dissertation_work/Scratch/Neural network diffeq.tex

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\title{Reformulation without f}
\author{Shakil Rafi}
\begin{document}
\maketitle  

\begin{lemma}
	Let $\Theta = \bigcup_{n\in\N}$, $d,m,\mathfrak{d} \in \N$, $T \in (0,\infty)$, $\act \in C \lp \R,\R\rp$, $\mathfrak{I}, \mathbf{F}, \mathbf{G} \in \neu$ satisfy $\lay\lp \mathfrak{I} \rp = \lp 1,\mathfrak{d},1\rp$, $\real_{\act} \lp \mathfrak{I} \rp = \mathbb{I}_1$, $\real_{\act} \lp \mathbf{F} \rp \in C\lp \R, \R \rp$, and $\real_{\act} \lp \mathbf{G} \rp \in C \lp \R^d, \R \rp$, for every $\theta \in \Theta$ let $\mathcal{U}^\theta: \lb 0,T \rb \rightarrow \lb 0,T \rb$ and $\mathcal{W}^\theta : \lb 0,T \rb \rightarrow \R^d$, be functions, for every $\theta \in \Theta$, $n\in \N_0$ let $U^\theta_n: [0,T] \times \R^d \rightarrow \R$ satisfy for all $t\in [0,T]$, $x \in \R^d$, that:
	\begin{align*}
		&U^\theta_n(t,x) = \frac{\mymathbb{1}_\N (n)}{M^n} \lb \sum^{m^n}_{k=1} \lp \real_{\act} \lp \mathbf{G}\rp \rp \lp x+\mathcal{W}^{\lp \theta,0,-k \rp}_{T-t} \rp \rb \\
		&+\sum^{n-1}_{i=1} \frac{T-t}{M^{n-i}} \lb \sum^{M^{n-i}}_{k=1} \lp \lp \real_{\act} \lp \mathbf{F}\rp \circ U_i^{(\theta,i,k)} \rp -\mymathbb{1}_\N \lp i \rp \lp \real_{\act} \lp \mathbf{F}\rp \circ U^{\lp \theta,-i,k \rp} \rp \rp \lp \mathcal{U}_t^{\lp \theta,i,k \rp},x+\mathcal{W}^{\lp\theta,i,k\rp}_{\mathcal{U}_t^{\lp \theta,i,k\rp}-t} \rp \rb 
	\end{align*}
	and let $\mathbf{U}^\theta_{n,t} \in \neu$, $t \in [0,T]$, $n\in \Z$, $\theta \in \Theta$, satisfy for all $\theta \in \Theta$, $n\in \N$, $t\in [0,T]$ that $\mathbf{U}^\theta_{0,t} = \lp \lp 0 \:0\:\cdots \:0 \rp,0 \rp \in \R^{1 \times d} \times \R$ and:
	\begin{align*}
		\mathbf{U}^\theta_{n,t} &= \lb \oplus^{M^n}_{k=1} \lp \frac{1}{M^n} \circledast \lp \mathbf{G}\bullet \aff_{\mathbb{I}_d, \mathcal{W}^{\lp \theta,0,-k \rp }_{T-t}} \rp \rp \rb \\ &\boxplus_{\mathfrak{I}} \lb \boxplus_{i=0,\mathfrak{I}} \lb \lp \frac{T-t}{M^{n-i}}\rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathfrak{I}}\lp \lp \mathbf{F}\bullet \mathbf{U}^{\lp \theta,i,k \rp}_{i,\mathcal{U}_t^{\lp \theta,i,k \rp}}\rp \bullet \aff_{\mathbb{I}_d,\mathcal{W}^{\lp \theta,i,k \rp}_{\mathcal{U}^{\lp \theta,i,k \rp}-t}} \rp \rp \rb \rb \\
		&\boxplus_{\mathfrak{I}}\lb \boxplus^{n-1}_{i=0,\mathfrak{I}}\lb \lp \frac{(t-T)\mymathbb{1}_\N \lp i \rp }{M^{n-i}} \rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathfrak{I}}\lp \lp \mathbf{F}\bullet \mathbf{U}^{\lp \theta,-i,k \rp }_{\max\{i-1,0\},\mathcal{U}_t^{\lp \theta,i,k \rp}} \rp \bullet \aff_{\mathbb{I}_d,\mathcal{W}^{\lp \theta,i,k \rp }_{\mathcal{U}_t^{\lp \theta,i,k \rp}-t}}\rp \rp \rb \rb 
	\end{align*}
\end{lemma}
\begin{theorem}\label{thm1}
