
Mixing Differential Equations and Neural Networks for Physics-Informed Learning - ChrisRackauckas
https://mitmath.github.io/18337/lecture15/diffeq_machine_learning
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eigenspace
Very interesting stuff, I'll have to spend some time digesting this as it's
relevant to some research interests of mine.

This might be a dumb question, but one thing I'm having trouble understanding
is your use of the phrase 'data driven' here.

In the section "Solving Ordinary Differential Equations" you describe a method
that seems to be able to (inefficiently) solve differential equations without
having to take in some 'data', but instead only needing to know `f(u, t)`.

In the later section, where you introduce the Physics Informed Neural Network,
it seems (but I could be misunderstanding) that you are assuming I have
measurements of the actual solution and we're training the network with that
data. Is this correct?

I was initially imagining a method where say I know the solution to the linear
part of the differential equation and then train the network just using the
non-linear part without having to have any data.

~~~
ChrisRackauckas
>In the later section, where you introduce the Physics Informed Neural
Network, it seems (but I could be misunderstanding) that you are assuming I
have measurements of the actual solution and we're training the network with
that data. Is this correct?

Yes, this is the PINN method M.Raissi, P.Perdikaris, and G.E.Karniadakis: use
a physical underpinning as a function to help the learning but relax towards
the data.

>I was initially imagining a method where say I know the solution to the
linear part of the differential equation and then train the network just using
the non-linear part without having to have any data.

These are the mixed neural differential equations that I have been showcasing
with DiffEqFlux.jl

[https://github.com/JuliaDiffEq/DiffEqFlux.jl#mixed-neural-
de...](https://github.com/JuliaDiffEq/DiffEqFlux.jl#mixed-neural-des)

So it's just different methods by different people / different groups (for
different purposes).

