{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/kernel-based-approximation-of-the-koopman","title":"Kernel-based approximation of the Koopman generator and Schrödinger operator","arxiv_id":"2005.13231","date":"2020-05-27","proceeding":null,"authors":["Stefan Klus","Feliks Nüske","Boumediene Hamzi"],"abstract":"Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schr\\\"odinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schr\\\"odinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.","url_abs":"https://arxiv.org/abs/2005.13231v3","url_pdf":"https://arxiv.org/pdf/2005.13231v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"kernel-based-approximation-of-the-koopman","repo_url":"https://github.com/sklus/d3s","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2005.13231","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2005.13231"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/sklus/d3s","reach":null}],"summary":{"ran_fixture":2},"by_repo_kind":{},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":2,"samples":[{"code_sha256_prefix":"8471eee6c309321c","entry":"amuse","repo":null,"repo_kind":null,"path":null,"file_url":null,"link_basis":"identical_code_first_harvested_elsewhere","language":"python","status":"ran_fixture","verification_level":1,"contract_check":"RAISES","metamorphic_tier":"none","behaviour_fingerprint":true,"licence":null,"inline_ok":false,"mcp_get_code":{"code_sha256":"8471eee6c309321c"}},{"code_sha256_prefix":"7363464ca8e7319e","entry":"dmd","repo":null,"repo_kind":null,"path":null,"file_url":null,"link_basis":"identical_code_first_harvested_elsewhere","language":"python","status":"ran_fixture","verification_level":1,"contract_check":"RAISES","metamorphic_tier":"deterministic","behaviour_fingerprint":false,"licence":null,"inline_ok":false,"mcp_get_code":{"code_sha256":"7363464ca8e7319e"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}