Section: Application Domains

Data-driven Numerical Modeling

Participants: Alessandro Bucci, Guillaume Charpiat, Cécile Germain, Isabelle Guyon, Flora Jay, Marc Schoenauer, Michèle Sebag

PhD and Post-doc: Victor Estrade, Loris Felardos, Adrian Pol, Théophile Sanchez, Wenzhuo Liu

Collaboration: D. Rousseau (LAL), M. Pierini (CERN)

As said (section 3.2), in domains where both first principle-based models and equations, and empirical or simulated data are available, their combined usage can support more accurate modelling and prediction, and when appropriate, optimization, control and design. This section describes such applications, with the goal of improving the time-to-design chain through fast interactions between the simulation, optimization, control and design stages. The expected advances regard: i) the quality of the models or simulators (through data assimilation, e.g. coupling first principles and data, or repairing/extending closed-form models); ii) the exploitation of data derived from different distributions and/or related phenomenons; and, most interestingly, iii) the task of optimal design and the assessment of the resulting designs.

The proposed approaches are based on generative and adversarial modelling [121], [106], extending both the generator and the discriminator modules to take advantage of the domain knowledge.

A first challenge regards the design of the model space, and the architecture used to enforce the known domain properties (symmetries, invariance operators, temporal structures). When appropriate, data from different distributions (e.g. simulated vs real-world data) will be reconciled, for instance taking inspiration from real-valued non-volume preserving transformations [85] in order to preserve the natural interpretation.

Another challenge regards the validation of the models and solutions of the optimal design problems. The more flexible the models, the more intensive the validation must be, as reminded by Leon Bottou. Along this way, generative models will be used to support the design of "what if" scenarios, to enhance anomaly detection and monitoring via refined likelihood criteria.

In the application case of dynamical systems such as fluid mechanics, the goal of incorporating machine learning into classical simulators is to speed up the simulations. Many possible tracks are possible for this; for instance one can search to provide better initialization heuristics to solvers (which make sure that physical constraints are satisfied, and which are responsible of most of the computational complexity of simulations) at each time step; one can also aim at predicting directly the state at $t+100$, for instance, or at learning a representation space where the dynamics are linear (Koopman - von Neumann). The topic is very active in the deep learning community. To guarantee the quality of the predictions, concepts such as Liapunov coefficients (which express the speed at which simulated trajectories diverge from the true ones) can provide a suitable theoretical framework.