Reverse SDE and probability-flow ODE

Consider a forward Itô process with drift f(x,t), scalar diffusion coefficient g(t) independent of state, and smooth marginal density p_t. Its score is the state gradient of log p_t, not a predicted clean sample unless the required parameterization conversion has been applied. The reverse-time formula uses the forward drift minus g squared times that score and integrates from terminal time T toward zero. In this convention dt is negative and backward Wiener increments have variance equal to the magnitude of dt. Reversing the sign twice changes the dynamics.

An equivalent increasing computational coordinate τ=T−t is possible, but its drift must be transformed consistently. Do not copy an update derived for decreasing t into an increasing step counter without changing the interpretation. Neural implementations commonly predict noise, clean data, or velocity instead of the score directly. Those predictions require schedule-dependent conversions and input scaling. A tensor with the right shape but the wrong prediction meaning is not a valid drift evaluation, and a scheduler class name cannot repair an incompatible model configuration.

The probability-flow ODE uses half the score correction in its drift. With the exact score, compatible initial marginal, and regularity conditions, its density evolution matches the forward SDE marginals; integrating it backward provides the corresponding transport. This is a distributional statement, not a pairing that makes each deterministic trajectory equal a reverse-SDE path. Randomness remains in the initial draw even when the subsequent evolution is deterministic. A deterministic map of random inputs need not produce a point mass.

In practice, the learned score has estimation error, terminal noise may only approximate the required p_T, and a finite-step solver adds numerical error. Replacing the exact score with a neural estimate does not automatically preserve the same marginal equivalence between two approximate processes. A DDIM-style deterministic update and a probability-flow interpretation also require their own parameterization and discretization conditions. Do not describe a twenty-step pipeline as exact sampling from the data distribution merely because an ideal continuous-time equation has an exact marginal identity.

Record the model revision, prediction parameterization, forward schedule, sampler update, terminal initialization, endpoint handling, conditioning, precision, and random draws. Distinguish numerical integration error from model-score and initial-law error when diagnosing artifacts. A smaller step may reduce one discretization component while leaving a poor learned field or incompatible parameterization untouched. Use controlled low-dimensional tests for drift signs and noise variances before trusting an image-level quality score to diagnose a mathematical mismatch.

The probability instrument contrasts a nondegenerate distribution with a degenerate one without pretending to integrate diffusion. Reset, set both p=0.5 and n=10, predict mean five and variance 2.5, then Play or Step. Separately Reset, set both p=1 and n=10, predict mean ten and variance zero, then Play or Step. These exact binomial bars are not Brownian paths, reverse-SDE samples, or ODE transport. They support the limited lesson that a mean alone does not characterize a stochastic law and that deterministic evolution must not be confused with absence of initial randomness.