Implicit integration methods for solving nonlinear systems

[chaoming0625]

Please:

• Check for duplicate requests.
• Describe your goal, and if possible provide a code snippet with a motivating example.

Implicit solvers are import to the (morphological) structure of neurons. It is essential to support Implicit integration methods.

[AnushkaShreyam]

Thank you for opening this issue @chaoming0625 ! Implicit solvers are indeed critical for accurately modeling stiff systems like neuronal structures. Here's my proposed plan to address this

• Implement core implicit methods such as Backward Euler and Crank-Nicholson, leveraging iterative solvers like Newton-Raphson or fixed-point iteration.
• Utilize JAX for efficient Jacobian computation and explore other JIT backends like Numba or Taichi to enhance solver performance.
• Design a user-friendly API to allow flexibility in choosing methods, setting tolerances, and configuring solver parameters.
• Validate the implementation with neural dynamics models and benchmark it for accuracy and performance compared to existing solvers.

Does this approach align with your expectations? Let me know if there are additional requirements or specific scenarios you'd like to prioritize. I'm happy to collaborate and contribute to this!