velocities at moving particle positions without adaptation.Each prediction changes the next query location, so errors can cause history-dependent query shifts and trajectory drift.
- Related work: Existing latent-particle, joint field–trajectory, and trajectory-reconstruction methods typically require Lagrangian supervision or particle-specific modeling, preventing zero-shot transfer.Eulerian neural operators do not directly address rollout at recursively evolving particle locations.
- Method: TLO uses one latent flow evolution with two query modes, combining spatially localized encoded observations, global flow context, and a shared latent processor with coordinate-conditioned decoding.The same trained model supports fixed-grid forecasting and moving particle queries because latent evolution is independent of the output query.
- Evaluation: TLO is evaluated on five fluid-dynamics benchmarks for Eulerian forecasting and three benchmarks for zero-shot Lagrangian particle rollout, including reference-path velocity prediction and closed-loop trajectories.Direct decoding is competitive with interpolation within Eulerian support and remains evaluable after particles leave it; decoder-only adaptation is also studied with sparse particle-velocity supervision.
Related Work
Prior work spans neural operators for PDE function-space mappings and Eulerian–Lagrangian methods that connect fields with particles. Existing approaches include coordinate-flexible neural operators, particle-grid transfers, latent particles, particle-observation inference, and particle-based simulators.
- Neural operators: Neural operators learn mappings between function spaces for PDE modeling.Representative architectures include spectral, coordinate-based, attention-based, and token-based operators.
- Neural operators: Several neural operators support arbitrary-coordinate evaluation but are generally evaluated on fixed or externally specified queries.
- Eulerian–Lagrangian modeling: Classical particle-grid methods couple Eulerian fields and Lagrangian particles through transfers between the two representations.
- Eulerian–Lagrangian modeling: Recent neural methods use latent particles for Eulerian forecasting, infer fields or trajectories from particle observations, or learn particle-based simulators.
Problem Setting
The problem setting represents fluid states as fields observed on fixed Eulerian coordinates, while requiring an operator that can query predictions at arbitrary coordinates. Eulerian-only supervision must therefore support both fixed-grid field forecasting and velocity-driven Lagrangian particle rollout, evaluated with distinct error measures.
- State and observations: Fluid states are vector-valued fields whose channels may include velocity, pressure, density, or other physical variables, with velocity extracted by projection.The velocity field is defined as v_t(x) := Π_v u_t(x).
- State and observations: Available data consist of temporal sequences of field values observed on the same fixed Eulerian coordinates.The continuous field is not directly available; coordinates remain fixed while values evolve.
- Operator objective: The learned operator maps a current Eulerian observation to the next field at any finite query set, with grid queries yielding Eulerian forecasts and particle queries yielding local velocities.This coordinate-flexible querying is the basis for using one operator in both representations.
- Eulerian supervision: Training uses consecutive Eulerian fields as supervision, so particle coordinates and trajectory labels are unnecessary for learning field evolution.Autoregressive queries on the fixed grid produce the predicted Eulerian field sequence and evolving flow state.
- Lagrangian rollout: Lagrangian rollout recursively updates particle positions using velocities queried from the predicted field at current positions, while enforcing the physical boundary condition.Particles determine evaluation locations only; the Eulerian state continues its own evolution.
- Evaluation: Performance is separated into Eulerian field, reference-path velocity, and closed-loop trajectory errors, denoted Eul, ERef, and Path.This distinction reflects that fixed-grid field error alone does not capture velocity accuracy or accumulated particle-position drift.
Transferable Latent Operator
TLO separates latent flow evolution from coordinate-dependent decoding through an encode–process–decode architecture. Its unified latent representation supports both Eulerian field prediction on fixed grids and Lagrangian rollout at evolving particle positions.
- Architecture: TLO decomposes its one-step operator into an encoder, latent processor, and coordinate-conditioned decoder.The encoder represents the observed flow, the processor advances it, and the decoder evaluates the predicted state at requested coordinates.
- Architecture: The encoder and processor are independent of query coordinates, while only the final readout depends on the output coordinates.Querying at fixed Eulerian coordinates produces the next field; querying at particle positions provides velocities for Lagrangian rollout.
- Patch Encoder: The hierarchical Patch Encoder combines local patches and global-domain context into a discretization-independent latent representation.This representation preserves local transport structures and global flow dependencies and can be queried on fixed Eulerian grids or evolving particle trajectories.
