# A DNN Framework for Learning Lagrangian Drift With Uncertainty

**Authors:** Joseph Jenkins, Adeline Paiement, Yann Ourmières, Julien Le Sommer, Jacques Verron, Clément Ubelmann, Hervé Glotin  
**arXiv:** [2204.05891](https://arxiv.org/abs/2204.05891)  
**Date:** 2022-04-12  
**Summary page:** https://www.summarizepaper.com/en/paper/2204.05891v1/

## Key points

- Data-driven framework for modelling probabilistic drift in flexible environments
- Focus on Lagrangian drift of objects lost at sea
- Existing approach to reconstructing Lagrangian drift is uncertain due to unresolved physical phenomena
- Stochasticity introduced into the drift to overcome uncertainty, but requires specific assumptions for modelling uncertainty
- Proposed framework trains a convolutional neural network (CNN) to predict temporal evolution of probability density maps of particle locations given input velocity field
- Groundtruth density maps generated based on ocean circulation model simulations by simulating uncertainty in initial position of particle trajectories
- Testing several loss functions for regressing predicted density maps and evaluating model on unseen velocities from a different year
- Model's outputs in good agreement with numerical simulations, indicating satisfactory generalization to different dynamical situations
- Review of existing approaches to modelling uncertain Lagrangian drift using stochastic trajectories through stochastic differential equations (SDEs)
- SDEs formulated as Fokker-Planck equations or fitted to simulated stochastic trajectories, but may not describe arbitrary sources of uncertainty and rely on specific physical quantities such as velocity information
- Previous works utilizing machine learning aim to model drift deterministically rather than probabilistically, lack generalization capabilities when spatio-temporal conditions deviate from training set
- Proposed data-driven framework uses CNN trained on groundtruth density maps generated from ocean circulation model simulations, allowing for probabilistic modeling without relying on specific physical quantities
- Model's ability to generalize demonstrated by evaluation on unseen velocities from a different year

## Summary

This article presents a data-driven framework for modelling probabilistic drift in flexible environments, specifically focusing on Lagrangian drift of objects lost at sea. The existing approach to reconstructing Lagrangian drift is often uncertain due to unresolved physical phenomena within the data. To overcome this uncertainty, stochasticity is introduced into the drift, but this requires specific assumptions for modelling uncertainty. The proposed framework removes this constraint by training a convolutional neural network (CNN) to predict the temporal evolution of probability density maps of particle locations given an input velocity field. The authors generate groundtruth density maps based on ocean circulation model simulations by simulating uncertainty in the initial position of particle trajectories. They test several loss functions for regressing the predicted density maps and evaluate their model on unseen velocities from a different year. The results show that the model's outputs are in good agreement with numerical simulations, indicating satisfactory generalization to different dynamical situations. In the previous works section, the authors review existing approaches to modelling uncertain Lagrangian drift. Stochastic trajectories through stochastic differential equations (SDEs) are commonly used to parameterize unresolved physics at subgrid scales. These SDEs can be formulated as Fokker-Planck equations or fitted to simulated stochastic trajectories. However, SDEs may not be able to describe arbitrary sources of uncertainty and rely on specific physical quantities such as velocity information. Previous works utilizing machine learning to predict Lagrangian drift aim to model drift deterministically rather than probabilistically. Some approaches train prediction models on individual instances of artificial simulated flows, while others learn from past observations of drifters at sea. However, these methods lack generalization capabilities when spatio-temporal conditions deviate from the training set. The proposed data-driven framework overcomes these limitations by using a CNN trained on groundtruth density maps generated from ocean circulation model simulations. This approach allows for probabilistic modeling of Lagrangian drift without relying on specific physical quantities. The authors demonstrate the model's ability to generalize to different dynamical situations by evaluating it on unseen velocities from a different year. Overall, this article presents a novel approach to modelling probabilistic drift in flexible environments, specifically focusing on Lagrangian drift of objects lost at sea.