Physical invariance in neural networks for subgrid-scale scalar flux modeling

In this paper we present a new strategy to model the subgrid-scale scalar flux in a three-dimensional turbulent incompressible flow using physics-informed neural networks (NNs). When trained from direct numerical simulation (DNS) data, state-of-the-art neural networks, such as convolutional neural networks, may not preserve well known physical priors, which may in turn question their application to real case-studies. To address this issue, we investigate hard and soft constraints into the model based on classical invariances and symmetries derived from physical laws. From simulation-based experiments, we show that the proposed physically-invariant NN model outperforms both purely data-driven ones as well as parametric state-of-the-art subgrid-scale model. The considered invariances are regarded as regularizers on physical metrics during the a priori evaluation and constrain the distribution tails of the predicted subgrid-scale term to be closer to the DNS. They also increase the stability and performance of the model when used as a surrogate during a large-eddy simulation. Moreover, the physically-invariant NN is shown to generalize to configurations that have not been seen during the training phase.

Variational Deep Learning for the Identification and Reconstruction of Chaotic and Stochastic Dynamical Systems from Noisy and Partial Observations

The data-driven recovery of the unknown governing equations of dynamical systems has recently received an increasing interest. However, the identification of the governing equations remains challenging when dealing with noisy and partial observations. Here, we address this challenge and investigate variational deep learning schemes. Within the proposed framework, we jointly learn an inference model to reconstruct the true states of the system from series of noisy and partial data and the governing equations of these states. In doing so, this framework bridges classical data assimilation and state-of-the-art machine learning techniques and we show that it generalizes state-of-the-art methods. Importantly, both the inference model and the governing equations embed stochastic components to account for stochastic variabilities, model errors and reconstruction uncertainties. Various experiments on chaotic and stochastic dynamical systems support the relevance of our scheme w.r.t. state-of-the-art approaches.

Detection of Abnormal Vessel Behaviours from AIS data using GeoTrackNet: from the Laboratory to the Ocean

The constant growth of maritime traffic leads to the need of automatic anomaly detection, which has been attracting great research attention. Information provided by AIS (Automatic Identification System) data, together with recent outstanding progresses of deep learning, make vessel monitoring using neural networks (NNs) a very promising approach. This paper analyses a novel neural network we have recently introduced -- GeoTrackNet -- regarding operational contexts. Especially, we aim to evaluate (i) the relevance of the abnormal behaviours detected by GeoTrackNet with respect to expert interpretations, (ii) the extent to which GeoTrackNet may process AIS data streams in real time. We report experiments showing the high potential to meet operational levels of the model.

Learning Variational Data Assimilation Models and Solvers

This paper addresses variational data assimilation from a learning point of view. Data assimilation aims to reconstruct the time evolution of some state given a series of observations, possibly noisy and irregularly-sampled. Using automatic differentiation tools embedded in deep learning frameworks, we introduce end-to-end neural network architectures for data assimilation. It comprises two key components: a variational model and a gradient-based solver both implemented as neural networks. A key feature of the proposed end-to-end learning architecture is that we may train the NN models using both supervised and unsupervised strategies. Our numerical experiments on Lorenz-63 and Lorenz-96 systems report significant gain w.r.t. a classic gradient-based minimization of the variational cost both in terms of reconstruction performance and optimization complexity. Intriguingly, we also show that the variational models issued from the true Lorenz-63 and Lorenz-96 ODE representations may not lead to the best reconstruction performance. We believe these results may open new research avenues for the specification of assimilation models in geoscience.

Joint learning of variational representations and solvers for inverse problems with partially-observed data

Designing appropriate variational regularization schemes is a crucial part of solving inverse problems, making them better-posed and guaranteeing that the solution of the associated optimization problem satisfies desirable properties. Recently, learning-based strategies have appeared to be very efficient for solving inverse problems, by learning direct inversion schemes or plug-and-play regularizers from available pairs of true states and observations. In this paper, we go a step further and design an end-to-end framework allowing to learn actual variational frameworks for inverse problems in such a supervised setting. The variational cost and the gradient-based solver are both stated as neural networks using automatic differentiation for the latter. We can jointly learn both components to minimize the data reconstruction error on the true states. This leads to a data-driven discovery of variational models. We consider an application to inverse problems with incomplete datasets (image inpainting and multivariate time series interpolation). We experimentally illustrate that this framework can lead to a significant gain in terms of reconstruction performance, including w.r.t. the direct minimization of the variational formulation derived from the known generative model.

Filtering Internal Tides From Wide-Swath Altimeter Data Using Convolutional Neural Networks

The upcoming Surface Water Ocean Topography (SWOT) satellite altimetry mission is expected to yield two-dimensional high-resolution measurements of Sea Surface Height (SSH), thus allowing for a better characterization of the mesoscale and submesoscale eddy field. However, to fulfill the promises of this mission, filtering the tidal component of the SSH measurements is necessary. This challenging problem is crucial since the posterior studies done by physical oceanographers using SWOT data will depend heavily on the selected filtering schemes. In this paper, we cast this problem into a supervised learning framework and propose the use of convolutional neural networks (ConvNets) to estimate fields free of internal tide signals. Numerical experiments based on an advanced North Atlantic simulation of the ocean circulation (eNATL60) show that our ConvNet considerably reduces the imprint of the internal waves in SSH data even in regions unseen by the neural network. We also investigate the relevance of considering additional data from other sea surface variables such as sea surface temperature (SST).

PDE-NetGen 1.0: from symbolic PDE representations of physical processes to trainable neural network representations

Bridging physics and deep learning is a topical challenge. While deep learning frameworks open avenues in physical science, the design of physically-consistent deep neural network architectures is an open issue. In the spirit of physics-informed NNs, PDE-NetGen package provides new means to automatically translate physical equations, given as PDEs, into neural network architectures. PDE-NetGen combines symbolic calculus and a neural network generator. The later exploits NN-based implementations of PDE solvers using Keras. With some knowledge of a problem, PDE-NetGen is a plug-and-play tool to generate physics-informed NN architectures. They provide computationally-efficient yet compact representations to address a variety of issues, including among others adjoint derivation, model calibration, forecasting, data assimilation as well as uncertainty quantification. As an illustration, the workflow is first presented for the 2D diffusion equation, then applied to the data-driven and physics-informed identification of uncertainty dynamics for the Burgers equation.

GeoTrackNet-A Maritime Anomaly Detector using Probabilistic Neural Network Representation of AIS Tracks and A Contrario Detection

Representing maritime traffic patterns and detecting anomalies from them are key to vessel monitoring and maritime situational awareness. We propose a novel approach-referred to as GeoTrackNet-for maritime anomaly detection from AIS data streams. Our model exploits state-of-the-art neural network schemes to learn a probabilistic representation of AIS tracks, then uses a contrario detection to detect abnormal events. The neural network helps us capture complex and heterogeneous patterns in vessels' behaviors, while the a contrario detection takes into account the fact that the learned distribution may be location-dependent. Experiments on a real AIS dataset comprising more than 4.2 million AIS messages demonstrate the relevance of the proposed method. Keywords: AIS, maritime surveillance, deep learning, anomaly detection, variational recurrent neural networks, a contrario detection.