dc.contributor.author
Nüsken, Nikolas
dc.contributor.author
Richter, Lorenz
dc.date.accessioned
2022-01-13T11:01:01Z
dc.date.available
2022-01-13T11:01:01Z
dc.identifier.uri
https://refubium.fu-berlin.de/handle/fub188/33499
dc.identifier.uri
http://dx.doi.org/10.17169/refubium-33220
dc.description.abstract
Optimal control of diffusion processes is intimately connected to the problem of solving certain Hamilton–Jacobi–Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate the potential of iterative diffusion optimisation techniques, in particular considering applications in importance sampling and rare event simulation, and focusing on problems without diffusion control, with linearly controlled drift and running costs that depend quadratically on the control. More generally, our methods apply to nonlinear parabolic PDEs with a certain shift invariance. The choice of an appropriate loss function being a central element in the algorithmic design, we develop a principled framework based on divergences between path measures, encompassing various existing methods. Motivated by connections to forward-backward SDEs, we propose and study the novel log-variance divergence, showing favourable properties of corresponding Monte Carlo estimators. The promise of the developed approach is exemplified by a range of high-dimensional and metastable numerical examples.
en
dc.format.extent
48 Seiten
dc.rights.uri
https://creativecommons.org/licenses/by/4.0/
dc.subject
Hamilton–Jacobi–Bellman PDEs
en
dc.subject
Forward-backward SDEs
en
dc.subject
Optimal control of diffusions
en
dc.subject
Divergences between probability measures
en
dc.subject
Rare event simulation
en
dc.subject
Deep learning
en
dc.subject.ddc
500 Naturwissenschaften und Mathematik::510 Mathematik::510 Mathematik
dc.title
Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space
dc.type
Wissenschaftlicher Artikel
dcterms.bibliographicCitation.articlenumber
48
dcterms.bibliographicCitation.doi
10.1007/s42985-021-00102-x
dcterms.bibliographicCitation.journaltitle
Partial Differential Equations and Applications
dcterms.bibliographicCitation.number
4
dcterms.bibliographicCitation.volume
2
dcterms.bibliographicCitation.url
https://doi.org/10.1007/s42985-021-00102-x
refubium.affiliation
Mathematik und Informatik
refubium.affiliation.other
Institut für Mathematik
refubium.funding
Springer Nature DEAL
refubium.note.author
Die Publikation wurde aus Open Access Publikationsgeldern der Freien Universität Berlin gefördert.
refubium.resourceType.isindependentpub
no
dcterms.accessRights.openaire
open access
dcterms.isPartOf.eissn
2662-2971