PhD Position F/M Multi-Fidelity Scientific Machine Learning with Heterogeneous Inputs

Created in 2008, the Inria Saclay Center is located at the heart of the Paris-Saclay scientific and technological excellence cluster, which alone accounts for 15% of French research. Serving the development of the Université Paris-Saclay and the Institut Polytechnique de Paris, the Inria Saclay center employs 80 people in research support services and 500 scientists of 54 nationalities.

Benefiting from continuous growth, the center now has a total of 42 project-teams and two in the process of being created, including 21 jointly with the Institut Polytechnique de Paris, 16 with the Université Paris-Saclay, as well as 7 Inria EPs, including one in collaboration with Onera and one with the Pôle Universitaire Centre Val de Loire. These research teams are spread over more than ten sites.

Environment

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.

Scientific context

Many engineering and scientific problems involve complex physical phenomena that are difficult—and sometimes impossible—to reproduce experimentally. Moreover, experimental campaigns are often costly in time, resources, and logistics. In this context, numerical simulation plays a central role for prediction, design, and decision support.

In modern applications, models are frequently multi-physics, involve strong couplings across scales, and require high-dimensional parameterizations. A single high-fidelity simulation of the full system is often too expensive to be used repeatedly, for instance in optimization, uncertainty quantification (UQ), calibration, or control. An illustrative example is computational hemodynamics. In this application, fully resolved simulations of blood flow in patient-specific arterial geometries require solving the three-dimensional, time-dependent Navier–Stokes equations, often coupled with vessel wall elasticity and boundary conditions inferred from clinical data. While such simulations provide detailed information, they are computationally intensive, which prevents their systematic use in large parametric studies. Consequently, simplified or surrogate models (e.g., 1D network models, reduced-order models, data-driven surrogates) are widely used to obtain fast, approximate predictions.

A major scientific challenge is therefore to combine information from models of different fidelity levels in a principled manner, in order to achieve the accuracy of high-fidelity simulations while maintaining computational tractability. This is the objective of multi-fidelity modeling.

This PhD position is funded through the MediTwin project, which aims at advancing patient-specific digital twins for medical applications by combining physics-based modeling, data assimilation,and efficient computational pipelines (https://www.3ds.com/fr/science/meditwin).

Multi-Fidelity Methods: State of the Art and Open Challenges

Multi-fidelity (MF) methods exploit correlations between low- and high-fidelity models to reduce the number of expensive evaluations required for prediction and optimization. A classical approach is co-Kriging (Kennedy--O'Hagan), which models the high-fidelity response through an autoregressive Gaussian process (GP) relationship with the low-fidelity response. Extensions include nonlinear information fusion with GPs, Bayesian multi-fidelity inference and deep probabilistic surrogates, as well as MF neural networks that learn nonlinear cross-fidelity correlations. 

More broadly, scientific machine learning methods such as physics-informed neural networks (PINNs) and operator learning (DeepONet, Fourier Neural Operator) provide scalable tools to learn mappings between function spaces, which is particularly relevant when model outputs are fields and discretizations differ. 

Despite substantial progress, a key limitation remains insufficiently resolved: heterogeneous inputs and outputs across fidelities. In many applications, the low- and high-fidelity models do not share the same parameterization, discretization, or state variables. Such heterogeneity prevents the direct application of standard MF frameworks that assume a shared input space and pointwise correspondence between outputs.

PhD Objectives

The objective is to design scalable multi-fidelity methods capable of merging information coming from models with mismatched parameterizations and heterogeneous data structures. 

The PhD will investigate i) how to define common latent representations linking heterogeneous input spaces across fidelities; ii) how to build multi-fidelity surrogates that remain consistent under such heterogeneities; iii) how to quantify and propagate uncertainty induced by limited high-fidelity data and representation mismatch, iv) how to design adaptive sampling strategies for selecting expensive high-fidelity evaluations.

Scientific Methodology and Work Plan

The project is structured around four main axes.

