@article{LelievreStoltzZhang2020, author = {Lelievre, Tony and Stoltz, Gabriel and Zhang, Wei}, title = {Multiple projection MCMC algorithms on submanifolds}, series = {IMA Journal of Numerical Analysis}, journal = {IMA Journal of Numerical Analysis}, doi = {10.1093/imanum/drac006}, year = {2020}, abstract = {We propose new Markov Chain Monte Carlo algorithms to sample probability distributions on submanifolds, which generalize previous methods by allowing the use of set-valued maps in the proposal step of the MCMC algorithms. The motivation for this generalization is that the numerical solvers used to project proposed moves to the submanifold of interest may find several solutions. We show that the new algorithms indeed sample the target probability measure correctly, thanks to some carefully enforced reversibility property. We demonstrate the interest of the new MCMC algorithms on illustrative numerical examples.}, language = {en} } @article{LelievrePigeonStoltzetal.2023, author = {Leli{\`e}vre, Tony and Pigeon, Thomas and Stoltz, Gabriel and Zhang, Wei}, title = {Analyzing multimodal probability measures with autoencoders}, series = {The Journal of Physical Chemistry}, journal = {The Journal of Physical Chemistry}, year = {2023}, abstract = {Finding collective variables to describe some important coarse-grained information on physical systems, in particular metastable states, remains a key issue in molecular dynamics. Recently, machine learning techniques have been intensively used to complement and possibly bypass expert knowledge in order to construct collective variables. Our focus here is on neural network approaches based on autoencoders. We study some relevant mathematical properties of the loss function considered for training autoencoders, and provide physical interpretations based on conditional variances and minimum energy paths. We also consider various extensions in order to better describe physical systems, by incorporating more information on transition states at saddle points, and/or allowing for multiple decoders in order to describe several transition paths. Our results are illustrated on toy two dimensional systems and on alanine dipeptide.}, language = {en} }