Deep Learning meets Numerical Analysis – DeepNuM

OBJECTIVE: DeepNuM aims at developing the interplay between two families of computational approaches, Deep Neural Networks (DNNs) and Partial Differential Equations (PDEs), with the goal of modeling complex dynamical systems arising from the observation of natural phenomena. Three central questions are considered: how to adapt and apply numerical analysis theory to DNNs for providing theoretical guaranties and improving their robustness; how to combine simulation and machine learning models into hybrid systems leveraging prior physics knowledge and information extracted from data; how to develop effective solvers where DNNs complement numerical solvers. The project targets fundamental aspects at the crossroad of machine learning and numerical analysis.

ORGANIZATION: The project is organized into 3 methodological WPs plus an application one. WP1 revisits the concepts of numerical analysis for DNNs along three directions: 1) consistency analysis and stability of the forward DNN pass, 2) optimal control theory for the training step, 3) model reduction perspective for DNNs. WP2 focuses on two machine learning challenges that condition the adoption of DNNs for the modeling of physical systems: 1) formal and algorithmic frameworks for combining numerical schemes and DNNs, 2) out of distribution generalization. WP 3 integrates the advances of WP 1 and 2 in the development of PDE solvers for high dimensional problems for two PDE families: fluid transport and reaction diffusion. WP4 is dedicated to the evaluation on three real use cases: air pollution forecasting, 2D shallow water equations approximating 3D Navier-Stokes, personalized cardiac electrophysiology.

OUTCOME: Project results will consist in 1) theoretical results regarding the adaptation of numerical analysis theory for PDEs to DNNs, 2) theoretical and algorithmic solutions for challenging ML problems concerning the modeling of spatio-temporal dynamics and related generalization issues, 3) PDE solvers integrating physics and data models, 4) evaluation on challenging real-world use cases for environmental and health science, 5) datasets and software components to be made available to the scientific community.

POSITIONING w.r.t. the call: The project is directly positioned in the axis 5.2 Artificial Intelligence: Machine Learning/Deep Learning and their mathematical foundations.

PARTNERS: DeepNuM gathers partners with complementary skills. INRIA-ANGE is specialized on modeling, analysis and simulation of geophysical flows. Its research has gradually included interactions with statistical approaches, e.g. dimensionality reduction, probabilistic forecasting and statistical meta-modeling. Sorbonne University- MLIA (SU-MLIA), is specialized in Statistical Machine Learning. It is one of the foremost and pioneer groups in Deep Learning in France. It has started to develop 3 years ago the modeling of dynamical physical systems with DNNs. INRIA-EPIONE is specialized in medical image analysis and in developing digital models of the human body. EPIONE focuses on methods to personalize such models to the available clinical data (e-patient). MLIA and EPIONE already have joint publications on DNNs for cardiac electrophysiology simulation.

SOCIETAL AND INDUSTRIAL IMPACTS: The project targets fluid transport and reaction-diffusion equations. This covers a wide range of problems in environment, climate, geophysics and health science. DeepNuM could have a strong potential for promoting the use of ML in these important domains. Several industrial domains are today in demand of solutions combining their numerical know-how and data or simulation results. As a concrete example, let us mention NUMTECH SME, a leading French company for the deployment of urban air quality systems. NUMTECH deploys ANGE data assimilation software for multiple cities and is eager to make use of the air quality model we could build in this project.