Context
Inverse problems, i.e. estimating underlying signals from corrupted observations, are ubiquitous in astrophysics, and our ability to solve them accurately is critical to the scientific interpretation of the data. Examples of such problems include inferring the distribution of dark matter in the Universe from gravitational lensing effects [1], or component separation in radio interferometric imaging [2].

Thanks to recent deep learning advances, and in particular deep generative modeling techniques (e.g. diffusion models), it now becomes not only possible to get an estimate of the solution of these inverse problems, but to perform Uncertainty Quantification by estimating the full Bayesian posterior of the problem, i.e. having access to all possible solutions that would be allowed by the data, but also plausible under prior knowledge.

Our team has in particular been pioneering such Bayesian methods to combine our knowledge of the physics of the problem, in the form of an explicit likelihood term, with data-driven priors implemented as generative models. This physics-constrained approach ensures that solutions remain compatible with the data and prevents “hallucinations” that typically plague most generative AI applications.

However, despite remarkable progress over the last years, several challenges still remain in the aforementioned framework, and most notably:

[Imperfect or distributionally shifted prior data] Building data-driven priors typically requires having access to examples of non corrupted data, which in many cases do not exist (e.g. all astronomical images are observed with noise and some amount of blurring), or might exist but may have distribution shifts compared to the problems we would like to apply this prior to.
This mismatch can bias estimations and lead to incorrect scientific conclusions. Therefore, the adaptation, or calibration, of data-driven priors from incomplete and noisy observations becomes crucial for working with real data in astrophysical applications.

[Efficient sampling of high dimensional posteriors] Even if the likelihood and the data-driven prior are available, correctly sampling from non-convex multimodal probability distributions in such high-dimensions in an efficient way remains a challenging problem. The most effective methods to date rely on diffusion models, but rely on approximations and can be expensive at inference time to reach accurate estimates of the desired posteriors.

The stringent requirements of scientific applications are a powerful driver for improved methodologies, but beyond the astrophysical scientific context motivating this research, these tools also find broad applicability in many other domains, including medical images [3].

PhD project
The candidate will aim to address these limitations of current methodologies, with the overall aim to make uncertainty quantification for large scale inverse problems faster and more accurate.
As a first direction of research, we will extend recent methodology concurrently developed by our team and our Ciela collaborators [4,5], based on Expectation-Maximization, to iteratively learn (or adapt) diffusion-based priors to data observed under some amount of corruption. This strategy has been shown to be effective at correcting for distribution shifts in the prior (and therefore leading to well calibrated posteriors). However, this approach is still expensive as it requires iteratively solving inverse problems and retraining the diffusion models, and is critically dependent on the quality of the inverse problem solver. We will explore several strategies including variational inference and improved inverse problem sampling strategies to address these issues.
As a second (but connected) direction we will focus on the development of general methodologies for sampling complex posteriors (multimodal/complex geometries) of non-linear inverse problems. Specifically we will investigate strategies based on posterior annealing, inspired from diffusion model sampling, applicable in situations with explicit likelihoods and priors.
Finally, we will apply these methodologies to some challenging and high impact inverse problems in astrophysics, in particular in collaboration with our colleagues from the Ciela institute, we will aim to improve source and lens reconstruction of strong gravitational lensing systems.
Publications in top machine learning conferences are expected (NeurIPS, ICML), as well as publications of the applications of these methodologies in astrophysical journals.

References
[1] Benjamin Remy, Francois Lanusse, Niall Jeffrey, Jia Liu, Jean-Luc Starck, Ken Osato, Tim Schrabback, Probabilistic Mass Mapping with Neural Score Estimation, https://www.aanda.org/articles/aa/abs/2023/04/aa43054-22/aa43054-22.html

[2] Tobías I Liaudat, Matthijs Mars, Matthew A Price, Marcelo Pereyra, Marta M Betcke, Jason D McEwen, Scalable Bayesian uncertainty quantification with data-driven priors for radio interferometric imaging, RAS Techniques and Instruments, Volume 3, Issue 1, January 2024, Pages 505–534, https://doi.org/10.1093/rasti/rzae030