	Let $p,q,r,L,C,\alpha_0,\alpha_1,\beta_0,\beta_1, T \in [0,\infty)$, $\mathfrak{q} \in [2,\infty)$, $\act \in C(\R,\R)$, $\mathfrak{I} \in \neu$. $\lp \mathbf{F}_{d,\ve} \rp _{\lp d,\ve \rp \in \N_0 \times (0,1]} \subsetneq \neu$. For every $d \in \N_0$ let $f_d \in C \lp \R^{\max\{d,1\}},\R \rp$, for every $d \in \N$ let $\nu_d: \mathcal{B} \lp \R^d \rp \rightarrow [0,1]$ be a probability measure, and assume for all $d \in \N_0$, $v,w \in \R$, $x \in \R^{\max\{d,1\}}$, $\ve \in (0,1]$ that $\lp \int_{\R^d}\left\|y\right\|^{pq\mathfrak{q}} \nu_d \lp dy\rp \rp ^{\frac{1}{pq\mathfrak{q}}}\les Cd^r$, $\hid\lp \mathfrak{I} \rp=1$, $\real_{\act} \lp \mathfrak{I} \rp = \id_\R $, $\real_{\act} \lp \mathbf{F}_{d,\ve} \rp \in C \lp \R^{\max\{d,1\}},\R \rp $, $\max \{ \left|f_0(v)-f_0(w)\right|, \left|\lp \real_{\act} \lp \mathbf{F}_{0,\ve} \rp \rp \lp x \rp - \lp \real_{\act} \lp \mathbf{F}_{0,\ve} \rp \rp \lp x \rp \right|\} \les L \left|v-w\right|$, $\ve^{\alpha_{\min\{d,1\}}}\dep \lp \mathbf{F}\rp_{d,\ve}+\ve^{\beta_{\min\{d,1\}}}\left\| \lay \lp \mathbf{F}_{d,\ve} \rp \right\|_{\max} \les C \lp \max \{x,1\}^p \rp$, and:
	\begin{align}
		\ve \left| \lp \real_{\act} \lp \mathbf{F}_{d,\ve}\rp \rp \lp x \rp \right| + \left|f_d\lp x\rp - \lp \real_{\act}\lp \mathbf{F}_{d,\ve} \rp \rp \lp x \rp \right| \les \ve C \lp \max \{x,1\}\rp ^p \lp 1+ \left\|x \right\| \rp ^{pq}  
	\end{align}
	It is then the case that for every $d \in \N$, there exists a $u_d \in C \lp \lb 0,T \rb \in \R^d,\R \rp$ with the following properties:
	\begin{enumerate}[label = (\roman*)]
		\item $u_d$ is polynomially growing.
		\item $u_d$ is a viscosity solution.
		\item $u_d$ is a solution to:
		\begin{align}
			\lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp +\frac{1}{2}\Trace\lp \sigma_d \lp x \rp \lb \sigma_d \lp x \rp \rb ^*\lp \Hess_x u_d \rp \lp t,x\rp \rp + \la u_d \lp x \rp, \lp \nabla_xu_d \rp \lp t,x \rp \ra + \alpha_d(x)u_d(t,x)= 0 \nonumber
		\end{align}
		with $u_d \lp T,x \rp =g_d \lp x \rp$ for $\lp t,x \rp \in \lp 0,T \rp \times \R^d$, and 
		\item there exist $\lp \mathbf{U}_{d,t,\ve} \rp _{ \lp d,t,\ve \rp \in \N \times \lb 0,T \rb \times \lp 0,1\rb}$ and $\eta \in \lp \eta_\delta \rp _{\delta \in \lp 0,\infty \rp}: \R \rightarrow \R$ such that for all $d \in \N$, $t \in \lb 0,T \rb$, $\ve \in (0,1]$, $\delta \in \lp 0,\infty \rp $ it holds that $\real_{\act} \lp \mathbf{U}_{d, t,\ve} \rp \in C \lp \R^d, \R \rp$, $\param \lp \mathbf{U}_{d,t,\ve} \rp \les \eta_\delta d^{p\lp 7+4q+ \lp 2+q \rp \delta \rp } \ve ^{-\lp 4+2\delta + \max\{\alpha_0,\alpha_1\}+2\max\{\beta_0,\beta_1\} \rp}$ and:
		\begin{align}
			\lp \int_{\R^d} \left|u_d\lp t,x \rp -\lp \real_{\act} \lp \mathbf{U}_{d,t,\ve} \rp \rp \lp x \rp \right|^\mathfrak{q} \nu_d \lp dx\rp \rp ^{\frac{1}{\mathfrak{q}}} \les \ve 
		\end{align}
	\end{enumerate} 
	
\end{theorem}

\begin{proof}
	The proof of Theorem \ref{thm1} is thus complete
\end{proof}






























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