# Non-parametric models as solutions to Partial Differential Equations

In the realm of scientific computing, there are a plethora of techniques developed to solve Partial Differential Equations (PDEs). Many of the popular methods are variants of common techniques such as Finite Differences, Finite Volume, Finite Elements and the theory behind where they work or not is pretty extensive. Each method has pretty clear pros and cons which might influence its use over another for any given research problem.

I was just now contemplating the idea of non-parametrically modeling a solution for some system of PDEs, say some hyperbolic system of PDEs to keep things interesting. The idea could be to essentially having some cloud of spacetime points with some solution coefficients that represent the approximate solution to the PDEs. Then a separate cloud of spacetime points could be used as collocation points used to "train" the model. Whenever we wish to evaluate our model and compute gradients at some spacetime location, we could perhaps do some least square fit based on data that is "close" to that spacetime location. The problem of actually solving the PDE might then be viewed as an optimization of the solution coefficients at the model's spacetime points such that the residual of the PDEs and boundary conditions are minimized.

In the context of hyperbolic PDEs, do you see such an approach as described working out? Do you see problems arising due to characteristics of the system of hyperbolic PDEs that might influence how one should evaluate the non-parametric model in spacetime? Are there any papers you know of covering similar concepts?

This idea came about thinking about some paper I read a while back about someone using a similar collocation approach but with a neural network as the approximate PDE solution instead. It seems to me a non-parametric model would be easier to optimize and would still be capable of capturing potential nonlinearities, at the expense of having to keep track of solution data at a likely good number of spacetime locations.

I haven't thought too hard about this so perhaps there is something obvious I am missing but any thoughts, comments, or references about similar material would be appreciated!