GRADSOLVE: fast exact gradients for ODE ensembles on GPUs

Imagine you're working on a simulation that relies on ordinary differential equations (ODEs) to model complex systems, like weather patterns or mechanical systems. You need to adjust parameters and see how those changes affect the outcomes, which requires calculating derivatives of the ODE solutions. Currently, the tools available for this task force you to choose between speed and the ability to differentiate: the fastest solvers can't provide gradients efficiently, while those that can are much slower. This trade-off can be frustrating when you're trying to optimize your models or run simulations at scale.

The approach taken here is to create a library called GRADSOLVE, which allows you to solve ODE ensembles and compute their gradients efficiently on NVIDIA GPUs. It does this by recording the steps taken by an adaptive solver and then differentiating a fixed-step replay of those steps, which is much faster than traditional methods. GRADSOLVE not only runs significantly faster than existing solvers like DiffEqGPU.jl but also computes gradients much more quickly once a record is established. This means that for anyone building simulations that require frequent adjustments and optimizations, GRADSOLVE could save a lot of time and computational resources compared to previous methods.

Introduces a new method for efficiently differentiating ODE solutions on GPUs.

Demonstrates significant speed improvements over existing methods with solid benchmarks.