PhD proposal -- Uncertainties in Physics-Informed Machine Learning

Proposal description

Recently, Physics-informed Neural Networks (PINN) have been proposed to explicitly introduce the Physics inside Machine Learning. On one side, Machine Learning is able to extract hidden information out of the data. On the other side, the conclusions are not always consistent with the physics. This is a strong limitation, but it offers the possibility to extend the capabilities of Machine Learning. The objective of the of PINN approaches is to update the weights of the neural network to fit the observed data as well as to obey the physics laws (Raissi et al., 2019).

The objective of the PhD work is to quantify the PINN uncertainties. From a starting initial solution, the network learning leads currently towards a deterministic solution. The stochastic gradient is more general, but a unique solution is determined, without uncertainties. Here, we propose to recast the PINN in a Bayesian context to derive a posterior probability.

The applications are related to seismic imaging. The traditional approach (without Machine Learning) is Full Waveform Inversion (Chauris, 2019). The Bayesian approach is here pertinent, as Full Waveform Inversion may easily converge towards a local minimum with the deterministic approaches. The objective is to demonstrate on synthetic and real data sets how the PINN strategy could avoid these local minima and to obtain the posterior probability density functions for different parameters (velocity, density) influencing the wave propagation.

References

Chauris, H. (2019). Full Waveform Inversion, in Seismic 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

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

Contact and location

Pr. Hervé Chauris

Email: herve.chauris@mines-paristech.fr

Tel.: +33 (0)1 64 69 49 13

MINES ParisTech - PSL Research University

Centre de Géosciences

35 rue Saint-Honoré

77305 Fontainebleau Cedex, France