PhD in Applied Mathematics: Neural Linear Solvers and Preconditioners for General Sparse Matrices

21 Mar 2024
Job Information

Organisation/Company

IFP Energies nouvelles (IFPEN)

Department

Applied Mathematics

Research Field

Mathematics » Computational mathematics

Researcher Profile

First Stage Researcher (R1)

Country

France

Application Deadline

31 Aug 2024 - 23:59 (Europe/Paris)

Type of Contract

Temporary

Job Status

Full-time

Hours Per Week

35

Offer Starting Date

4 Nov 2024

Is the job funded through the EU Research Framework Programme?

Not funded by an EU programme

Is the Job related to staff position within a Research Infrastructure?

No

Offer Description

Numerical simulation is a strategic tool crucial for research in several scientific fields, from Computational Fluid Dynamics (CFD) for aeronautics design to Darcy flow in porous media for CO2 storage and geothermal energy. The performance of numerical simulators is a critical factor directly impacting both result quality and the ability to perform large-scale computations. Adapting industrial codes to fully harness the capabilities of modern supercomputers (including computational accelerators such as GPUs) is a major challenge.

In many numerical simulators, the resolution of ill-conditioned linear systems represents the most time-consuming step. The objective of this work is therefore to accelerate the convergence of preconditioned linear system resolution through Machine Learning. Two main approaches will be investigated especially for the coarse operator of two-level methods:

• First, we will focus on learning preconditioners in order to solve the preconditioned linear system. The general idea is to consider the matrix A of the linear system as the adjacency matrix of a graph 𝒢, where the edges represent the coefficients a_{ij} for i ≠ j, and the nodes represent the coefficients a_{ii}. We then train a Graph Neural Network (GNN) to predict without supervision using a loss function that represents the expectation of the difference between the new matrix and identity.
• Second, we will directly predict the solution to the linear system as the node values of the output graph 𝒢. The GNN can then be trained either in a supervised manner with a loss function expressing the difference between the predicted solution and the reference solution, or in an unsupervised manner with a loss function associated with the residual norm of the original equations.

In order to obtain convergence guarantees on the result, we will also investigate hybrid algorithms where these trained models are directly plugged in classical iterative
preconditioned linear system solvers such as PGMRES or PBiCGStab. The developed hybrid approaches will be assessed in terms of CPU/GPU cost and the number of matrix-vector products required to achieve convergence in representative test cases drawn from typical IFPEN and ONERA applications.

Keywords: Linear Solver, Preconditioner, Machine Learning, Graph Neural Networks

Academic supervisor: Frédéric NATAF (Laboratoire Jacques-Louis Lions and INRIA)

Industrial supervisors: Ani ANCIAUX-SEDRAKIAN (IFPEN) and Emeric MARTIN (ONERA)

Requirements

Research Field

Mathematics » Computational mathematics

Education Level

Master Degree or equivalent

Skills/Qualifications

Master's degree or equivalent with a strong background either in Machine Learning, Mathematics or Computer Science

Languages

ENGLISH

Level

Excellent

Additional Information
Work Location(s)

Number of offers available

1

Company/Institute

Laboratoire Jacques-Louis Lions

Country

France

City

Paris

Postal Code

75005

Street

4 place Jussieu

Geofield

Where to apply

E-mail

[email protected]

Contact

City

Rueil-Malmaison

Website

http://www.ifpenergiesnouvelles.com/

Street

4 avenue de Bois-Préau

Postal Code

92852

STATUS: EXPIRED