Réunion

Oral #1 (09h30-09h50)

Beyond PCA by explicitly taking into account system symmetries

Simon Kneer (Imperial College London), Taraneh Sayadi (Institut Jean-Le Rond D'Alemebert), Denis Sipp (ONERA/DAAA), Peter J. Schmid (KAUST / Arabie Saoudite), Georgios Rigas (Imperial College London)

Linear principal component analysis (PCA) experiences an increase in the dimensionality of the latent space when it is applied to configurations that exhibit symmetries. In this study, we introduce a novel machine learning embedding, which uses spatial transformer networks and Siamese networks to account for continuous and discrete symmetries, respectively. This embedding, which we term symmetry-aware PCA, will be applied to three configurations: Burger's equation exhibiting a continuous translation symmetry, flow in sudden expansion, a discrete reflexional symmetry, and Kolmogorov Flow which combines discrete shift-reflect and continuous translation symmetries. We will show a drastic increase in the number of modes required to represent given trajectories.

Oral #2 (09h50-10h10)

Modélisation continue d'un écoulement de particules par apprentissage: couplage CFD/IA

DARVES-BLANC Quentin, MARTIN Sylvain, GAVET Yann, KAURIC Guilhem, MACQUERON Corentin, NDIAYE Abibatou, BONNEFOY Olivier

Mines Saint-Etienne, Univ Lyon, CNRS, Orano

Se comportant tantôt comme des solides, des liquides ou des gaz tout en gardant leurs spécificités, les milieux granulaires présentent une dynamique complexe. De nombreuses méthodes permettent de simuler leur comportement, avec des degrés de précision différents.

Ainsi, la DEM (Discrete Element Method) inspirée par la dynamique moléculaire, permet une analyse précise, mais reste limitée à la fois au nombre de particules dans le système, et à la géométrie de ces dernières [1]. Elle ne permet donc pas de simuler le nombre de particules qu?on pourrait trouver aux échelles industrielles. Quant à elles, les méthodes continues ne sont pas concernées par ces problèmes. Pour autant, elles ont l?inconvénient de ne pas couvrir l?entièreté des régimes d?écoulement. Ce problème est dû au fait qu?il n?existe, pour l?instant, pas d?équation constitutive permettant de relier le tenseur des contraintes ?? aux autres grandeurs, qui couvre les différents régimes [2,3].

L?approche que nous nous proposons d?étudier consiste en un couplage de deux solveurs. Le premier utilise les méthodes de CFD (Computational Fluid Dynamics) pour simuler l?écoulement granulaire du point de vue continu. Le second, basé sur des méthodes d?apprentissage automatique (Réseau de neurones), est chargé de fournir les informations relatives au tenseur ?? au premier solveur.

Cet algorithme IA est entraîné à partir de données issues de simulations DEM qui auront été transformées (homogénéisées) de manière à les rendre continues. La quantité importante, ainsi que la diversité des données fournies par la DEM rendent l?entraînement d?un réseau de neurones capable de prédire ?? en fonction des autres grandeurs qu?elles soient continues ou discrètes.

La capacité de généralisation ainsi que la rapidité de restitution de ce type d?algorithme permettraient aux simulations d?être plus rapides que les méthodes classiques, et ce, aux échelles dites « industrielles ».

Nous proposons ici une présentation de la démarche générale, des outils utilisés et des premiers résultats.

Keynote #1 (10h10-11h10)

Coffee break (11h10-11h30)

Oral #3 (11h30-11h50)

Hyperspectral Image Unmixing with Neural Networks: Integration of Physics- Based and Data-Driven Models

Xiuheng Wang, *Min Zhao, *Jie Chen, ?Cédric Richard

?Laboratoire Lagrange, Université Côte d?Azur, 06000, Nice, France xiuheng.wang@oca.eu, cedric.richard@unice.fr

*Northwestern Polytechnical University, 710072, Xi'an, China dr.jie.chen@ieee.org, minzhao@mail.nwpu.edu.cn

Spectral unmixing is one of the most important quantitative analysis tasks in hyperspectral data processing. Conventional physics-based models are characterized by clear interpretation. However they may not be suitable for analyzing scenes with unknown complex physical characteristics. Data-driven methods have developed rapidly in recent years, in particular deep learning methods because they possess superior capability in modeling complex and nonlinear systems. Simply transferring these methods as black-boxes to conduct unmixing may lead to low physical interpretability and generalization ability. In this presentation, we shall show how hyperspectral unmixing methods can benefit from the advantages of both physics-based models and data-driven methods by means of deep neural network structures design, prior design and loss design. Most of these methods derive from a common mathematical optimization framework, and combine good interpretability with high accuracy.

