Simulating Physical PDEs Efficiently with Deep learning – SPEED

This project addresses every aspects of the learning workflow, from acquiring information to enforcing guarantees on the predictions.
Methods and tools will be first developed with low-dimensional dynamical systems but will then be illustrated and demonstrated on a full-scale turbulent fluid flow numerical simulation.

The project naturally adopts an organization in 3 methodology-oriented tasks addressing related, yet distinct, aspects of the proposed work. These methodological efforts call for computationally agile yet relevant and credible synthetic environments mimicking the complex systems this projects focuses on. To this end, a toward-realistic demonstrator will be implemented to validate and show-case the different methodological outcomes.
The four main axes of the project are:
- The crux of learning: acquiring training data
- Inferring the model structure
- Accuracy bottlenecks
- Benchmarks & Applications

The project started a few months ago and we cannot yet report significant results.

This project will carefully revisit each step of an ML procedure, thoroughly identifying requirements concerning the training data along with the limitations of the resulting model, e.g., in the form of confidence bounds for the prediction. As such, this project will discuss and build evaluation frameworks for ML approaches and will be a valuable contribution toward good ML practice as well as reliable and explainable ML in applications.

In Engineering domain the validation and certification are of critical importance for acceptability and innovation at large. Therefore, application domains under consideration in this project radically differ from the famous and widely known AI applications.
This project proposes to lay the foundation of physics-oriented evaluations of ML progress. In this perspective, its challenges are meant to provide more than a leaderboard to compare Deep-Learning architectures. Moreover, the scientific community, and especially the ML part, calls for more reliable evaluation criteria and practices as breakthrough to spread the recent progress in AI.

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This project is a trans-disciplinary effort toward inference and prediction of complex physical systems. Such systems are often ineffectively described by first principles models and should be modeled via a data-driven approach. However, difficulties arise from the high dimensional and multi-scale nature of these systems. Further, only limited and poorly informative observations are typically available. Prototypical of these situations is subsurface ocean inference or the prevention of seizures in neurosciences. For many applications however, some degree of expertise is available.

The goal of this project is to leverage both the theoretical and the data science pillars to infer computable models informed from the existing prior knowledge (Physics first principles and theories) and providing new hints into the principles satisfied by the proposed abstractions, amenable to interpretation and refutation. More precisely, building upon the pluridisciplinary expertise of the team in the domains of Fluid Mechanics and Deep Neural Networks (DNN), the goal of the proposal is:
- i) to make the model space (neural architecture and computational flow) compliant with the known Physics of the system under consideration,
- ii) to exploit the data and inference tools to train efficient models built on first principles, thereby enhancing their robustness and reducing their data-hunger,
- iii) to form and inspect the abstractions built by the DNN systems, to check whether they satisfy the expected properties and understand the properties they satisfy.

Methods and tools will be first developed with low-dimensional dynamical systems but will then be illustrated and demonstrated on a full-scale turbulent fluid flow numerical simulation.