Sampler discretization and local error

Use an increasing computational coordinate τ and write a deterministic ODE dx/dτ=F(x,τ). Explicit Euler evaluates the field at the current state and advances by a positive step h. If the original sampling equation used decreasing diffusion time t, transform its drift before applying this convention. A numerical fixture with an analytic solution can expose sign, scaling, and endpoint mistakes independently of a neural model. It cannot by itself predict quality on a high-dimensional learned image manifold.

A local defect starts each trial at the exact reference state and asks how one numerical step differs from the exact evolution over that step. Global error instead includes propagation and accumulation of previous mistakes over a fixed horizon. Lipschitz and smoothness assumptions control this propagation; stiffness, endpoint singularities, or a poorly behaved learned field can invalidate a naive comparison. Stochastic strong and weak convergence have separate definitions and orders, so a deterministic Euler order statement must not be transferred unchanged to an SDE sampler.

For a sufficiently smooth ODE on a fixed finite interval with suitable Lipschitz bounds, Euler’s one-step state defect is second order in h and its global endpoint error is first order. The constants and stability region still matter. Higher-order diffusion solvers exploit special structure and require their own derivation; the DPM-Solver paper analytically handles a linear component while approximating a neural-network integral. An exact integral representation does not make a finite quadrature of that integral exact for every learned field.

A solver can use several network evaluations per step, reuse previous evaluations, or perform additional corrector calls. Report neural function evaluations, batch size per invocation, guidance branches, and total wall time rather than equating the loop counter with cost. Reducing function evaluations can improve latency but may change discretization error, conditioning adherence, diversity, or artifacts. Distilled or consistency-trained models have a different learned objective and supported sampler contract; they are not evidence that arbitrary pretrained models can lose most evaluations without quality consequences.

Use a scalar or small-vector reference to verify an implementation’s claimed order and schedule indexing. Then use matched model, prompts, seeds, precision, and output processing to measure task quality and latency. A tighter solver tolerance may increase evaluations without improving a metric dominated by model error. Conversely, a numerically different trajectory can appear preferable under one perceptual metric. Report that empirical observation honestly instead of converting it into a theorem that numerical error and image quality always move together.

The optimization instrument exactly reproduces Euler updates for the test equation x′=−x. Reset, set both η=0.25 and x₀=1, predict x₄=0.31640625 at computational horizon one, then Step four times. Separately Reset, set both η=0.1 and x₀=1, predict x₁₀≈0.34867844 at the same horizon, then Step ten times. Here η plays the Euler step h and each Step advances one iteration, not h units of the UI clock. This fixed quadratic-descent model has no neural score, stochastic diffusion, or image-quality measure.