Oral #4 (11h50-12h10)

Recognition models to learn dynamics from partial observations with neural ODEs

Mona Buisson-Fenet, Val ?ery Morgenthaler, Sebastian Trimpe, Florent Di Meglio

Mines Paris, Université PSL

Predicting the behavior of complex systems is of great importance in many fields. In engineering for instance, designing controllers for robotic systems requires accurate predictions of their evolution. The behavior of such systems often follows a certain structure, which can be inferred, for example from the laws of physics. Mathematically, this structure is often captured by differential equations. However, even an accurate model cannot account for all aspects and physical parameters can only be measured with uncertainty. Data-driven methods promise to enhance our predictive capabilities for ever more complex systems based on experimental data. In the case of dynamical systems, not all system states can usually be measured, so this experimental data is only partial and also noisy. In general, learning the dynamics is an ill-posed problem: given observations, many different state-space representations can explain them. Physical knowledge can help reduce the number of possible models and improve the performance of data-driven approaches.

Learning latent representations of the available data is an active research area in machine learning. In particular, neural ordinary differential equations (NODEs) were introduced by [1] and have since sparked significant interest, e.g., [2]. The aim is to approximate a vector field that generates the observed data following a continuous-time ODE with a neural network. However, as for most machine learning methods, little insight on the desired latent representation is usually provided. This leads to difficulties with the interpretability of the obtained models, and their capability to generalize.

We show that the NODE formulation is well-suited for learning dynamical systems from partial measurements [3]. It is both general enough to avoid needing a new design for each new system, but can also enforce a wide range of physical insight that may be available, allowing for a physically meaningful and interpretable model. The NODE optimization problem can directly include physical knowledge in the form of ?soft? priors (extra cost terms) or ?hard? priors (constraints).

However, in the case of partial observations, the unknown latent state must be estimated from data jointly with its dynamics. We propose to design recognition models to map the observations to the initial latent state. We discuss several approaches: a direct method that learns the mapping between the first few observations and the initial state, several designs based on nonlinear observer theory and in particular KKL observers [4] inspired by [5], and existing designs based on Recursive Neural Networks as in [1, 2]. We show that the KKL-based recognition models perform well and have desirable properties such as their filtering ability and the known size of the latent state. Such recognition models can then be embedded in the NODE formulation or any other optimization-based system identification algorithm, and enable learning dynamics from partial observations while enforcing prior knowledge.

Lunch break (12h10-14h00)

Poster session (14h00-15h00)

Keynote #2 (15h00-16h00)

Oral #5 (16h00-16h20)

Controlling earthquake-like instabilities using control theory and reinforcement learning

Efthymios Papachristos and Ioannis Stefanou

Ecole Centrale de Nantes

Earthquakes nucleate when large amounts of elastic energy are suddenly released due to abrupt sliding over seismic faults. Besides physical causes, this energy release can be also triggered by injecting large amounts of fluids in the earth's crust. Indeed, recent experience shows that injections can reactivate existing seismogenic faults and induce/trigger important earthquakes. However, one can see fluid injections from another perspective. The dependence of fault friction on fluid pressure can be used as an input for stabilization and control. Our mathematical derivations prove earthquake control is possible, provided that fault friction is bounded. Based on the mathematical theory of control, we derive a new class of controllers which achieve stabilization and asymptotic tracking of the underlying complex physical system which is uncertain, strongly non-linear and underactuated.

An alternative to the mathematical theory of control is reinforcement learning. With the rapid growth of machine learning, prediction-control problems are all the more tackled by function approximation models that learn how to control a specific task. Here, we show, for the first time, the possibility of controlling earthquake-like instabilities with deep reinforcement learning techniques. The controller is trained using a reduced model of the physical system, which embodies the main dynamics leading to earthquakes. Its robustness to unmodeled dynamics is explored through a parametric study.

