RapidMiner is a framework implemented in Java that integrates with Weka and R. The only caveat to be mindful of here is integration into web services is a premium service and not in the community version of the system.
As for something you could throw together and run directly in a browser without plugins or premium services there is Brain which can do Neural Betworks, Naive Bayes, and Cross Validation. There are numerous other implementations for the web (e.g. javascript of K-means, Neural Networks, Genetic Algorithms, etc.) most of these though are for generally smaller sets of algorithms instead of larger collections you tend to find in R packages or in systems like Mahout and Weka. So you might end up having to have to piece a complete library yourself depending on what you're looking for.
Since your question is fairly generic its hard to provide more details, but that should be an adequate starting point. If you have a more detailed question feel free to ask, I hope this helps.
Towards Data Science isn't a reliable website, and the text you've quoted is, unfortunately, nonsense.
For any Optimization problem with respect to Machine Learning, there can be either a numerical approach or an analytical approach. The numerical problems are Deterministic, meaning that they have a closed form solution which doesn’t change. [...] These closed form solutions are solvable analytically. But these are not optimization problems.
What they meant to say, I hope, is that "analytical problems are Determinstic [...]", etc.
I won't explain the difference between analytic and numeric approaches here, because there are lots of good sources, but going by this paragraph I'm going to say the post you read isn't one of them.
EDIT: OK, I'll explain a bit
Part of the problem is that there are a lot of partially overlapping terms. Very roughly speaking, you have:
- Models where you can directly calculate the parameters: AKA closed-form solutions, analytical or analytic solutions, or sometimes algebraic solutions.
- Models where you have to use an iterative algorithm to fit the parameters. All such models are numerical, but
- They might be deterministic (no randomness), like batch gradient descent with fixed starting points, or stochastic (random), like stochastic gradient descent.
- They might always reach the best value (convex optimisation), or might have a risk of getting stuck at local optima (non-convex optimisation)
There are plenty of other ways to slice this up, but these should be plenty to get started!
Best Answer
Absolutely!
Here is information on the "shooting method". (link)
For much harder problems than the example given, the "root finding" takes more work. It is useful to stick some machine learning on top of the output in order to determine which initial conditions are appropriate for the solution of interest.
EDIT:
Neural Networks (NN) are used to (profoundly) improve computation time for combustion. The networks are trained on the thermo-chemical model and approximate the chemical reactions so that instead of solving (insane) complexity coupled fluid-dynamic and chemistry differential equations, the numeric solver has a reduced set of solves, and the NN with its very short run time, fills in the gaps "well enough". Here is a link. Here is another.