{"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_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_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_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_n_ff27b07c4577", "title": "Fluid dynamics as the behavior of the axes", "shelf": "concepts", "surface": "secular", "snippet": "Matt: 'Fluid dynamics as behavior of axes? something along that line.' The axes are not static — they FLOW and redistribute as the map grows. Fluid dynamics may describe their behavior: a turbulent fl"}, {"id": "card_c_3437884ad1bb", "title": "Fluid probability dynamics related to the spectral (Fourier) lens", "shelf": "connections", "surface": null, "snippet": "Fourier eigenmodes of the Laplacian are exactly how the diffusion/continuity operator is solved."}, {"id": "card_c_3cc9f3e8e74f", "title": "Fokker–Planck & Liouville instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "Statistical mechanics is the stochastic instance of the probability-continuity equation."}, {"id": "card_c_108666f40f9f", "title": "Fluid probability dynamics resonates with Fluid dynamics of the axes", "shelf": "connections", "surface": null, "snippet": "Same conserved-flow skeleton, read on the map's own axes — RESONANCE, a map-aid not a proof."}, {"id": "card_n_810d857d886e", "title": "Madelung hydrodynamics — quantum mechanics as a probability fluid", "shelf": "science", "surface": "secular", "snippet": "Write ψ = √ρ·e^(iS/ħ) and the Schrödinger equation becomes fluid dynamics of the probability\ndensity ρ = |ψ|²: a continuity equation with velocity v = ∇S/m, plus an Euler equation carrying a\n'quantum "}, {"id": "card_c_576d28632055", "title": "Madelung hydrodynamics instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "Quantum mechanics rewritten as the fluid dynamics of |ψ|²."}, {"id": "card_n_0a41c0e3298d", "title": "Pheromones — the molecule is the message", "shelf": "science", "surface": "secular", "snippet": "A pheromone is chemistry acting as language: a molecule IS a signal. Sealed: a\n20-component blend encodes 2^20 = 1,048,576 distinguishable messages (~4.3 bits per component),\nand an ant's trail evapor"}, {"id": "card_n_7d5ce7aee5ff", "title": "Easton: Lily", "shelf": "dictionary", "surface": "secular", "snippet": "The Hebrew name shushan or shoshan, i.e., “whiteness”, was used as the general name of several plants common to Syria, such as the tulip, iris, anemone, gladiolus, ranunculus, etc. Some interpret it, "}, {"id": "card_n_e7be41668d40", "title": "Aurelius, Meditations §aur_08_liv: The sun seemeth to be shed abroad.", "shelf": "classics", "surface": "secular", "snippet": "The sun seemeth to be shed abroad. And indeed it is diffused but not effused. For that diffusion of it is a τάσις or an extension. For therefore are the beams of it called ἀκτῖνες from the word ἐκτείν"}, {"id": "card_n_36166748f8de", "title": "Westminster Shorter Q36", "shelf": "codex", "surface": "witness", "snippet": "What are the benefits which in this life do accompany or flow from justification, adoption, and sanctification?\n\nThe benefits which in this life do accompany or flow from justification, adoption, and "}, {"id": "card_n_901a38e7d2bd", "title": "Pilgrim's Progress §242: Now I saw that there stood behind the multitude a chariot and a couple of horses, waiti...", "shelf": "classics", "surface": "secular", "snippet": "Now I saw that there stood behind the multitude a chariot and a couple of horses, waiting for Faithful, who (so soon as his adversaries had despatched him) was taken up into it, and straightway was ca"}, {"id": "card_n_dabe5995cdb5", "title": "Augustine, Confessions §aug_conf_11_039: But because Thy loving-kindness is better than all lives, behold, my life is ...", "shelf": "classics", "surface": "witness", "snippet": "But because Thy loving-kindness is better than all lives, behold, my life is but a distraction, and Thy right hand upheld me, in my Lord the Son of man, the Mediator betwixt Thee, The One, and us many"}, {"id": "card_n_61bff7847701", "title": "Pilgrim's Progress §397: This done, they compassed them round on every side; some went before, some behind, and ...", "shelf": "classics", "surface": "secular", "snippet": "This done, they compassed them round on every side; some went before, some behind, and some on the right hand, some on the left, (as it were to guard them through the upper regions), continually sound"}, {"id": "card_n_c19201015c9f", "title": "Pilgrim's Progress §169: Now, as Christian went on his way, he came to a little ascent, which was cast up on pur...", "shelf": "classics", "surface": "secular", "snippet": "Now, as Christian went on his way, he came to a little ascent, which was cast up on purpose that pilgrims might see before them. Up there, therefore, Christian went, and looking forward, he saw Faithf"}]}