Our studies are a first step towards minimizing seismicity in industrial projects (geothermal energy, hydrogen storage and CO2 sequestration) and for inspiring techniques for natural earthquake control and prevention.

Oral #6 (16h20-16h40)

Constrained Physical-Statistics Models for Dynamical System Identification and Prediction (ICLR 2022)

Marie Déchelle, Patrick Gallinari

ISIR, Sorbonne Université

Modeling dynamical systems combining prior physical knowledge and machine learning (ML) is promising in scientific problems when the underlying processes are not fully understood, e.g. when the dynamics is partially known. The modeling of such systems traditionally rely on ordinary or partial differential equations (ODE/PDE), and their resolution via numerical solvers and data assimilation. In real world applications, two main pitfalls occur: first the dynamics may only be partially known and thus do not fully represent the studied phenomena; second, the system state may only be partially observed as in ocean models. ML has become a complementary approach to traditional physics based models (denoted MB for model based).

In that perspective, recent lines of work tackle the learning of hybrid models relying on prior physical knowledge and machine learning. Efficiently learning such decompositions actually means solving two different tasks: system identification, i.e. estimating the parameters of the physical model, and prediction, i.e. recovering the trajectories associated to the dynamics. Both are essential for hybrid MB/ML models of dynamical systems. Whereas prediction aims at robust extrapolation, identification accounts for physical interpretability of the MB/ML model. The combination of physical models and deep learning is still an open area of research: without any prior knowledge, the recovered estimates of a dynamical system state are not physically plausible despite accurate predictions.

Such observations highlight the need to incorporate physically motivated constraints in the learning of hybrid models, e.g. through regularization penalties. To complete prior dynamical knowledge with a data-driven component and ensure interpretability of the decomposition, we work out a principled framework that generalizes previous attempts in the regularization of hybrid models. We introduce a novel way to recover well-posedness and interpretability in the learning of hybrid MB/ML models via the control of an upper bound, and propose a novel alternate-optimization algorithm to learn hybrid models. Finally, we extend our framework to incorporate auxiliary data when available to handle complex real-world data and experimentally evidence the soundness of our approach on complex settings of increasing difficulty including challenging real world problems, namely ocean surface currents modeling.

Oral #7 (16h40-17h00)

Filtered CoPhy: Unsupervised Learning of Counterfactual Physics in Pixel Space (ICLR 2022)

Steeven Janny*, Fabien Baradel+, Natalia Neverova%, Madiha Nadri§, Greg Mori$, Christian Wolf+

* INSA-Lyon, LIRIS ; + Naver Labs Europe ; % Meta AI; § Université Claude Bernard Lyon 1, LAGEPP; $ Simon Fraser University, Vancouver, Canada

ArXiv : https://arxiv.org/pdf/2202.00368.pdf

Website : https://filteredcophy.github.io/

Causal discovery is at the core of human cognition. It enables us to reason about the environment and make counterfactual predictions about unseen scenarios that can vastly differ from our previous experiences. We consider the task of causal discovery from videos in an end-to-end fashion without supervision on the ground-truth graph structure. In particular, our goal is to discover the structural dependencies among environmental and object variables: inferring the type and strength of interactions that have a causal effect on the behavior of the dynamical system. Our model consists of (a) a perception module that extracts a semantically meaningful and temporally consistent keypoint representation from images, (b) an inference module for determining the graph distribution induced by the detected keypoints, and (c) a dynamics module that can predict the future by conditioning on the inferred graph. We assume access to different configurations and environmental conditions, i.e., data from unknown interventions on the underlying system; thus, we can hope to discover the correct underlying causal graph without explicit interventions. We evaluate our method in a planar multi-body interaction environment and scenarios involving fabrics of different shapes like shirts and pants. Experiments demonstrate that our model can correctly identify the interactions from a short sequence of images and make long-term future predictions. The causal structure assumed by the model also allows it to make counterfactual predictions and extrapolate to systems of unseen interaction graphs or graphs of various sizes.

