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As a result, the population can only be monitored with periodic snapshots, obtained by sampling a few particles that are sacrificed in exchange for measurements. Given only access to these snapshots, can we reconstruct likely individual trajectories for all other particles? We propose to model these trajectories as collective realizations of a causal Jordan-Kinderlehrer-Otto (JKO) flow of measures: The JKO scheme posits that the new configuration taken by a population at time $t+1$ is one that trades off an improvement, in the sense that it decreases an energy, while remaining close (in Wasserstein distance) to the previous configuration observed at $t$. In order to learn such an energy using only snapshots, we propose JKOnet, a neural architecture that computes (in end-to-end differentiable fashion) the JKO flow given a parametric energy and initial configuration of points. We demonstrate the good performance and robustness of the JKOnet fitting procedure, compared to a more direct forward method.","url_abs":"https://arxiv.org/abs/2106.06345v4","url_pdf":"https://arxiv.org/pdf/2106.06345v4.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":"jkonet-proximal-optimal-transport-modeling-of","repo_url":"https://github.com/bunnech/jkonet","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"jkonet-proximal-optimal-transport-modeling-of","repo_url":"https://github.com/gjhuizing/wsingular","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2106.06345","atlas_url":"https://app.syntology.ai/?focus=2106.06345","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2106.06345"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+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. 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