Robust inversion with quantile surrogate models

The Inria center at Université Côte d'Azur includes 42 research teams and 9 support services. The center’s staff (about 500 people) is made up of scientists of different nationalities, engineers, technicians and administrative staff. The teams are mainly located on the university campuses of Sophia Antipolis and Nice as well as Montpellier, in close collaboration with research and higher education laboratories and establishments (Université Côte d'Azur, CNRS, INRAE, INSERM ...), but also with the regional economic players.

With a presence in the fields of computational neuroscience and biology, data science and modeling, software engineering and certification, as well as collaborative robotics, the Inria Centre at Université Côte d'Azur  is a major player in terms of scientific excellence through its results and collaborations at both European and international levels.

This internship project is part of a collaborative project between the Acumes project-team from Inria Research Center at Université Côte d’Azur, Sophia Antipolis and IFP Energies nouvelles (IFPEN), Rueil-Malmaison.

The Acumes project-team is a joint team between Inria and the Jean-Alexandre Dieudonné Mathematics Laboratory (LJAD) of the Côte d'Azur University. The research carried out focuses on the analysis and optimization of systems governed by partial differential equations, with multi-disciplinary applications ranging from the mechanics of fluids and structures to the modeling of biological phenomena, road and pedestrian traffic. The team is also focusing on deep learning methods to effectively combine data and physical models.

In the context of wind turbine design, the considered simulators are subject to uncertainties due to the random nature of the wind. For a set of control variables (pitch control, etc.) or design variables (tower geometry, materials, etc.) of the wind turbine, we aim to estimate a quantity of interest f (electricity power output) subject to uncertainties in the environmental parameters (wind, etc.).

Thus, given the same set of control and design parameters, n_L simulator launches n_L give different values of f. From these observations, an empirical estimator of the quantile of f can be deduced. However, the simulators are computationally expensive and the simulation budget is limited. In general, to limit the numerical cost, a substitution model is built to predict the quantile, such as in [1] with a Gaussian process model.

This model may be used to select new observations sequentially with a dedicated infill criterion, such as in Bayesian optimization [2]. Here the challenge is to propose sequential infill criteria to efficiently learn safe regions [3], for possibly large number of control and design variables [4]. Besides dedicated modeling options, areas for improvement involve leveraging the accuracy of the quantile estimation.

[1] Picheny, V., Moss, H., Torossian, L., Durrande, N., 2022. Bayesian quantile and expectile optimisation . Uncertainty in Artificial Intelligence, 1623–1633.

[2] R. Garnett. Bayesian Optimization. Cambridge University Press (2023)

[3] El Amri, M. R., Helbert, C., Lepreux, O., Zuniga, M. M., Prieur, C., & Sinoquet, D. (2020). Data-driven stochastic inversion via functional quantization. Statistics and Computing, 30, 525-541.

[4] Binois, M., & Wycoff, N. (2022). A survey on high-dimensional Gaussian process modeling with application to Bayesian optimization. ACM Transactions on Evolutionary Learning and Optimization, 2(2), 1-26.

The objectives of this internship are:

• Study the state-of-the-art in infill criteria for quantile inversion,
• Propose a scalable approach for safe region representation with many variables,
• Validation of strategies on analytical test cases and application to a real case.

This internship might be followed by a PhD thesis as part of a collaboration with IFPEN and DGA.

Technical skills and level required :

• Master 2 or engineering degree in applied mathematics - specializing in statistics and data science

• Good knowledge of Python/R programming languages

• Experience in IT development (internships, projects)

Relational skills : team worker (verbal communication, active listening, motivation and commitment)

Other valued appreciated : good level of spoken and written English

• Subsidized meals
• Partial reimbursement of public transport costs
• Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
• Possibility of teleworking and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities
• Access to vocational training
• Contribution to mutual insurance (subject to conditions)

Traineeship grant depending on attendance hours.