[3] Zaccharie Ramzi, Benjamin Remy, Francois Lanusse, Jean-Luc Starck, Philippe Ciuciu, Denoising Score-Matching for Uncertainty Quantification in Inverse Problems, https://arxiv.org/abs/2011.08698

[4] François Rozet, Gérôme Andry, François Lanusse, Gilles Louppe, Learning Diffusion Priors from Observations by Expectation Maximization, NeurIPS 2024, https://arxiv.org/abs/2405.13712

[5] Gabriel Missael Barco, Alexandre Adam, Connor Stone, Yashar Hezaveh, Laurence Perreault-Levasseur, Tackling the Problem of Distributional Shifts: Correcting Misspecified, High-Dimensional Data-Driven Priors for Inverse Problems, https://arxiv.org/abs/2407.17667

Context
In 2016, the announcement of the first direct detection of gravitational waves ushered in an era in which the universe will be probed in an unprecedented way. At the same time, the complete success of the LISA Pathfinder mission validated certain technologies selected for the LISA (Laser Interferometer Space Antenna) project. The year 2024 started with the adoption of the LISA mission by the European Space Agency (ESA) and NASA. This unprecedented gravitational wave space observatory will consist of three satellites 2.5 million kilometres apart and will enable the direct detection of gravitational waves at undetectable frequencies by terrestrial interferometers. ESA plans a launch in 2035.
In parallel with the technical aspects, the LISA mission introduces several data analysis challenges that need to be addressed for the mission’s success. The mission needs to prove that with simulations, the scientific community will be able to identify and characterise the detected gravitational wave signals. Data analysis involves various stages, one of which is the rapid analysis pipeline, whose role is to detect new events and characterise the detected events. The last point concerns the rapid estimation of the position in the sky of the source of gravitational wave emission and their characteristic time, such as the coalescence time for a black hole merger.
These analysis tools form the low-latency analysis pipeline. As well as being of interest to LISA, this pipeline also plays a vital role in enabling multi-messenger astronomy, consisting of rapidly monitoring events detected by electromagnetic observations (ground-based or space-based observatories, from radio waves to Gamma rays).

PhD project
The PhD project focuses on the development of event detection and identification tools for the low-latency alert pipeline (LLAP) of LISA. This pipeline will be an essential part of the LISA analysis workflow, providing a rapid detection of massive black hole binaries, as well as a fast and accurate estimation of the sources’ sky localizations as well as coalescence time. These are key information for multi-messager follow-ups as well as for the global analysis of the LISA data.
While rapid analysis methods have been developed for ground-based interferometers, the case of space-based interferometers such as LISA remains a field to be explored. Adapted data processing will have to consider how data is transmitted in packets, making it necessary to detect events from incomplete data. Using data marred by artefacts such as glitches or missing data packages, these methods should enable the detection, discrimination and analysis of various sources: black hole mergers, EMRIs (spiral binaries with extreme mass ratios), bursts and binaries from compact objects. A final and crucial element of complexity is the speed of analysis, which constitutes a strong constraint on the methods to be developed.
To this end, the problems we will be tackling during this thesis will be:
1. The fast parameter inference of the gravitational waves, noticeably, the sky position, and the coalescence time. Two of the main difficulties reside in the multimodality of the posterior probability distribution of the target parameters and the stringent computing time requirements. To that end, we will consider different advanced inference strategies including:
(a) Using gradient-based sampling algorithms like Langevin diffusions or Hamiltonian Monte Carlo methods adapted to LISA’s gravitational wave problem,
(b) Using machine learning-assisted methods to accelerate the sampling (e.g. normalising flows),
(c) Using variational inference techniques.
2. The early detection of black hole mergers.
3. The increasing complexity of LISA data, including, among others, realistic noise, realistic instrument response, glitches, data gaps, and overlapping sources.
4. The online handling of the incoming 5-minute data packages with the developed fast inference framework.
This thesis will be based on applying Bayesian and statistical methods for data analysis and machine learning. However, an effort on the physics part is necessary, both to understand the simulations and the different waveforms considered (with their underlying hypotheses) and to interpret the results regarding the detectability of black hole merger signals in the context of the rapid analysis of LISA data.