PhD opportunities

Physics-Informed Machine Learning in the context of seismic imaging

Thesis proposal

Area of expertiseMathmatiques numriques, Calcul intensif et Donnes
Doctoral SchoolSFA - Sciences Fondamentales et Appliques
SupervisorM. Elie HACHEM
Research unitCentre for material forming
Starting dateOctober 1st 2020
KeywordsMachine learning, Physics-informed Machine Learning (PINN), deep neural networks, seismic imaging, wave propagation
AbstractIn 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].

How to apply:

Please send by email to and before the 29th of May 2020 (in pdf format):

A resume
A motivation letter
At least a recommendation letter (or the name of a person to contact)
A copy of a research report (e.g. master report)
ProfileThe candidate should have a strong background in maths and physics. He/she should have a clear interest for Machine Learning and for geophysical applications, in particular in the context of seismic imaging. He/she should have a strong experience in scientific programming. It is appreciated if he/she also has some knowledge on high performance computing (HPC). It is essential to be fluent in English speaking and writing.
FundingFinancement par crdits ANR
PDF document