Oral #8 (17h00-17h20)

Deriving Systems' Observability from Simulators --- A Case Study in Turbine Engines' Health Analysis

Lennart GULIKERS, Yosra MARNISSI, Michel NOCTURE, Sebastien RAZAKARIVONY, Dong Quan VU

SAFRAN

Bayesian learning is an extremely active machine learning domain with various applications such as medical diagnosis, robotics, semantic search, etc. Bayesian methods often involve making predictions/decisions by incorporating the prior belief and incrementally updating when new evidence is available. Such methods are particularly useful in studying physics systems that involve uncertainty factors (especially, in systems dealing with time-series data such as navigation in robotics, prognostic for health monitoring, etc.). Re- cursive Bayesian filtering is a robust tool for estimating systems states that are hidden (or not directly observed). Importantly, Bayesian filters allow us to analyse the systems? observability; in other words, it addresses the question ?which information of the system can be derived from given observations??

In this work, we apply Bayesian filters to analyse the conditions of gas-path (aircraft) turbine engines which are captured by several health indicators (particularly, they indicate efficiencies of specific modules in engines). In practice, these indicators depend on complex and non-deterministic factors and hence, they are not directly observed; instead, only the measurements from several sensors installed on the engine are accessible. For this reason, it is natural to model the dynamics of the engine conditions by a hidden Markov model (health indicators and sensor measurements are Markov processes where the former is hidden and the latter can be observed). It is worth noting that in this application scenario, we encounter an under-determined problem: by physical constraints, we can only operate a limited number of sensors which is typically smaller than the hidden states? dimension (i.e., the number of health indicators). This leads to the challenge in determining the observability of the system (from given measurements).

To solve the challenge mentioned above, we first build a simulator taking the engine conditions and operating conditions as inputs and generating synthetic sensor data. We then run several Bayesian filters with data obtained from different degradation models and of different combinations of sensors. Particularly, we apply the Unscented Kalman Filter (UKF) ? a classical Bayesian filter that is capable of dealing with non-linear systems ? to estimate the engines? health indicators. From these results, we conduct two observability analyses: (i) first, we use the covariance matrices estimated by UKF to detect several non- observable directions (i.e., the combinations of different indicators that cannot be observed given the installed sensors) and (ii) we determine the set of indicators that when being fixed, allows observable systems.

Observability analysis via Bayesian filters allows us to have better models for determining health conditions of aircraft engines. Our work can be extended to have a more thorough comparison with other observability-analysis approaches as well as to determine more explicitly the relation between estimation and observability.

Posters:

Data-driven Model Generation Process for Thermal Monitoring of Wind Farm Main Components through Residual Indicators Analysis

Théodore Raymond*, Alexis Lebranchu*, Sylvie Charbonnier+, Christophe Bérenguer+

*Valmeo, +Gipsalab

Most of the SCADA fault indicators proposed in the literature to detect a fault that induces a temperature increase of the physical components of a wind turbine are temperature residuals. Temperature residuals measure the difference between the current value of the temperature of a component and its prediction by a normal behavior model. In the literature, normal behavior models built from variable selection algorithms are ad-hoc models, designed to correctly predict the temperature of a specific component of a specific turbine of a specific wind farm. In practice, these models cannot be used to predict the temperature of a component of another turbine, let alone a turbine in a different wind farm, because the sensors used by wind turbine manufacturers are not the same. It is therefore impossible for an industrial wind farm manager to deploy a residual-based fault detection system on a wind farm scale.

In order to make it possible to deploy these methods in an industrial context, we propose in this paper a methodology to automatically build linear models capable of predicting all temperatures of any component of any turbine of a given wind farm. The method is designed to be easy to implement, interpretable by the operator, and fast to execute to meet industrial constraints. The set of models obtained allows to build a network of thermal state indicators, which can be used for fault isolation. The method is applied to the monitoring of the thermal condition of a real French fleet of wind farms composed of turbines from seven different manufacturers.

