SciMLTutorials.jl VS 18337

This was previously the https://github.com/mitmath/18337 course website, but now in a new iteration of the course it is being reset. To avoid issues like this in the future, we have moved the "book" out to its own repository, https://github.com/SciML/SciMLBook, where it can continue to grow and be hosted separately from the structure of a course. This means it can be something other courses can depend on as well. I am looking for web developers who can help build a nicer webpage for this book, and also for the SciMLBenchmarks.

I would say Fortran is a pretty great language for teaching beginners in numerical analysis courses. The only issue I have with it is that, similar to using C+MPI (which is what I first learned with, well after a bit of Java), the students don't tend to learn how to go "higher level". You teach them how to write a three loop matrix-matrix multiplication, but the next thing you should teach is how to use higher level BLAS tools and why that will outperform the 3-loop form. But Fortran then becomes very cumbersome (`dgemm` etc.) so students continue to write simple loops and simple algorithms where they shouldn't. A first numerical analysis course should teach simple algorithms AND why the simple algorithms are not good, but a lot of instructors and tools fail to emphasize the second part of that statement.

On the other hand, the performance + high level nature of Julia makes it a rather excellent tool for this. In MIT graduate course 18.337 Parallel Computing and Scientific Machine Learning (https://github.com/mitmath/18337) we do precisely that, starting with direct optimization of loops, then moving to linear algebra, ODE solving, and implementing automatic differentiation. I don't think anyone would want to give a homework assignment to implement AD in Fortran, but in Julia you can do that as something shortly after looking at loop performance and SIMD, and that's really something special. Steven Johnson's 18.335 graduate course in Numerical Analysis (https://github.com/mitmath/18335) showcases some similar niceties. I really like this demonstration where it starts from scratch with the 3 loops and shows how SIMD and cache-oblivious algorithms build towards BLAS performance, and why most users should ultimately not be writing such loops (https://nbviewer.org/github/mitmath/18335/blob/master/notes/...) and should instead use the built-in `mul!` in most scenarios. There's very few languages where such "start to finish" demonstrations can really be showcased in a nice clear fashion.

And why not take a course while you're at it.

Here's a good one. It's in Julia but it should do the trick. The main instructor is the most prolific Julia dev in the world.

NeuralPDE.jl fully automates the approach (and extensions of it, which are required to make it solve practical problems) from symbolic descriptions of PDEs, so that might be a good starting point to both learn the practical applications and get something running in a few minutes. As part of MIT 18.337 Parallel Computing and Scientific Machine Learning I gave an early lecture on physics-informed neural networks (with a two part video) describing the approach, how it works and what its challenges are. You might find those resources enlightening.

It's a very thriving field. If you are interested in methods research and want to learn some of the techniques behind it, I would recommend taking a dive into my lecture notes as I taught a graduate course at MIT, 18.337 Parallel Computing and Scientific Machine Learning, specifically designed to get new students onboarded into this research program.