{"query": "Diffusion models — the probability-flow ODE behind generativ", "count": 20, "results": [{"id": "card_n_2be7f1c3f6e9", "title": "Diffusion models — the probability-flow ODE behind generative AI", "shelf": "science", "surface": "secular", "snippet": "Score-based generative models noise data toward a Gaussian and learn to reverse it. The\n'probability-flow ODE' is the deterministic fluid whose time-marginals match the noising SDE — the\nsame continui"}, {"id": "card_c_d7daafda1482", "title": "Fluid probability dynamics — one continuity  ↔ Diffusion models — the probability-flow ODE ", "shelf": "connections", "surface": null, "snippet": "machine learning (the probability-flow ODE under diffusion models)  — a concord the card itself states; mined + verified."}, {"id": "card_n_8530c72a2201", "title": "Fluid probability dynamics — one continuity equation across physics, geometry, and ML", "shelf": "science", "surface": "secular", "snippet": "One equation wears many clothes. Probability is a conserved fluid: the continuity equation\ndρ/dt + ∇·J = 0 (current J = ρv) says density is never created or destroyed — it only flows. The\nsame skeleto"}, {"id": "card_c_e510a4e8b7f4", "title": "Diffusion models instantiate Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "The probability-flow ODE is the same continuity equation with a learned velocity."}, {"id": "card_n_041763374eca", "title": "Fokker–Planck & Liouville — probability flow in statistical mechanics", "shelf": "science", "surface": "secular", "snippet": "The stochastic instantiation of probability-as-fluid. Fokker–Planck sets the current\nJ = μρ − D∇ρ (drift + diffusion); its steady state is the Boltzmann distribution ρ ∝ e^(−U/kT),\nwhich makes the cur"}, {"id": "card_src_word_conditional_probability", "title": "conditional probability", "shelf": "dictionary", "surface": "secular", "snippet": "conditional probability: (noun) the probability that an event will occur given that one or more other events have occurred — syn: contingent probability"}, {"id": "card_src_word_contingent_probability", "title": "contingent probability", "shelf": "dictionary", "surface": "secular", "snippet": "contingent probability: (noun) the probability that an event will occur given that one or more other events have occurred — syn: conditional probability"}, {"id": "card_c_c1d4f07c75cd", "title": "Fokker–Planck & Liouville — probability flow ↔ Fluid probability dynamics — one continuity ", "shelf": "connections", "surface": null, "snippet": "The stochastic instantiation of probability-as-fluid. ... In Hamiltonian phase space, Liouville's theorem gives dρ/dt = 0 — the probability fluid is incompressible.  — a concord the card itself states"}, {"id": "card_src_word_air_flow", "title": "air flow", "shelf": "dictionary", "surface": "secular", "snippet": "air flow: (noun) the flow of air — syn: airflow, flow of air"}, {"id": "card_sys_effort_flow", "title": "The effort–flow form — the master pattern", "shelf": "systems", "surface": "secular", "snippet": "Every physical system runs on a pair: EFFORT (the push) and FLOW (the response). Electrical: voltage & current. Fluid: pressure & volumetric flow. Mechanical: force & velocity. Thermal: temperature & "}, {"id": "card_src_word_joint_probability", "title": "joint probability", "shelf": "dictionary", "surface": "secular", "snippet": "joint probability: (noun) the probability of two events occurring together"}, {"id": "card_v_e25edb860b45", "title": "E[Binom(10, ½)] = n·p = 5 · P(X=2) = C(10,2)·½¹⁰ = 45/1024 ≈ 0.04395", "shelf": "probability", "surface": "secular", "snippet": "Verdict: HOLDS.\n• probability.binomial_mean: E[Binom(10,0.5)] = 5.0\n• probability.binomial: P(X=2 | n=10, p=0.5) = 0.0439453\nSealed and independently re-checkable: https://narrowhighway.com/s/e50261aa"}, {"id": "card_src_word_probability_theory", "title": "probability theory", "shelf": "dictionary", "surface": "secular", "snippet": "probability theory: (noun) the branch of applied mathematics that deals with probabilities — syn: theory of probability"}, {"id": "card_c_e655b3fcef54", "title": "Optimal transport instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "The geometry of moving probability — Fokker–Planck as a Wasserstein gradient flow."}, {"id": "card_src_word_theory_of_probability", "title": "theory of probability", "shelf": "dictionary", "surface": "secular", "snippet": "theory of probability: (noun) the branch of applied mathematics that deals with probabilities — syn: probability theory"}, {"id": "card_c_m_c148666fff7a", "title": "The one diffusion equation instance of The effort–flow form — the master pattern", "shelf": "connections", "surface": "secular", "snippet": "the flow form in space and time"}, {"id": "card_src_lex_g20932", "title": "G20932 — περιαγκωνίζω (periagkōnizō): to tie the hands behind the back", "shelf": "lexicon", "surface": "secular", "snippet": "περιαγκωνίζω (periagkōnizō), Strong's G20932: to tie the hands behind the back. tie the hands behind the back"}, {"id": "card_theory_kolmogorov_probability_axioms", "title": "Kolmogorov probability axioms", "shelf": "theories", "surface": "secular", "snippet": "Kolmogorov probability axioms — an engine domain that can touch it: probability. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and seal them. norma"}, {"id": "card_n_d59ca677ea24", "title": "Optimal transport — the geometry of moving probability", "shelf": "science", "surface": "secular", "snippet": "Benamou–Brenier: the distance between two distributions is the least kinetic energy of a\nfluid that carries one to the other, subject to the continuity equation. Jordan–Kinderlehrer–Otto\n(1998): the F"}, {"id": "card_src_rfc_7015", "title": "RFC7015 — Flow Aggregation for the IP Flow Information Export (IPFIX) Protocol", "shelf": "rfcs", "surface": "secular", "snippet": "RFC7015: Flow Aggregation for the IP Flow Information Export (IPFIX) Protocol (September 2013). Status: PROPOSED STANDARD."}]}