Multi-scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators

Léon Migus, Ahmed Mazari, Patrick Gallinari, Yuan Yin

We would like to present our work on Multi-scale Physical Representations for Approximating Partial Differential Equation (PDE) Solutions with Graph Neural Operators. It is an accepted ICLR 2022 workshop paper. Our goal is to incorporate numerical scheme ideas in Graph Neural Operators (GNOs) architectures to solve physics problems. More precisely, our main inspiration comes from numerical schema. Solving differential equations (DEs) analytically is intractable in most practical cases. To circumvent that, one could seek for an approximated solution through numerical analysis. When the phenomenon involves information and energy exchange in different ranges, methods with multi-scale modeling, e.g. multi-grid (multiresolution) methods, are proposed for solving DEs. Interactions at different scales are modeled with a pyramidal discretization [Bergot and Durufl´e, 2013, O?Malley et al., 2018]. With the multi-scale modeling, the solvers converge faster than single-scale methods [Lie et al., 2017, Passieux et al., 2010]. Hence, these multi-scale modeling tools are widely used in the numerical analysis community.

In Deep Learning (DL), many methods have been proposed for approximating PDE solutions on a regular grid at a single scale [Thuerey et al., 2020, Um et al., 2020]. However, in real-world applications, the domain of PDEs is often discretized on meshes. They are represented by Euclidean graphs where vertices are points in an Euclidean domain and the edges the distance between those points. The nodes and the edges represent respectively the physical states and their interaction. In this case, we use Graph Neural Networks (GNNs) instead, e.g. Pfaff et al. [2021], Xu et al. [2021]. Some methods use U-net [Ronneberger et al., 2015], e.g. Wandel et al. [2021], to enable long-range interactions on regular grid data. Recently, Multipole Graph Neural Operator (MGNO, Li et al., 2020) introduces a new graph-based method in this category. It learns an operator mapping between two function spaces by a Message Passing Graph Neural Networks (MPGNNs) with a multilevel graph. However, Li et al. [2020] only focus on reducing computation cost of long-range correlation, inspired by fast multipole methods.

In this work, we explore new ways to extend the multi-scale modeling capacity with neural networks. We would like to shed light on different numerical schema and understand the reason for the choice of multi-scale schema from the lens of DL. We observe in practice that the multi-scale DL models share a similar structure with some of the numerical schemes. For example, both composed of a straight-through downscaling and upscaling process, U-net shares a very similar structure with V-cycle schema in multi-scale numerical analysis [Jaysaval et al., 2016]. We then draw inspiration from discretized multi-resolution schema [Jaysaval et al., 2016] such as W-cycle and F-cycle to propose new architectures.

In this work, we propose new multi-scale DL architectures, based on the original MGNO, for learning representation of multi-scale physical signals by approximating the functional spaces of PDEs. They are tested with steady and unsteady PDEs-based physical systems. This opens perspectives to rethink the architecture design for multi-scale problems and to help practitioners using U-net to include these numerical schema variants in their study.

Online greedy identification of linear dynamical systems

Matthieu Blanke, Marc LeLarge

Inria

This work addresses the problem of exploration in an happen that the controller knows B, in which case ? = A unknown environment. For linear dynamical systems, and q = m. we use an experimental design framework and introduce an online greedy policy where the control maximizes the Contributions information of the next step. In a setting with a limited number of experimental trials, our algorithm has low In practice, systems have complex dynamics and can

complexity and shows experimentally competitive performances compared to more elaborate gradient-based methods.

Learning to estimate UAV created turbulence from scene structure observed by onboard cameras

Quentin Possamaï*, Steeven Janny*, Madiha Nadri§, Laurent Bako$, Christian Wolf+

* INSA-Lyon, LIRIS ; § Université Claude Bernard Lyon 1, LAGEPP ; $ Ecole Centrale de Lyon, AMPERE ; + Naver Labs Europe

ArXiv: http://arxiv.org/abs/2203.14726

Controlling UAV flights precisely requires a realistic dynamic model and accurate state estimates from onboard sensors like UAV, GPS and visual observations. Obtaining a precise dynamic model is extremely difficult, as important aerodynamic effects are hard to model, in particular ground effect and other turbulences. While machine learning has been used in the past to estimate UAV created turbulence, this was restricted to flat grounds or diffuse in-flight air turbulences, both without taking into account obstacles. In this work we address the complex problem of estimating in-flight turbulences caused by obstacles, in particular the complex structures in cluttered environments. We learn a mapping from control input and images captured by onboard cameras to turbulence. In a large-scale setting, we train a model over a large number of different simulated photo-realistic environments loaded into the Habitat.AI simulator augmented with a dynamic UAV model and an analytic ground effect model. We transfer the model from simulation to a real environment and evaluate on real UAV flights from the EuRoC-MAV dataset, showing that the model is capable of good sim2real generalization performance.

