PhD Position - Interpretable and Robust Machine Learning for Physics-Based Numerical Simulations

Position Overview

We are seeking a motivated PhD candidate to work on the development of interpretable and robust machine learning models for accelerating numerical simulations of complex physical systems within the Engineering Digital Twins (EDT) program within Catalyst: the Reliable Hybrid Model Forge. .

Machine learning methods are increasingly used to emulate and accelerate numerical solvers for complex time-dependent physical problems. In many applications, neural networks can act as response surfaces, reproducing with high accuracy the behavior of physical phenomena that would otherwise require computationally expensive numerical simulations.

However, these models face two major limitations. First, they behave largely as black-box models, whose internal decision mechanisms remain difficult to interpret. This lack of interpretability limits the level of confidence that can be placed in their predictions. Second, these models are highly sensitive to the quality and representativeness of training data, and their performance can deteriorate significantly when used outside the domain explored during training, especially in the presence of uncertainty.

The goal of this PhD is to develop advanced statistical methods to address these limitations and improve the reliability of machine learning surrogates used in physics-based simulations.

Research Focus

The research will investigate statistical tools to enhance both the interpretability and the robustness of machine learning models used in scientific computing. A central objective of this research is to develop quantitative indicators of model reliability, allowing users to determine whether a machine learning surrogate can be safely reused in a new simulation context.

In particular, the work will explore the integration of high-dimensional statistical analysis techniques into machine learning workflows for physical simulations.

The main research directions include:

• Global Sensitivity Analysis
  Identifying the most influential parameters of neural network models (such as layer components, weights, or connections) with respect to model outputs.

• Optimal Transport Methods
  Analyzing the structure of the learned distributions and their evolution in parameter space in order to better understand how models generalize.

• Statistical Depth and Geometric Distances
  Developing metrics to quantify the confidence in model predictions and detect situations where the prediction lies far from the training data distribution.

These approaches will help characterize the reliability domain of machine learning models, identify extrapolation situations, and improve uncertainty quantification.

Ultimately, the research aims to support the development of hybrid modeling approaches that combine the predictive power of machine learning with the interpretability and rigor of statistical methods. Such models could enable the design of more reliable, explainable, and generalizable simulation tools for complex physical systems. These methods will help identify when a prediction is performed within or outside the domain of validity of the learned model, enabling more reliable reuse of machine learning surrogates in digital twin workflows.

Key Responsibilities

• Conduct research on statistical methods for interpretable machine learning
• Develop techniques for sensitivity analysis and uncertainty quantification in neural network models
• Investigate optimal transport and statistical depth methods for model reliability assessment
• Implement and test methods on physics-based simulation datasets
• Collaborate with researchers working on numerical simulations and digital twin models
• Publish results in international conferences and journals
• Participate in project meetings and collaborative activities within the EDT program

Research Environment

The PhD will be conducted within the Engineering Digital Twins (EDT) research program, which brings together multiple institutions working on the foundations of next-generation digital twin technologies.

The candidate will work in an interdisciplinary environment combining expertise in:

• machine learning and statistical modeling
• uncertainty quantification and high-dimensional statistics
• numerical simulation of physical systems
• scientific computing and digital twin engineering

Facilities and Resources

• Access to high-performance computing resources
• Collaboration with experts in statistics, machine learning, and physics-based simulation
• Real-world numerical simulation datasets
• Opportunities for international research collaborations

Qualifications

Required

• Master’s degree in Applied Mathematics, Statistics, Data Science, or Computer Science
• Strong background in probability, statistics, or machine learning
• Experience with scientific programming (Python, R, Julia, or similar)
• Interest in numerical simulations and uncertainty quantification
• Strong analytical and problem-solving skills
• Good written and oral communication skills in English

Preferred

• Knowledge of sensitivity analysis or uncertainty quantification
• Familiarity with optimal transport theory
• Experience with neural networks or deep learning
• Background in scientific computing or physics-based modeling

Application Process

Please submit the following documents:

1. Cover Letter describing your motivation and research interests
2. Curriculum Vitae
3. Academic Transcripts
4. Short Research Statement (1–2 pages)
5. Contact information for two references

Contact Information

Main supervisor Supervisor: Fabrice Gamboa Institution: Université de Toulouse Email: fabrice.gamboa@math.univ-toulouse.fr

Co Supervisors Supervisor: VALLEE Mathieu and Clément Gauchy Institution: CEA LITEN and CEA ISAS Email: mathieu.vallee@cea.fr and clement.gauchy@cea.fr

For scientific questions regarding the PhD topic, please contact the main supervisor directly.

Funding and Benefits

• Duration: 3 years
• Salary: Competitive PhD stipend according to French standards
• Benefits: Social security, health coverage, research travel support
• Travel: Support for conference attendance and research collaborations

Selected References

• Cholaquidis et al., Weighted lens depth: Some applications to supervised classification, Canadian Journal of Statistics, 2023.
• Da Veiga et al., Basics and Trends in Sensitivity Analysis, SIAM, 2021.
• Hallin et al., Distribution and quantile functions in dimension d: a measure transportation approach, Annals of Statistics, 2021.
• Nguyen et al., Combining statistical depth and Fermat distance for uncertainty quantification, NeurIPS, 2024.
• Yahiaoui et al., Reliable neural network model for accelerating coupled thermodiffusion simulation, ICAPP, 2025.

About the EDT Program

The Engineering Digital Twins (EDT) program is a major French research initiative dedicated to advancing the science and engineering of digital twin technologies through interdisciplinary collaboration between mathematics, computer science, and engineering.