Thank you. Thank you for the introduction and thank you for the invitation of the

committee. And good afternoon everyone. And today I will talk about physics-informed neural networks

for Nesmos PD-constrained optimization problems. And first I will introduce some backgrounds and

motivation. And then I will take two typical examples about Nesmos PD-constrained optimization

to illustrate the main ideas of the design of pins to solve this kind of problems. And finally,

there are some conclusions and perspectives. And first, it is well known that PDs model various

physical phenomena. And for some applications, we need not only to model certain physical process,

but also to control the system or optimize the considered process to meet certain goals. And for

this purpose, a given objective functional has to be minimized subject to a PD or a system of coupled

PD systems, usually with other additional constraints. And this kind of additional

constraints are used to guarantee some realistic requirements. And we get PD-constrained optimization

problems. And this is an example of control the head distribution of a metal bar. And mathematically,

a PD-constrained optimization problem can be written in this form. And here, U and Y are

assumed to be binary spaces and UAD and YAD are closed convex sites. And these two sites are used

to impose some control of state constraints. And here, G is objective functional to be minimized.

And E represents a PD or PD systems. Here, Y describes the state of the considered system

modeled by the PD. And the variable U is a parameter from the source term or some coefficient that

should be adapted in an optimal way and to minimize the objective functional. And the

constraints, U belongs to UAD and Y belongs to YAD, describe some physical restrictions on

the realistic requirements. And this model is somehow very abstract. And this kind of PD-constrained

optimization problem can cover very many important applications in optimal control, optimal design,

and inverse problems. And here, we focus on some non-smooth cases. And we talk about two types of

non-smoothness. The first type of non-smoothness is we consider a non-smooth objective functional.

That means the objective functional can be written in this form. And here, the functional G consists

of a data fidelity term and a possible smooth regularization. For example, the L2 norm. And here,

the non-smooth functional R is used to capture some prior information on the variable U. For example,

the boundedness, sparsity, or the discontinuity. And to impose this kind of prior information,

we use different types of non-smooth regularizations. And as we mentioned, to expose our ideas clearly,

here we focus on a concrete example. It's a parabolic sparse optimal control problem. And that means we

would like to promote the sparsity of the control of this problem. And for this purpose, we use the

L1 norm of the control variable as the regularization term. And here, the sparse or the sparsity means

that the support of the control variable is only a subset of the domain. That means in some parts of

the domain, the control variable is zero. And here, the state equation is a parabolic equation. And here,

UED is used to impose some pointwise boundedness to the control variable. And the second type of

non-smoothness is we consider some non-smooth PDE. And here, we focus on interface problems. Interface

problems are piecewise defined PDEs in different regions coupled with together interface conditions.

And the solutions are non-smooth or even discontinuous. And here is an example of the geometry of an

interface problem. Here, the gamma is the interface, and it divides the whole domain into

omega minus and omega plus. And then we define the PDEs piecewise in omega minus and omega plus.

And again, we focus on a concrete example. And here, we consider an elliptic interface optimal

problem. Here, we consider a smooth object functional. And the state equation is an elliptic interface

problem. And here, beta is a piecewise constant. And the jump discontinuity across the interface is

defined here by the limit from different sides of the interface. And this term means the interface

gradient condition. That is the condition, the gradient of the solution across the interface.

And next, we will focus on these two typical examples to expose our ideas.

And before we introduce our algorithms, we first present a brief literature review.

Okay, there are many works about the theoretical analysis of the applications of the numerical

methods for PD constrained optimization problems. And for the numerical methods,

they mainly consist of optimization algorithms plus some numerical discretization.

And the optimization algorithms include semi-smooth Newton methods, primordial active side methods,