Implementing Advanced Differential Equation Solvers and Neural ODEs with Diffrax and JAX

This tutorial outlines the process of utilizing the Diffrax library for solving differential equations and creating neural ordinary differential equation (ODE) models. We start by establishing a suitable computational environment, ensuring that essential libraries like JAX, Diffrax, Equinox, and Optax are properly installed. The guide demonstrates solving ordinary differential equations through adaptive solvers and showcases dense interpolation techniques to retrieve solutions at specific time points. We delve into advanced features of Diffrax, such as addressing classical dynamical systems, utilizing PyTree-based states, and executing batched simulations with JAX's vectorization capabilities. Additionally, we simulate stochastic differential equations and generate data from a dynamical system, which will be instrumental in training a neural ODE model. Throughout the tutorial, we provide practical examples, including logistic growth, the Lotka-Volterra model, and a spring-mass-damper system, illustrating the versatility of Diffrax in handling various differential equations.