Re-themed in 2026-10 with a color-blindness simulation (deuteranopia, protanopia, tritanopia); the old red/green pair for conditional vs marginal was ΔE 9 for deuteranopes. Every color also has a second, color-free cue.
#E0E0E0 = noise: x0, p0 = N(0,1) (p0 is p_t at t = 0, so it shares the neutral density color)#009E73, dots = data: x1, q = ½δ(−2) + ½δ(+2)#E69F00, DASHED lines = per-sample (conditional) straight lines, their densities, and their target x1 − x0#9CC8FF, SOLID lines = marginal velocity u_t(x), its field and its (curved) trajectories#B07CFF = the network v_θ and the sampler that queries it (Euler steps)Times below are sentence starts from the current audio/cues.json.
Title top (y≈3.4). Main stage: (t, x) plot on the left, t∈[0,1] horizontal, x∈[−3,3] vertical, centered at (−2.8, −0.35). Right panel (x≈1.6…6.9) for formulas and tables. Captions bottom (y≈−3.45).
[0] 0.05 “How do you turn noise into data?” title; grey 2-D Gaussian cloud fades in center; faint green target clusters left-up / right-down [1] 2.35 “…velocity field.” purple arrow grid (field at t=0) grows in; label “velocity field v_θ(x,t)” [2] 5.09 “…at every point and time…” t counter appears; t sweeps 0→0.7→0 and arrows change with it (dots fixed) [3] 11.20 “…start from noise and follow the arrows.” t 0→1: dots flow along the field into the two green clusters [4] 14.72 “But there is a puzzle.” everything dims to 0.25 [5] 15.96 “…target it can never match exactly.” loss formula, target (x1 − x0) orange, “min > 0” note [6] 19.81 “…what it learns, and why that works.” three keyword chips: “what it learns · why it works · how to sample”
[0] 0.05 “…simplest case, one dimension.” title “Setup: 1-D, straight paths”; axes Create [1] 2.81 “Time runs from zero to one…” t-axis and labels 0, 1 Indicate; small dot sweeps along t axis [2] 6.34 “…noise, x0, standard Gaussian.” grey sideways density at t=0 grows; label x0 ~ N(0,1) [3] 11.41 “At time one we have data, x1.” green dots at (1, ±2) pop in; label x1 [4] 13.75 “…two points, ±2, prob ½.” right panel: q = ½δ₋₂ + ½δ₊₂; dot labels “½” [5] 19.89 “Pick one noise sample and one data point…” grey dot x0=−0.8 at t=0; (1,2) indicated; orange line Create [6] 25.31 “At time t, the point is (1−t)x0 + t x1.” formula x_t = (1−t)x0 + t x1; white dot slides along line 0→0.6 [7] 31.84 “Its velocity … x1 − x0.” formula dx_t/dt = x1 − x0 = 2.8 (orange); orange tangent arrow on the dot [8] 36.21 “…same at every time…constant speed.” dot+arrow slide 0.6→1→0.2; arrow never changes; “same at every t”
[0] 0.05 “Training uses this velocity as target.” previous plot state; title “Training: conditional flow matching” [1] 3.67 “Sample a time, a noise point, a data point.” right panel: t~U[0,1], x0~N(0,1), x1~q appear one by one; dashed t=0.6 line [2] 6.92 “Form the point x_t on their line.” white x_t dot on the orange line at t=0.6 [3] 9.24 “The network sees x_t and t … squared error.” loss formula (v_θ purple, target orange); purple arrow from x_t (prediction) next to orange arrow (target) [4] 16.14 “This is the conditional FM loss.” “L_CFM” box Indicate [5] 19.04 “Now draw many such pairs.” 14 more orange lines Create (lagged), single example dims [6] 20.86 “The straight lines cross.” small white circles flash at several crossings [7] 22.85 “…x = 0 at time one half.” white dot at (½, 0) with dashed guides [8] 26.29 “…slope +4 … slope −4.” two bold orange lines (−2→+2 and +2→−2) through it; others dim; labels +4, −4 [9] 30.99 “Same input, two targets.” right: “(x=0, t=½) → +4 or −4” [10] 33.48 “Loss can never reach zero.” right: “⇒ min_θ L_CFM > 0”
[0] 0.05 “What does the network learn?” title; small plot (left) with point (½,0) and the ±4 lines [1] 1.51 “best prediction = average of targets” argmin_c E(Y−c)² = E[Y]; light-blue flat arrow at the point “avg = 0” [2] 6.46 “…all lines through x at time t.” both lines through the point Indicate [3] 10.69 “…marginal velocity u_t(x).” light-blue u_t(x) appears [4] 14.00 “…conditional expectation…” u_t(x) = E[x1 − x0 | x_t = x] [5] 20.25 “Now expand the loss around it.” plot fades; L_CFM moves top; expanded square with (v_θ − u_t) + (u_t − (x1−x0)) [6] 22.23 “First term: distance to u_t.” term 1 boxed LIGHT-BLUE, tag “L_FM: what we want” [7] 26.27 “Second term: variance, no θ.” term 2 boxed GREY, tag “variance, no θ” [8] 30.90 “Cross term zero…” cross term appears then gets “= 0” with reason E[x1−x0 | x_t] = u_t [9] 34.90 “same gradients as marginal loss” ∇_θ L_CFM = ∇_θ L_FM boxed
