PhD Position - Structural Methods for Mixed Model/Data Digital Twin Engineering

Position Overview

We are seeking a highly motivated PhD candidate to contribute to the Engineering Digital Twins (EDT) program within Catalyst: the Reliable Hybrid Model Forge. The research focuses on developing new methods, algorithms, and tools that help designers correct model/data mismatches in digital twins of physics-dominated systems.

Modern modeling languages and tools allow engineers to build large-scale models directly from first principles of physics. Languages such as Modelica enable scalable modeling of complex cyber-physical systems, often using modeling paradigms such as port-Hamiltonian systems.

While assembling models from component libraries is relatively straightforward, practitioners often face major challenges in:

• parameter identification
• consistent model initialization
• fine-tuning of model dynamics
• integrating empirical models for poorly understood subsystems

This PhD aims to develop scalable methods that combine physics-based modeling with data-driven approaches to improve the reliability and accuracy of digital twin models.

Research Focus

A key challenge in digital twin engineering is the integration of experimental or simulated data into complex physics-based models.

Existing approaches typically rely on optimization-based data assimilation techniques, including:

• data reconciliation
• system identification
• deep learning approaches such as autoencoders

Although powerful, these methods often do not scale well to large dynamical systems involving thousands of variables.

This PhD proposes to address this challenge using structural analysis techniques for differential-algebraic equation (DAE) systems.

The core research idea is to transform the problem of data integration into the analysis of structurally overdetermined models. By leveraging structural analysis algorithms, it becomes possible to compute Minimal Structurally Overdetermined (MSO) subsystems, which can act as parity spaces to detect inconsistencies between model predictions and observed data.

These MSO subsystems can be solved using measured data, and the resulting residuals provide indicators of model inconsistencies. This approach enables the localization of model/data mismatches and supports targeted model corrections. Ultimately, the goal is to assist designers in discovering structural deficiencies in equation-based models, by identifying where the available data cannot be explained by the current set of equations.

The PhD research will investigate:

• Structural Analysis for Digital Twin Models: Adapting DAE structural analysis algorithms for model/data integration
• Parity Space Construction using MSOs: Identifying subsystems suitable for mismatch detection
• Scalable Algorithms for Large Systems: Leveraging graph-based algorithms with polynomial complexity
• Model Diagnosis and Correction: Using statistical analysis of residuals to localize inconsistencies

The proposed methods are particularly attractive because they scale well to large sparse systems, potentially containing millions of equations, and do not require prior knowledge of the reachable state space.

Key Responsibilities

• Conduct research on structural analysis methods for differential-algebraic equation systems
• Develop algorithms to detect and localize model/data mismatches
• Implement prototype tools supporting model validation in digital twin workflows
• Evaluate scalability on large-scale engineering models
• Collaborate with researchers working on modeling languages and digital twin platforms
• Publish results in international conferences and journals
• Participate in EDT consortium activities and collaborative research meetings

Research Environment

The PhD will be conducted within the Engineering Digital Twins (EDT) program, a national initiative aiming to advance the science and engineering of digital twin systems.

The candidate will work in a research environment combining expertise in:

• modeling languages such as Modelica
• hybrid systems and dynamical systems
• structural analysis of differential-algebraic equations
• large-scale simulation and digital twin architectures

Facilities and Resources

• Access to advanced modeling and simulation tools
• Collaboration with researchers specializing in Modelica and hybrid systems
• Opportunities to test methods on real-world engineering models
• Participation in international scientific collaborations

Qualifications

Required

• Master’s degree in Computer Science, Applied Mathematics, Automatic Control, or related fields
• Strong background in dynamical systems, numerical methods, or scientific computing
• Interest in modeling languages and digital twin technologies
• Strong analytical and problem-solving skills
• Good communication skills in English

Preferred

• Knowledge of differential-algebraic equations (DAEs)
• Experience with modeling tools such as Modelica
• Background in control systems, system identification, or data assimilation
• Familiarity with graph algorithms or structural system analysis

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 academic references

Funding and Benefits

• Duration: 3 years
• Salary: Standard French PhD funding
• Benefits: Health insurance, social security, travel support for conferences

Contact Information

Supervisor: Benoit Caillaud
Institution: Inria, Hycomes
Email: benoit.caillaud@inria.fr

Selected References

• P. Fritzson, “Principles of Object-Oriented Modeling and Simulation with Modelica 3.3: A Cyber-Physical Approach”, Wiley, 2014
• A. van der Schaft, D. Jeltsema, “Port-Hamiltonian Systems Theory: An Introductory Overview”, Foundations and Trends in Systems and Control: Vol. 1: No. 2-3, pp 173-378, 2014.
• B. Bachmann, P. Aronsson, P. Fritzson, “Robust Initialization of Differential Algebraic Equations”, 5th International Modelica Comnference, 2006.
• L. C. Mesa-Moles, A. Jardin, J.-P. Argaud, “How data assimilation can help in the initialization of Modelica models?”, 18th ModProd workshop, 2024.
• G. F. Machado and M. Jones, “Sparse Identification of Nonlinear Dynamics with Side Information (SINDy-SI),” 2024 American Control Conference (ACC), Toronto, ON, Canada, 2024, pp. 2879-2884, doi: 10.23919/ACC60939.2024.10644812.
• J Bakarji, K Champion, J Nathan Kutz, SL Brunton, “Discovering governing equations from partial measurements with deep delay autoencoders”, Proceedings of the Royal Society A, 2023.
• J. D. Pryce, “A Simple Structural Analysis Method for DAEs”, BIT, Numerical mathematics, 41(2):364-394, 2001.
• M. Krysander and E. Frisk, “Sensor Placement for Fault Diagnosis,” in IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, vol. 38, no. 6, pp. 1398-1410, Nov. 2008.
• B. Caillaud, M. Malandain, J. Thibault. Implicit structural analysis of multimode DAE systems. HSCC 2020 - 23rd ACM International Conference on Hybrid Systems: Computation and Control, Apr 2020, Sydney, Australia.
• B. Caillaud, A. Benveniste, M. Malandain. Benchmarking the Modular Structural Analysis Algorithm. 2025 - 16th International Modelica & FMI Conference, Sep 2025, Lucerne, Switzerland.

About the EDT Program

The Engineering Digital Twins (EDT) program is a major French research initiative bringing together leading research institutions to advance digital twin engineering. The program aims to develop new scientific foundations, software platforms, and industrial applications for next-generation digital twin systems.