Physics-Based Machine and Deep Learning for PDE Models

Machine learning has been successfully used for over a decade for applications in engineering. It has recently started to attract attention for scientific computing in domains dominated up to now by the classical mechanistic modeling paradigm. It is particularly promising for domains involving complex processes, only partially known and understood or when existing solutions are computationally not feasible. This is the case for the modeling and simulation of complex dynamical physical systems. Classical modeling relies on PDEs, and simulation is central to engineering and physical science with applications in domains such as earth systems, biology, medicine, mechanics and robotics. Traditional simulation problems involve computational fluid dynamics and turbulence modeling, mechanistic design and many other domains. Such numerical models are also used intensively in industrial systems design, in simulation for decision support, or in safety studies. They are used for inversion, data assimilation and forecasting. Despite extensive developments and promising progress, this classical paradigm suffers from limitations. It is often impossible or too costly to carry out direct simulations at the scale required for natural or industrial problems. The physics may be too complex or unknown, leading to incomplete or inaccurate models.

The availability of increasingly large amounts of data, either from observations or from simulations, and the successes witnessed by ML methods on large size or large dimensional problems has opened the way for exploring the data driven modeling of complex dynamical physical phenomena. ML based techniques may accelerate simulations, acting, for example, as reduced models. More generally, a promising direction consists in integrating physics-based models with machine learning. This raises several challenges such as how to perform such decompositions, how to train such combined systems, how to handle discretization errors or guarantee numerical stability of the solutions, how to handle out-of-sample scenarios, and how to ensure physical consistency of the solutions.

An additional challenge is the shift from academic case studies to realistic problems representing complex phenomena. Current solutions are most often demonstrated on simulated problems and there is still a large gap between academic and real-world developments.

This Special Issue, therefore, aims to gather specialists from different disciplines and to enable the dissemination of their recent research at the crossroad of model based and data based dynamical physical system modeling and on “physically inspired” ML models for dynamic systems.

The topics of interest for publication include but are not limited to:

• Deep learning;
• Gaussian processes;
• Uncertainty quantification;
• Data-driven techniques;
• PDE solving;
• Spatio-temporal forecasting;
• Simulation;
• Computational fluid dynamics;
• Graphics;
• Robotics.

Dr. Nicolas Bousquet
Prof. Dr. Patrick Gallinari
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

• deep learning
• machine learning
• PDE
• neural networks
• uncertainty quantification
• physics-inspired meta-models
• Gaussian processes