Readout Times Are Not Solver Nodes: A Two-Mesh API for Generative ODE Surrogates

Patrick Reichherzer

Keywords: Generative ODE samplers, Score-to-flow, Flow matching, Scientific surrogates, Dense output, Readout-mesh invariance

TL;DR: Generative ODE samplers should expose solve_at and save_at separately — coupling them silently turns requested readouts into solver nodes and costs accuracy at fixed budget.

Abstract: As generative diffusion and flow models move into scientific surrogate roles---for physical processes, sensor arrays, video, and other time-indexed measurements---their sampler interfaces face a question classical generation largely avoids: where should outputs be reported, and where should the solver step? Scientific surrogate samplers should satisfy \emph{readout-mesh invariance}: changing where a user asks to observe a sample should not silently change the numerical mesh used to generate it. Many fixed-schedule diffusion samplers conflate these, exposing one schedule for both. A scoped static audit of eleven representative sampler interfaces finds no strict off-mesh dense-readout interface. We show this conflation has measurable numerical cost. In score-to-flow coordinates, a physical variance law $V(\tau) \sim \tau^a$ induces a boundary cusp for $a < 2$, so uniform external readouts produce singular internal solver meshes with worst-step scaling $h_R = \Theta(K^{-a/2})$. The repair is standard in scientific ODE software: a two-mesh API that integrates on an internal mesh and evaluates readouts by dense output. On a heat-equation virtual sensor diagnostic, decoupling reduces sensor RMSE from $8.21 \times 10^{-4}$ to $1.84 \times 10^{-6}$ at the same 128 vector-field evaluations, a 446-fold reduction. Learned MNIST score-to-flow fields show 32- to 51-fold lower fixed-field readout error.

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Submission Number: 263