Utilisation des réseaux de neurones génératifs pour les méthodes inverses

T. Santos (LGL-TPE, CRAL, Lyon), T. Bodin (LGL-TPE, Lyon), F. Soulez (CRAL, Lyon)

Les problèmes inverses consistent à rechercher les causes (les paramètres d'intérêt) étant donnés les effets (les observations) et leurs liens de causalité (le modèle direct). Cette approche se retrouve dans de nombreuses disciplines comme l?Astrophysique et la Géophysique, par exemple pour l?imagerie de la terre interne à partir d'observations faites en surface. Dans ces disciplines, le modèle direct consiste souvent en des simulations numériques coûteuses, produisant des images complexes, et dépendant d?une multitude de paramètres physiques. Une approche naïve pour estimer ces paramètres est de comparer les observations avec une grille de modèles. Malheureusement, cette approche est computationnellement intensive et produit un échantillonnage restreint des paramètres.

Nous proposons une méthode approchée basée sur des réseaux de neurones pour synthétiser rapidement des images complexes proches des simulations numériques. En effet, les réseaux génératifs permettent de synthétiser à volonté de nouvelles images reproduisant les caractéristiques d'un jeu d'images d?entraînement. Un modèle génératif peut alors être utilisé comme une bonne approximation de la simulation physique. Cela a deux intérêts principaux : (i) il permet de paramétriser le problème dans un espace de plus petite dimension plus facile à échantillonner et (ii) ce modèle est suffisamment rapide pour être utilisé dans des méthodes itératives en ayant de plus accès aux variations des paramètres par rapport aux observations.

Dans ce travail, nous entraînons un Variational Auto-Encoder (VAE) [1], utilisé comme modèle génératif, sur des simulations. Une fois entraîné, il remplace le simulateur dans les méthodes inverses par échantillonnage (MCMC) ou optimisation continue (BFGS). Les résultats préliminaires de cette méthode ont été obtenus dans le cas simple d?un problème-jouet : un anneau 3D projeté en 2D, dont l?image a été bruitée. Le problème inverse à résoudre est donc de déterminer l?image originale d?anneau projeté en n?ayant accès qu?à l?image bruitée.

Modèles d?apparence et apprentissage en stéréo-photométrie

C. Joubert, N. Prouteau, B. Bringier, M. Khoudeir

XLIM, UMR CNRS 7252, Université de Poitiers

De nos jours, la vision par ordinateur permet d?estimer la géométrie d?une surface par l?utilisation de modèles qui traduisent ses propriétés physiques. Ainsi, l?interaction de la lumière et de la matière est décrite par la BRDF (Bidirectional Reflectance Distribution Function). La stéréo-photométrie l?utilise pour reconstruire la géométrie d?une surface (carte de normales) à partir de plusieurs photographies. Pour chaque élément de surface (traitement pixel par pixel), sa dérivée partielle est estimée à partir d?un modèle de BRDF et d?une mesure par direction d?éclairage. Les modèles de BRDF réalistes sont des fonctions non linéaires qui dépendent de nombreux paramètres (dimensionnalité du problème). Ainsi pour la stéréo-photométrie, les méthodes traditionnelles de résolution des problèmes inverses donnent généralement des résultats satisfaisants. Cependant il est nécessaire d?avoir un grand nombre de directions d?éclairage (mesures), ce qui rend le temps d?acquisition et de calcul coûteux. Chen et al. [1] montrent qu?il est possible d?utiliser un réseau de neurones de type convolutif prenant en entrée des images acquises complètes pour estimer la carte de normales. Pour cette méthode, le modèle de BRDF est implicite dans la base de données d?apprentissage et l?utilisation des images complètes provoque un lissage de la géométrie. Ikehata [2] conserve une approche par élément de surface pour réduire la dimensionnalité du problème. La fonction de BRDF est alors représentée par une carte d?observation analysée par un réseau de neurones convolutif. En raison de la parcimonie de cette représentation lorsque le nombre de mesures est limité, cette approche n?est pas pertinente pour des systèmes utilisant peu de directions d?éclairage (coût du système et temps d?acquisition d?une surface).

