phd position 07 – msca cofund, ai4thesciences (psl, france) - “physics-informed machine learning in...
Universit PSL

phd position 07 – msca cofund, ai4thesciences (psl, france) - “physics-informed machine learning in...

France 26 Feb 2021


Universit PSL
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Host Country
26 Feb 2021
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“Artificial intelligence for the Sciences” (AI4theSciences) is an innovative, interdisciplinary and intersectoral PhD programme, led by Université Paris Sciences et Lettres and co-funded by the European Commission. Supported by the European innovation and research programme Horizon 2020- Marie Sklodowska-Curie Actions, AI4theSciences is uniquely shaped to train a new generation of researchers at the highest academic level in their main discipline (Physics, Engineering, Biology, Human and Social Sciences) and master the latest technologies in Artificial Intelligence and Machine Learning which apply in their own field.

26 doctoral students will join the PSL university's doctoral schools in 2 academic cohorts to carry out work on subjects suggested and defined by PSL's scientific community. The 2020 call will offer up to 15 PhD positions on 24 PhD research projects. The candidates will be recruited through HR processes of high standard, based on transparency, equal opportunities and excellence.

Description of the PhD subject: “Physics-Informed Machine Learning in the context of seismic imaging”

Context - Motivation

Within the MINDS project (Mines Initiative for Numerics and Data Science) developed at Mines ParisTech-PSL, the objective is to fill the gap between Machine Learning and Physics-based approaches. Machine Learning is growing very rapidly. After a possible learning step, the objective is to let the data speak. These approaches tend to forget the more traditional physics-based approaches. The objective of the work is to develop, in the context of seismic imaging, an intermediate approach to preserve the physics [1]. Currently, the main contributions of Machine Learning to seismic processing are related to pre-processing steps (de-noising, picking, ...) but not really yet to the imaging part (determining the Earth's properties from surface measurements, a highly non-linear problem). The explicit introduction of physics within Machine Learning should fill this gap.If successful, the project will have a large impact in the way industrial companies handle seismic data.

In 2019, Raissi et al., demonstrated how it is possible to combine Machine Learning approaches with more traditional physics approaches (Physics-Informed Neural Networks, PINN) [3]. The applications are related to the resolution of partial differential equations (i.e. direct problems) as well as to the resolution of inverse problems (determining the main parameters controlling the physical phenomena, for example the wave propagation, from a set of observations). The later approach will be developed here.

On the one hand, deep neural networks are able in theory to describe any functions. Learning is usually a complex task and in physics-related problems, observations are rare and expensive to acquire. On the other hand, Machine Learning does not usually consider physics-based equations, a very useful source of information. As proposed in [3], a modified loss function in the neural networks contains several terms to ensure that the data predict the observations and that the laws of physics are fulfilled. This second term can be seen as a regularisation term, essential in practice to avoid any over-fitting in the case of noisy data. The auto-differentiation (back-propagation of the errors) within the neural networks provides a way to estimate the optimal parameters.This approach is very attractive and will be extended and modified to be applicable in the context of seismic imaging. Seismic acquisition consists of activating a seismic source and of recording acoustic / elastic waves. The objective is to determine seismic velocity wave fields and any other parameters controlling the wave propagation within the sub-surface. In comparison with the first PINN applications, seismic imaging offers some particular aspects to be properly considered:

  • Seismic wave are mainly propagative waves, meaning that the wave field is not smooth. In order to check that the wave field obeys the wave equation, the number of controlling points is a priori much larger than for a diffusive problem with a more regular solution;
  • The traditional loss function in seismic imaging contains a large number of local minima. How does the PINN approach behave? How is it possible to take advantage of the frequency content of the data? In the classical approaches, the model estimation first relies on the low frequencies and then enlarges the frequency spectrum, in order to avoid local minima. How could the neural network benefit from this approach (e.g. a proxi for the modelling part)?
  • Finally, the number of unknowns (number of parameters to be estimated) is potentially very large (thousands or much more, as the parameters depend on the spatial coordinates). In the first articles, only a few values were determined. How to play with the neural network to address this issue? The Generative Adversarial Networks (GAN) could be very useful to determine the optimal parameterisation [2].

Main references:

[1] Chauris, H. (2019). Full Waveform Inversion, inSeismic Imaging, a practical approach, J-L. Mari and M. Mendes (Eds.), EDP Sciences, chapter 5, 23 p., ISBN (ebook):978-2-7598-2351-2, doi:10.1051/978-2-7598-2351-2.c007

[2] Goodfellow, I., J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville and Y. Bengio (2014). Generative Adversarial Networks.Proceedings of the International Conference on Neural Information,arXiv:1406.2661

[3] Raissi, M., P. Perdikaris, G.E. Karniadakis (2019).Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics,378, 686-707

Scientific Objectives, Methodology & Expected results

The objective of the PhD thesis is thus to develop a novel Physics-informed Machine Learning approach in the context of seismic imaging. The validations will be performed on synthetic and real data sets, provided by the industrial company. The company will also co-supervise the work (under a person with “HDR” degree) and propose internship for the application to real data, such that the company can really benefit from the work. As the identified company is international, the real data may come from abroad, with collaboration with local teams. The main academic supervisor has a long experience in seismic imaging and inverse problems. He supervised more than15 PhD students.

International mobility

The candidate is expected to spend around 6 months to work within the company for the application to real data and for the transfer of technology. Summer schools in industrial companies will be highly supported.

Thesis supervision

Hervé Chauris and Elie Hachem


Created in 2012, Université PSL is aiming at developing interdisciplinary training programmes and science projects of excellence within its members. Its 140 laboratories and 2,900 researchers carry out high-level disciplinary research, both fundamental and applied, fostering a strong interdisciplinary approach. The scope of Université PSL covers all areas of knowledge and creation (Sciences, Humanities and Social Science, Engineering, the Arts). Its eleven component schools gather 17,000 students and have won more than 200 ERC. PSL has been ranked 36th in the 2020 Shanghai ranking (ARWU).

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