The Williams GTDR Formula: A Gauss-TaylorDerivative Reconstruction Integrator for Low-NFEContinuous-Time Machine Learning

We introduce the Williams GTDR Formula (Gauss-Taylor Derivative Reconstruction Integrator), an explicit numerical integrator for ordinary differential equations (ODEs) achieving third-order local accuracy with only 3 function evaluations per step. A Romberg extrapolation patch raises the scheme to effective fourth-order accuracy, matching classical Runge-Kutta 4 (RK4) while reducing per-step function evaluations by 25%. In Neural ODEs and score-based diffusion models, each function evaluation is a full neural network forward pass — making evaluation count the dominant computational cost. Benchmarks confirm 25–33% NFE reduction versus RK4 at equivalent accuracy, with a clear implementation path in existing Neural ODE and diffusion model frameworks.