[0] 0.05 “Why is u_t the field we want?” title; axes + grey p0 + green dots [1] 1.84 “…distribution of x_t at a few times.” white sideways density profile appears at t=0 [2] 5.05 “…one Gaussian, two bumps, two points.” profile slides t: 0→0.95, leaving faint copies at 0.25, 0.5, 0.75 [3] 10.51 “Each line moves its own conditional distribution.” at t=½: two orange dashed component bumps (½N(±1, ½²)) under the white sum [4] 17.10 “…weighted by how likely each line is” formula u_t(x) = Σ w(x1|x,t) u_t(x|x1), w posterior [5] 23.79 “Continuity equation linear in p·u” conditional eq (orange) → average over x1 → marginal eq (light-blue) [6] 28.14 “…moves the whole noise distribution…” p0 (grey) —u_t→ p1 (green) labels; formulas dim [7] 33.05 “Following u_t from noise gives data.” five light-blue trajectories drawn from grey to green
[0] 0.05 “…real numbers, at time one half.” title; axes; dashed t=½ line [1] 3.30 “At x = 0, targets +4 and −4.” point (½,0); two orange lines and orange arrows +4/−4; table row [2] 8.40 “Equally likely, marginal = 0.” weights ½, ½; u = ½·4 + ½·(−4) = 0; light-blue flat arrow [3] 13.28 “…a value it never saw as a target.” light-blue 0 Indicate; tag “never a target” [4] 17.52 “Move to x = ½.” old arrows fade; dot slides to (½, ½) [5] 19.85 “Toward +2: target +3.” orange line (0,−1)→(1,2), arrow +3 [6] 22.80 “Toward −2: target −5.” orange line (0,3)→(1,−2), arrow −5 [7] 25.98 “Posterior weights not equal.” two conditional densities at t=½ (sideways); heights at x=½ marked; w ∝ N(½; ±1, ½²) [8] 28.25 “≈ 98% and 2%.” weights 0.982 / 0.018; −5 line fades to 0.15 [9] 31.20 “Marginal ≈ 2.86 toward +2.” u = 0.982·3 + 0.018·(−5) ≈ 2.86; light-blue arrow at the point
[0] 0.05 “…whole marginal field.” light-blue slope-field arrows grow over the plane [1] 3.74 “Closed form: (E[x1|x_t=x] − x)/(1 − t)” formula, plus this data’s 2·tanh(2tx/(1−t)²) [2] 9.72 “Follow the field from many noise samples.” 13 light-blue trajectories Create from grey x0’s [3] 12.47 “Curved, do not cross.” field dims; trajectories thicken [4] 16.14 “…hesitate, then split.” the two trajectories nearest 0 highlighted; circle at the split region [5] 20.07 “Compare the training lines.” faint orange straight lines fade in [6] 21.74 “Straight, but they cross.” crossing circles flash on orange lines [7] 24.00 “Deterministic map noise → data.” “x0 > 0 ↦ +2, x0 < 0 ↦ −2” tag; orange lines fade
[0] 0.05 “Draw x0 from the Gaussian.” title; axes; faint light-blue field; grey dot at (0, 0.4) [1] 3.00 “Solve dx/dt = v_θ(x,t) from 0 to 1.” ODE formula (purple v_θ); faint light-blue exact trajectory from 0.4 [2] 11.26 “Euler’s method.” x_{k+1} = x_k + h·v_θ(x_k, t_k) [3] 13.89 “Step of size h…evaluate again” (scaffold arrow) “h” brace on t axis [4] 19.21 “x0 = 0.4, four steps of ¼.” table k | t_k | x_k with first row 0 | 0 | 0.40 [5] 24.57 “0.30, 0.37, 1.09, 2.” 4 purple segments drawn in turn, table rows filled [6] 31.38 “Lands on +2.” final dot on (1,2) Indicate
[0] 0.05 “Only one step?” title; axes; grey density; green dots [1] 1.79 “At t=0, x_t carries no info about data.” t=0 line highlighted; “x_0 ⟂ x_1” [2] 5.71 “u_0(x) = E[x1] − x.” formula [3] 9.49 “Mean 0 → velocity −x0.” = −x; light-blue arrows on t=0 line pointing to 0 [4] 13.69 “One step sends every sample to 0.” 7 purple straight segments from x0’s to (1,0); dot at (1,0) [5] 18.02 “Not a data point.” white cross + “mean, not data” [6] 19.73 “Straight lines ≠ straight marginal paths.” curved light-blue trajectories over faint orange lines [7] 23.16 “Curvature → several steps.” the 4-step purple path from Sampling reappears, lands at 2
[0..6] one line per sentence, stacked left-aligned, colored by legend: v_θ(x,t) · target x1−x0 on x_t=(1−t)x0+t x1 · v* = u_t(x)=E[x1−x0|x_t=x] · u_t moves p0 → p1 · min L_CFM = E Var > 0 · sample dx/dt = v_θ, t: 0→1