- Latent Processor: The Latent Processor evolves latent tokens using attention-based blocks and operates purely in latent space, independent of spatial coordinates.The resulting dynamic representation can be decoded at arbitrary Eulerian locations or evolving Lagrangian particle positions.
- Query Decoder: The Query Decoder recovers physical quantities through coordinate-conditioned cross-attention between query features and neighboring-patch latent tokens.This readout decouples latent dynamics evolution from spatial discretization.
Error Analysis
The error analysis links TLO’s velocity prediction error to Lagrangian particle-position error under forward-Euler rollout. It shows that local velocity errors accumulate and can be amplified when position deviations change subsequent query locations.
- Rollout error mechanism: Forward-Euler rollout updates each predicted particle position by querying the predicted velocity at its current location and applying the boundary bounce operator.The update is byi,t+1 = B(byi,t + ∆t bvt(byi,t)).
- Error definitions: The analysis distinguishes position error δi,t from velocity error ri,t, with velocity error evaluated at the reference particle position.These errors are defined using domain distance and the L2 norm of predicted-versus-reference velocity.
- Proposition 1: Proposition 1 assumes matching initial positions, a non-expansive bounce operator, and a Lipschitz predicted velocity field over the region visited by both particles.Under these assumptions, the proposition bounds trajectory error over the rollout.
- Error propagation: At each step, velocity error ri,t causes an immediate position error of approximately ∆t ri,t.The resulting position deviation affects later rollout steps because the updated position becomes the next query coordinate.
- Error propagation: Once the predicted trajectory deviates from the reference path, subsequent velocities are evaluated at different locations, introducing further error through the factor 1 + ∆t bLt.This factor captures amplification associated with the Lipschitz behavior of the predicted velocity field.
Experiments
Experiments evaluate TLO across five fluid-dynamics benchmarks using autoregressive Eulerian rollouts and reference-path or closed-loop particle errors. TLO maintains strong fixed-grid forecasting, transfers zero-shot from Eulerian supervision to Lagrangian rollout, improves with decoder-only adaptation, and supports off-grid particle queries.
- Benchmarks: TLO is evaluated on five benchmarks spanning particle-based and grid-based simulations, plus real-world ocean reanalysis data.The benchmarks include DAM2D, TGV3D, NS2D, Burgers3D, and SEA, covering two- and three-dimensional flows.
- Evaluation protocol: All models train on one-step Eulerian field prediction and are evaluated autoregressively with fixed-grid, reference-path, and closed-loop particle-position errors.Eul measures fixed-grid field error, Ref measures velocity error along reference trajectories, and Path measures closed-loop particle-position error.
- Eulerian rollout: TLO achieves strong Eulerian rollout performance across all five benchmarks while supporting moving-particle queries through the same learned representation.The result holds across SPH-derived and grid-native data, two- and three-dimensional systems, and simulated and reanalysis flows.
- Model ablations: The 12×12 patch lattice yields the lowest error, while Local+Global consistently outperforms Local under matched total token budgets.Local+Global reaches its lowest error at 160 tokens, supporting complementary local spatial information and global domain context.
- Eulerian-to-Lagrangian transfer: In the Eulerian-only setting, TLO achieves the lowest Ref and Path errors among compared methods on all three Lagrangian benchmarks without particle-based training supervision.Particle positions are introduced only as decoder queries at inference, and queried velocities are recursively integrated into closed-loop rollouts.
- Decoder adaptation and off-grid rollout: Decoder-only fine-tuning with sparse particle-coordinate velocity samples provides additional dataset-dependent improvements and reduces off-grid rollout error.TLO-Direct remains evaluable beyond the Eulerian grid domain, unlike standard interpolation without an added extrapolation rule.
Conclusion
The paper formulates zero-shot Eulerian-to-Lagrangian generalization and develops TLO to test whether fixed-grid Eulerian learning supports closed-loop particle rollout. TLO separates latent flow dynamics from coordinate-conditioned decoding, enabling both field prediction and rollout at evolving particle locations.
- Conclusion: TLO addresses zero-shot Eulerian-to-Lagrangian generalization using flow dynamics learned only from fixed-grid Eulerian data.The goal is closed-loop Lagrangian particle rollout without relying on Lagrangian training data.
- Conclusion: TLO separates query-independent latent dynamics from coordinate-conditioned decoding.This design decouples flow evolution from the coordinates used to query predictions.
- Conclusion: The same representation supports fixed-grid field prediction and rollout at recursively evolving particle locations.Across five field-prediction and three particle-rollout benchmarks, the passage reports strong Eulerian accuracy, but provides no complete quantitative value.