I am working on a research project involving a system of differential equations with one ordinary differential equation (ODE) and two partial differential equations (PDEs).
I would like to discuss methods and approaches for solving this system efficiently using Numerical method and Machine Learning.
Can you recommend Python/Matlab code for using numerical techniques and ML
Any guidance or references would be greatly appreciated.
That's a good question. I rarely do that so not answering this myself but you can find some nice tutorials for that from experts in this field.
Tutorials:
https://www.youtube.com/watch?v=p9p2YAjIgUU
https://www.youtube.com/watch?v=LQ33-GeD-4Y
https://www.youtube.com/watch?v=U38hnksTVdE
Sachin Chandrasekara Coupled systems of ODEs and PDEs can be challenging to solve, but here are some recommendations:
For numerical methods:
- Discretize the PDEs using finite difference, finite element or finite volume methods to convert into a system of ODEs.
- Integrate the full system of ODEs simultaneously using multistep or multistage numerical integrators like Backward Differentiation Formula or Runge-Kutta methods.
- Stability and convergence depends on step size, boundary conditions, and consistency of discretizations.
For machine learning:
- Discretize the spatial dimensions and reconstitute as a large system of ODEs.
- Use neural ODE techniques with tradeoffs on network size vs accuracy.
- Alternatively use random forests or XGBoost on discretized spatiotemporal snapshots.
Python has excellent ML packages like PyTorch and SciKit-Learn. MATLAB also has toolboxes like Deep Learning, PDEPE for such systems.
Let me know if you need help implementing a specific numerical scheme or ML technique.
Only using ODEs (Converted PDEs) can we use machine learning. Do you have machine learning algorithm for this?
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