Dans nos travaux, nous proposons une nouvelle représentation de la BRDF basée sur les cartes de réflectance en entrée d?un réseau de neurones (PS-Refmap-CNN). Cette approche reste efficace même pour un nombre limité d?éclairages et peut être entraînée à partir d?un modèle (simulation numérique) ou de données mesurées de BRDF. Néanmoins, la génération d?une image représentant le modèle physique pour chaque élément de surface et l?utilisation d?un réseau convolutif pour leur interprétation a un coût de calculs important. Une autre approche basée sur une architecture de réseau de neurones dite ?entièrement connectée? permet de résoudre ce problème (PSGL-Net). Elle utilise l?hypothèse qu?un modèle de BRDF peut être séparé en deux fonctions : une partie spéculaire et une partie diffuse. Ainsi le réseau de neurones a pour objectif d?isoler la partie diffuse, partie où seule la normale est l?inconnue du problème. L?estimation de cette dernière peut alors se faire simplement par l?utilisation d?une méthode des moindres carrés peu coûteuse en temps d?exécution. Une comparaison avec les méthodes de l?état de l?art est exposée dans la Figure 1. Nos deux approches montrent qu?une utilisation efficace de la fonction physique au travers d?un réseau de neurones permet d?optimiser la précision des estimations et le temps d?exécution.

Thermodynamics-based Artificial Neural Networks for multiscale modeling of inelastic materials and structures

Ioannis Stefanou and Filippo Masi

Ecole Centrale de Nantes

The mechanical behavior of inelastic materials with microstructure is very complex and hard to grasp with heuristic, empirical constitutive models. For this purpose, multiscale, homogenization approaches are often used for performing reliable and accurate predictions of the macroscopic mechanical behavior of solids and structures. Nevertheless, the calculation cost of these approaches is extremely high and prohibitive for real-scale applications involving inelastic materials.

Here, we propose the so-called Thermodynamics-based Artificial Neural Networks (TANN) for the constitutive modeling of materials with inelastic and complex microstructure. Our approach includes thermodynamics-aware dimensionality reduction techniques and thermodynamics-based deep neural networks to identify, in an autonomous way, the underlying constitutive laws and discover the internal state variables of complex inelastic materials. The ability of TANN to deliver accurate and physically consistent predictions is demonstrated through several examples, both at the microscopic and macroscopic scale. Moreover, TANN manages to identify the internal state variables that characterize the inelastic deformation of the complex microstructural fields. These internal state variables are then used to reconstruct the microdeformation fields of the microstructure at a given state. Finally, a double-scale homogenization scheme (FEMxTANN) is used to solve a large-scale boundary value problem. The high performance of the homogenized model using TANN is illustrated through detailed comparisons with microstructural calculations at large scale. An excellent agreement is shown for a variety of monotonous and cyclic stress-strain paths.

Nouvelle stratégie d?apprentissage des Modèles de Markov Cachés pour l?estimation de l?état de santé de systèmes industriels

Nesrine Keltoum KHODJA, Florent DUCULTY, Stéphane BEGOT, Manuel AVILA.

PRISME, IUT de l?Indre

Les chaînes de fabrication industrielles utilisent des systèmes technologiques de plus en plus avancés. Seulement, les politiques de maintenance ne sont pas adaptées aux nouvelles exigences ainsi qu?aux attentes de compétitivité demandées par les dirigeants. Dans notre étude, nous utilisons des Modèles de Markov Cachés (MMC) comme un outil de pronostic pour l?aide à la décision. Le but est de modéliser le niveau de dégradation d?un système industriel [1]. Pour la modélisation d?un tel processus, nous utilisons des MMCs de topologie gauche-droite. Dans cette étude, nous nous sommes focalisés, sur le cas particulier, de la mise en compétition de plusieurs modèles. Une nouvelle stratégie d'apprentissage a été développée. Cette stratégie s'inspire des principes de non-occurrence (NBA) [2].

