The French National Institute for Research in Digital Science and Technology (Inria) is inviting applications for a fully funded PhD position focused on developing hybrid numerical methods combining neural networks and classical approaches for solving high-dimensional Partial Differential Equations (PDEs). This interdisciplinary project lies at the intersection of machine learning, numerical analysis, and physics-informed computation.
Important Information
| Field | Details |
|---|---|
| Title | PhD Position (F/M): Numerical Methods Mixing Neural Networks and Classical Approaches for High-Dimensional PDEs |
| Organization/Publisher | Inria (Institut National de Recherche en Informatique et en Automatique) |
| Work Location | Paris / Strasbourg, France |
| Research Field | Numerical Analysis, Machine Learning, Scientific Computing |
| Funding Info | Fully funded PhD position (Fixed-term contract, 3 years) |
| Application Deadline | November 28, 2025 |
| Posted Date | October 2025 |
| Country | France |
| Researcher Profile | PhD Candidate |
| Apply Button | Apply on Inria Official Portal |
| Required Qualification | Master’s degree (M2) or equivalent in Mathematics, Physics, or Scientific Computing |
| Required Experience | Strong background in PDEs and numerical methods; experience with Python; machine learning knowledge preferred |
| Salary Details | Approx. €2300 gross/month |
| Supervisor | Dr. Emmanuel Franck (emmanuel.franck@inria.fr) |
| Starting Date | October 1, 2026 |
| Contract Type | Fixed-term (3 years) |
| Location (Inria Center) | Centre Inria de l’Université de Lorraine, Strasbourg |
| Theme/Domain | Numerical Schemes, Simulations, Scientific Computing |
About the Project
Partial Differential Equations (PDEs) are central to scientific computing in domains like astrophysics, plasma physics, and fluid dynamics. However, traditional numerical methods often face the curse of dimensionality, making high-dimensional simulations computationally expensive.
This PhD project aims to overcome these challenges by designing hybrid numerical schemes that integrate neural networks (PINNs) with classical Discontinuous Galerkin (DG) methods to efficiently solve high-dimensional PDEs.
The goal is to develop both explicit and semi-Lagrangian schemes that merge the robustness of deterministic solvers with the adaptability of neural models, thereby drastically reducing computational cost while preserving physical accuracy.
Key Research Objectives
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Develop hybrid numerical schemes combining DG methods and neural PINN approaches.
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Capture large-scale dynamics via neural components while applying DG solvers for fine-scale correction.
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Construct reduced-order models (ROMs) using local and non-local closures.
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Utilize machine learning and data-driven training to design physically informed closure models.
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Analyze convergence and stability both theoretically and numerically.
This research will build a strong foundation in differentiable programming, physics-informed neural networks, and reduced-order modeling for PDEs.
Main Duties and Responsibilities
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Design and implement new hybrid numerical schemes.
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Develop reduced-order models based on physical principles.
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Conduct numerical experiments and analyze convergence/stability.
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Implement models in JAX, leveraging differentiable programming.
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Extend code for multi-GPU and multi-patch computations.
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Publish results in scientific journals and present findings at conferences.
Candidate Profile
Required Skills
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Strong background in Partial Differential Equations (PDEs) and numerical analysis.
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Proficiency in Python and scientific programming.
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Excellent analytical and problem-solving abilities.
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Fluent in English (written and spoken).
Preferred Skills
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Familiarity with machine learning or neural PDE solvers.
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Knowledge of JAX, TensorFlow, or PyTorch.
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Experience with high-performance computing (HPC) environments.
Benefits and Employment Conditions
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Monthly salary: ~€2300 gross/month.
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Contract duration: 3 years (Fixed-term).
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Leave: 7 weeks annual leave + 10 RTT days.
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Work flexibility: Option for teleworking after 6 months.
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Additional benefits: subsidized meals, transport reimbursement, vocational training, and access to cultural/sports programs.
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Social security and pension coverage included.
Diversity and Inclusion
Inria is committed to promoting diversity and equal opportunities. All qualified applicants, including individuals with disabilities, are encouraged to apply.
How to Apply
Applications must be submitted online through the official Inria recruitment portal:
👉 https://jobs.inria.fr
Required documents:
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Motivation letter
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Curriculum Vitae
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Academic transcripts (Bachelor’s + Master’s)
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Proof of English proficiency
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Contact details of two academic referees
📧 PhD Supervisor: Dr. Emmanuel Franck
✉️ Email: emmanuel.franck@inria.fr
⚠️ Applications submitted outside the Inria website may not be processed.
🗓️ Application deadline: November 28, 2025
About Inria
Inria is France’s premier research institute for digital science and technology, employing over 2,600 researchers and engineers. With 200 collaborative project teams and 900 support staff, Inria is at the forefront of global innovation in areas like AI, computation, and data science, developing technologies that drive progress across industries and academia.
Description
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