2023-06019 - Post-Doctorant F/H Design of Neural Operator based on PINNs and application to wave and fluid dynamics
Contract type : Fixed-term contract
Level of qualifications required : PhD or equivalent
Fonction : Post-Doctoral Research Visit
ContextSince few years several methods have been introduced to approximate PDEs with deep learning methods. Among them, the Physics-Informed Neural Networks (PINNs) deserve a particular attention. They are implemented by formulating the solution of the considered PDE as an optimization problem along with a Monte-Carlo estimation. This approach allows solving only initial and boundary conditions by training. Many variants of this method have been introduced since like WPINNs 1 specific for hyperbolic problems or CPINNS/Hp-VPINNs 2-3 which involve domain decomposition. PDEs generally depend on parameters and one strength of PINNs approach is to allow solving families of PDEs at each training using the good property of the neural networks for large dimension problem. Another approach is called Neural Operator. The aim is to approximate directly the inverse operator of the PDE. This allows a large number of initial conditions and parameters to be processed with one training. We can mention networks based on a spectral approach (FNO 4, LNO 5, etc), on a formalism closed to the function basis DeepONet 8 or on dimension reduction (PCA Net 6, NOMAD 7).
1 wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws , Tim De Ryck, Siddhartha Mishra, Roberto Molinaro, 2022 2 Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems , Computer Methods in Applied Mechanics and Engineering, Ameya D. Jagtap, Ehsan Kharazmi, George Em Karniadakis 3 hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition , Ehsan Kharazmi, Zhongqiang Zhang, George Em Karniadakis, Computer Methods in Applied Mechanics and Engineering 2021 4 Fourier Neural Operator for Parametric Partial Differential Equations , Li, Z. and Kovachki, N. and Azizzadenesheli, K. and Liu, B. and Bhattacharya, K. and Stuart, A. and Anandkumar, A. (2020) 5 Laplace neural operator for complex geometries , G. Chen, X. Liu, Y. Li, Q. Meng, L. Chen, Arxiv, (2023) 6 Model Reduction and Neural Networks for Parametric PDEs , K. Bhattacharya, B. Hosseini, N. B. Kovachki, A. M. Stuart. SMAI Journal of computational mathematics, Tome 7 (2021). 7 NOMAD: Nonlinear Manifold Decoders for Operator Learning, J. H. Seidman, G. Kissas, P. Perdikaris, G. J. Pappas, Arxiv 2022. 8 Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators , L. Lu, P. Jin, G. Pang, Z. Zhang and G. Em Karniadakis.
AssignmentThe scientific objective is to propose new strategies to obtain neural operator and to apply these new methods to specific wave and compressible fluid problems that interests the INRIA teams involved in the project. If successful, these approaches can be coupled with more classical numerical approaches to obtain efficient hybrid methods with convergence guarantees.
The person recruited will be able to transfer the techniques learned to the members of the different teams and will be the driving force behind this new collaboration between the INRIA centers.
Colaboration : This work will be in collaboration with E. Franck, V.M. Dansac, V. Vigon and J. Aghili for the INRIA Strasbourg, F. Faucher and H. Barucq for INRIA Pau and S. Lanteri for INRIA Sophia Antipolis.
Main activitiesThe first work will consist of proposing a new physical informed Neural Operators based on a coupling of PINNs with deep dimension reduction methods in order to treat very general meshes (as inputs and outputs), to be compatible with some variants of PINNs and to encode particular structures of the physical equations inside the neural operator. The training will be made using the residual loss of the PINNs approach and a classical loss on the data. This method will be validated on elliptic and time-dependent acoustic and electromagnetic wave problems in heterogeneous medium and compared to the classical methods of literature. This approach could be coupled with interesting variants of PINNs like Hp-VPINNs or others.
The second work will explore another approach where we will also combine parametric PINNs or SINDy approach (parametric sparse nonlinear model) to compute a specific basis and obtain a reduced model by a Galerkin projection on this basis. The aim is to have a model which has better interpretability and accuracy than the classical Neural Operator. For this approach we will investigate different strategies of learning and try to select the less costly and the most accurate ones. This method will also be validated on elliptic and time-dependent wave problems.
SkillsThe ideal profile for this post-doc is someone with a thesis in applied mathematics specialized in numerical analysis/computational science with very good code skills, especially in Python, and experience in machine learning (ML). However, this position is also open to a Ph.D. in applied mathematics with expertise in numerical analysis/computational science and very good coding skills, especially in Python, and an interest in statistical methods. Experience in ML is indeed a plus but not essential. Doctors in mathematics specialized in ML can also apply, but it is required to have obtained a master's degree in which knowledge in PDE and numerical analysis has been acquired.
Benefits package2746€ brut/mois (2746€ gross/month)
General InformationTheme/Domain : Numerical schemes and simulations Scientific computing (BAP E)
Town/city : strasbourg
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