Nos travaux consistent à essayer d?estimer la durée de vie résiduelle (RUL) d?un processus industriel, en améliorant la partie apprentissage des MMCs. Pour cela, nous avons modélisé un système réel qui génère des données de synthèse, afin de produire des observations similaires à un véritable système industriel. Une architecture gauche-droite de MMC a été appliquée pour modéliser ces données. Cette architecture avait été jugée la plus proche du système réel dans les travaux de Bernard Robles [3]. La topologie est représentée par des automates à états cachés constitués d?une variable non observable. Cette variable représente l?état du système à modéliser. Seules les variables de sortie sont observables. Elles représentent les séquences des observations, sous forme de chaînes de Markov. A partir de quatre modèles générés avec Matlab, représentant différents modes de défaillance, nous avons constitué deux corpus : base d?apprentissage et base de test. Nous avons fait en sorte que chaque modèle apprenne avec la portion apprentissage correspondante à l?aide de l?algorithme BaumWelch. Les tests ont été effectués en calculant le maximum de vraisemblance.

Dans une deuxième phase de notre étude, nous avons considéré les échantillons des autres classes pour apprendre ce que nous avons qualifié de non-modèle (Figure 1). Dans la phase de test visant à identifier les échantillons inconnus, nous avons recherché le minimum de vraisemblance.

Les premiers essais effectués sur des modèles de synthèse ont montré qu?il y a de l?information dans les « non-classes » qui permet d?identifier la classe ou le mode de défaillance.

An extensible Benchmarking Graph-Mesh dataset for studying Steady-State Incompressible Navier-Stokes Equations (ICLR 2022 Workshop)

Florent Bonnet, Jocelyn Ahmed Mazari, Thibaut Munzer, Pierre Yser, patrick gallinari

Sorbonne, ENS, Extrality, Criteo

Recent progress in Geometric Deep Learning (GDL) has shown its potential to provide powerful data-driven models. This gives momentum to explore new methods for learning physical systems governed by Partial Differential Equations (PDEs) from Graph-Mesh data. However, despite the efforts and recent achievements, several research directions remain unexplored and progress is still far from satisfying the physical requirements of real-world phenomena. One of the major impediments is the absence of benchmarking datasets and common physics evaluation protocols. In this paper, we propose a 2-D graph-mesh dataset to study the airflow over airfoils at high Reynolds regime (from and beyond). We also introduce metrics on the stress forces over the airfoil in order to evaluate GDL models on important physical quantities. Moreover, we provide extensive GDL baselines.

Architecture de cellule récurrente structurellement stable pour la simulation dynamique en temps réel

Louen Pottier, Anders Thorin

La simulation en interaction avec un utilisateur humain nécessite de résoudre les équations de la physique i) suffisamment rapidement pour tenir le temps réel, ii) sans connaître à l?avance l?évolution temporelle complète des sollicitations appliquées par l?utilisateur. Le présent travail se concentre sur la dynamique des structures impliquant des matériaux dissipatifs élastiques ou hyperélastiques pouvant subir de grandes transformations. Pour de telles structures, la contrainte i) peut être très restrictive, d?où la nécessité de remplacer la résolution coûteuse des équations de la physique par l?évaluation d?un modèle de substitution. La contrainte ii) élimine un large éventail de méthode classique de réduction de modèles, telles que la Décomposition Propre Généralisée (PGD) ou les réseaux neuronaux à information physique (PINN), mais elle est satisfaite par l?utilisation de réseaux de neurones récurrents (RNN). On propose une famille d?architectures de cellules récurrentes conçue de manière à respecter les propriétés de stabilité connues a priori du système physique original, et étant en particulier structurellement incapable d?apprendre des orbites stables. Notre architecture est validée sur des problèmes unidimensionnels simples et sur un modèle d'éléments finis d?une poutre en grande transformation avec flambement. Nos résultats montrent que les cellules RNN proposées sont capables d?apprendre correctement l?ensemble des points d?équilibres stables d?un système, qui sont parfois mal appris par les cellules GRU (Gated Recurrent Unit) pour lesquelles l?apprentissage accidentel d?orbites stables non physiques n?est pas impossible.