Model Reduction Techniques for Parameter Calibration Problems

The work of the PhD candidate will be supervised by P.M. Congedo, E. Denimal Goy and Olivier Le Maître, experts in uncertainty quantification methods.

The work will be conducted in the Platon team, a joint research group between Ecole Polytechnique and CNRS, hosted by the Center for Applied Mathematics (CMAP) of ´Ecole Polytechnique. The Platon project-team focuses on developing innovative methods and algorithms for uncertainty mangament in numerical models, including advanced calibration strategies from data (observations, measurements, other model predictions) and uncertainty reduction.

Context

The calibration of model parameters is a central challenge when using high-fidelity numerical simulations to represent complex physical or biological systems. In particular, the emerging field of digital twins,virtual replicas of individual patients or organs, requires accurate and robust parameter identification to ensure predictive reliability.

However, the combination of high computational cost, large parameter spaces, and limited, noisy observations makes traditional calibration techniques (e.g. deterministic optimization or Monte Carlo sampling) impractical for real-time or clinical use. There is therefore a strong need for advanced methodologies that combine machine learning (ML), model reduction, and uncertainty quantification (UQ) to address these challenges [1,2,3].

One way to reduce the complexity of calibration problems is to decrease the dimensionality of either the observation space and/or the parameter space. Reducing the observation space consists in projecting or compressing the available data (e.g. via Proper Orthogonal Decomposition, autoencoders, or spectral methods), so that the inverse problem is solved using a reduced set of informative measurements. Alternatively, reducing the parameter space aims at representing the set of admissible parameters in a low-dimensional manifold, either through linear approaches (e.g. Principal Component Analysis, Active Subspaces) or nonlinear embeddings (e.g. manifold learning, neural encoders). Both strategies lead to a simpler and more tractable calibration problem, enabling faster computations and more robust inference.

The objective of this internship is to investigate such linear and nonlinear reduction techniques in the parameter space for improving the efficiency of parameter calibration in high-fidelity models, with potential applications in digital twin frameworks.

Objectives and Work Plan

The internship will focus on developing and assessing reduction strategies to facilitate parameter calibration in computational models, with a particular emphasis on reducing the dimensionality of the parameter space. The student will begin by conducting a literature review on model calibration, inverse problems, and dimensionality reduction techniques, in order to identify the most relevant approaches for parameter-space reduction. Building on this groundwork, a reference calibration pipeline will be implemented. Linear techniques such as Principal Component Analysis (PCA), Singular Value Decomposition (SVD), and Active Subspaces will first be explored to identify low-dimensional structures in the parameter space that capture the dominant modes of model variability. In a second phase, nonlinear reduction strategies, for example based on autoencoders or kernel mappings, will be developed to capture more complex parameter dependencies. These reduced representations will be integrated into a calibration framework, using optimization or Bayesian inference methods, to assess their impact on both computational cost and estimation accuracy. The results will be analyzed on synthetic test cases.

Bibliography

[1] K. Bruynseels, D. Santoni de Sio, and J. van den Hoven, Digital twins in health care: Ethical implications of an emerging engineering paradigm, Frontiers in Genetics, 9:31 (2018).

[2] M. Viceconti, J. McCulloch, and L. Pappalardo, The digital twin of the human body: A scientific grand challenge, npj Digital Medicine, 4, 119 (2021).

[3] A. Guha and K. Willcox, Learning reduced-order models for parameterized systems with control and calibration applications, SIAM J. Sci. Comput., 43(4):A26612685 (2021).

[4] P. G. Constantine, Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies, SIAM Spotlights, 2015.

[5] T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning, Springer, 2009.

Candidates should be enrolled in a Master’s program in engineering, applied mathematics or a related discipline, and a specialization in machine learning, uncertainty quantification, optimization or related fields.

The 6-months internship may be followed by a PhD thesis funded through the Meditwin project, which is devoted to the development of digital twins for healthcare applications. Applicants should submit a CV, a covering letter as a single document detailing the knowledge, skills and experience you think make you the right candidate for the job, a list of your MSc courses and grades. For further details and applications, please contact PM Congedo (pietro.congedo [at] inria.fr).