The first axis is around the building of controlled benchmark settings where fidelity levels differ in parameterizations and/or discretizations (e.g., reduced vs.\ full-order models, coarse vs.\ fine meshes, sparse sensors vs.\ full fields). The goal is to formalize sources of heterogeneity (input mismatch, output mismatch, partial observability, missing variables) and define the problem mathematically.

Secondly, a central component of the PhD will be to learn mappings between heterogeneous spaces through latent-variable models and representation learning. Some methods that will be explored rely on the following techniques: i) Autoencoders / Variational Autoencoders (VAE) to embed high-dimensional inputs (fields, images, mesh-based signals) into low-dimensional latent coordinates; ii) Cross-modal alignment (e.g., canonical correlation analysis and Deep CCA) to align heterogeneous parameterizations and modalities in a shared latent space, iii) Operator learning (DeepONet, FNO) to learn mappings between function spaces and provide a bridge between heterogeneous inputs and field outputs.

The emphasis will be on representations that preserve physical meaning, support generalization, and remain compatible with downstream MF inference and uncertainty quantification.

Building on learned representations, the PhD will develop MF surrogate models that fuse low- and high-fidelity information. In particular, some techniques will be explored and compared: 

-  Latent-space co-Kriging: define GP models in a shared latent space and propagate uncertainty induced by the embedding.
- Multi-fidelity neural surrogates: residual learning and hierarchical neural architectures conditioned on fidelity indicators and latent variables.
- Hybrid probabilistic--deep models: combine neural representations with probabilistic heads (GPs, Bayesian neural networks) for calibrated uncertainty estimates.


Finally, the PhD candidate will focus on \textbf{active learning / adaptive design} for MF settings with heterogeneous inputs. The goal is to decide where to run expensive high-fidelity simulations to maximize information gain. 

[1] M. C. Kennedy and A. O’Hagan, Predicting the output from a complex computer code when fast approximations are available, Biometrika, 87(1), 1–13 (2000).

[2] P. Perdikaris, M. Raissi, and G. E. Karniadakis, Nonlinear information fusion algorithms for data-efficient multi-fidelity modeling, Proc. Roy. Soc. A, 473(2198):20160751 (2017).

[3] B. Peherstorfer, K. Willcox, and M. Gunzburger, Survey of multifidelity methods in uncertainty propagation, inference, and optimization, SIAM Review, 60(3):550–591 (2018).

[4] Diederik P Kingma and Max Welling, Auto-Encoding Variational Bayes, International Conference on Learning Representations (ICLR) 2014 ArXiv. http://arxiv.org/abs/1312.6114.

[5] X. Meng and G. E. Karniadakis, A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems, JCP, 401, 109020 (2020).

[6] M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear PDEs, J. Comput. Phys., 378:686–707 (2019).

[7] Y. Yang, P. Perdikaris, Conditional deep surrogate models for stochastic, high-dimensional, and multi-fidelity systems, arXiv:1901.04878 (2019).

8] L. Lu, J. Pengzhan, P. Guofei, G. Karniadakis, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nature Machine Intelligence, 3-3, 218–229, (2021).

[9] Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, Fourier Neural Operator for Parametric Partial Differential Equations, arXiv:2010.08895, 2020.

[10] B. Shahriari, K. Swersky, Z. Wang, R. P. Adams and N. de Freitas, Taking the Human Out of the Loop: A Review of Bayesian Optimization, Proceedings of the IEEE, 104-1, 148-175, 2016.

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.

Expected skills

• Proficiency in Matlab/Python/Julia
• Oral presentation skills: progress meetings, team meetings
• Good writing skills: report writing, article writing
• Ability to work in an international team

• Subsidized meals
• Partial reimbursement of public transport costs
• Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
• Possibility of teleworking and flexible organization of working hours
• Professional equipment available (videoconferencing, loan of computer equipment, etc.)
• Social, cultural and sports events and activities
• Access to vocational training
• Social security coverage

Gross Salary per month: 2300€