Original title: GRADSOLVE: fast exact gradients for ODE ensembles on GPUs
Alessio Spurio Mancini · September 2026
Scientists use ordinary differential equations, or ODEs, to describe changing systems like orbits or chemical reactions. Often, researchers need gradients—which show how tiny tweaks to settings change the outcome—across many simulations running at once on graphics processing units (GPUs). Existing tools force a bad choice: fast solvers cannot find these gradients quickly, while solvers built for gradients run slowly. To solve this, researchers built GRADSOLVE. It runs a simulation first and saves only the time steps the solver accepted. Next, it replays those exact steps in reverse as a simple, fixed-length chain to calculate the exact gradient, dodging the heavy math of adaptive loops. In tests, GRADSOLVE ran forward simulations 2.8x faster than the tool DiffEqGPU.jl. When calculating gradients from a saved run, it performed 5.6-14.1x faster than the library Diffrax across three GPU generations.
Imagine planning a long road trip where you need to choose between several highways, and traffic conditions on each route change unpredictably as you drive. Testing one route doesn't tell you what will happen on another, so planning is slow and frustrating. In this paper, simulating an ODE is like driving through time, and computing gradients is like knowing how small changes in your starting conditions affect arrival time. GRADSOLVE records the exact path taken and replays it with tiny adjustments to quickly figure out the effect, instead of driving the whole route again each time.
Scientists and engineers use ordinary differential equations to model systems like climate or circuits, and they need derivatives to optimize parameters. Faster gradient computation means they can test and improve models more quickly, saving time and resources in research and development.
The advantage in speed narrows for large ensembles and may reduce to parity for stiff systems at tight accuracy settings. It specifically targets low-dimensional ODE ensembles on NVIDIA GPUs and does not address higher-dimensional problems or other hardware.
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GRADSOLVE: fast exact gradients for ODE ensembles on GPUs — Alessio Spurio Mancini
Abstract licence: CC0
Explanation generated by Synthesis from the abstract — it is not the original paper. This is a preprint, so it hasn't necessarily been peer reviewed.