Introduction. So I'd like to thank the invitation and it's a pleasure to give the talk at the

seminar. So today the talk is really like an overview talk and most of it is not about

my own research, it's more about what I think are the important things about the neural

network and also partial differential equations. And in fact, the style of the talk is like,

they will contain quite a bit of materials. It's kind of like one slide per topic. So

I'll probably run quite fast, but to give you the overall high level ideas. So if you

have questions about the details, I'll be happy to answer at the end of the talk, or

maybe you can interrupt me in the middle of the talk.

So the title of the talk is about the neural network and partial differential equations.

And so we're all familiar. I mean, I think most of us are coming from a numerical analysis

of computational math background. So we all know PD very well. And neural network is something

which gained a lot of attention in the past 10 years. So it has made a lot of progress

in machine learning and AI. So from a mathematical point of view, I think the most important

thing about the neural network, it is that it's a very flexible representation for high

dimensional functions, maps, and also distribution. Because in modern computational science, more

and more we start to represent work with high dimensional objects. So that's why the neural

network becomes quite useful. So here I list a few common architectures for neural network.

The first is the fully connected neural network, where you see you have the input coming from

the left hand side and gradually being processed by the so-called neurons or hidden layers

in the middle, and then eventually produce output. So the second example is CNN, which

is a convolutional neural network. This is probably one of the most successful, mostly

commonly used neural network architectures. It has played a lot of roles in vision and

also image processing. And we also have a recurrent neural network, which is developed

by INA. So here you can really think about this like a Markov chain. So the data feeding

to the system, this block A at every step, and it at the same time also produce one piece

of output. But the real network is you should think about this whole thing where you have

incoming sequence and it produces an outgoing sequence. So finally, this so-called ResNet,

it's very simple. Typically it's combined with CNN, but the important thing about ResNet

is that it uses so-called skip connections, which essentially what it's doing is doing

this transformation gradually. You can think about like an ODE. Now, these are the famous

examples of a neural network, at least some of the famous examples. And you can already

see that it's strong connection with mathematics, like convolutional operators, Markov chains,

ODE, and dynamic systems. So really mathematics and PDEs, sorry, neural network PDEs are interwined

together. So in this talk, what I'm trying to share with you is that what I think that

how neural network can help solve partial differential equations. And in the other direction

is how PDE can help explain also develop neural network architectures. So as I said, this

talk will be like one slide per topic. So in the first part, neural network for PDEs,

I will try to touch on a few topics. So again, as I said, the main reason that neural network

will be used for PDE is because neural network can represent high dimensional functions and

high dimensional maps. Now, what can high dimensional maps and functions appear? The

first place is high dimensional PDEs. There's a PDE that we work with in scientific computation

as a really inherent high dimension. So these such high dimensional map can also appear

in low dimensional PDEs, especially in the case of the inverse problems and parametric

PDEs. So finally, I will also touch on the case of reducing from high dimensional to

low dimensional. This is a model reduction. Okay, let's touch on the first part of the

first part, which is a neural network for high dimensional PDEs. So here the neural

network is typically just used to represent the solutions of the PDEs. And I think the

best way is to just give you three important examples for high dimensional PDEs. The first

example is the so-called many-body quantum mechanics. So we all know that on a micro

scale our world is governed by quantum mechanics. And because we have many particles, typically