\chapter{A modified Multi-Level Picard and Associated Neural Network}
We now look at neural networks in the context of multi-level Picard iterations.
\begin{lemma}
	Let $\alpha, \beta, M \in \lb 0,\infty \rp$, $U_n \in \lb 0,\infty \rp$, for $n\in \N_0$ satisfy for all $n\in \N$ that:
	\begin{align}\label{7.0.1}
		U_n \les \alpha M^n + \sum^{n-1}_{i=0}M^{n-i} \lp \max \left\{ \beta, U_i\right\} + \mathbbm{1}_{\N} \lp i \rp \max \left\{ \beta, U_{\max \left\{i-1,0 \right\}} \right\} \rp 
	\end{align}
	It is then also the case that for all $n\in \N$ that $U_n \les \lp 2M+1 \rp^n \max \left\{\alpha,\beta \right\}$.
\end{lemma}
\begin{proof}
	Let:
	\begin{align}\label{7.0.2}
		S_n = M^n + \sum^{n-1}_{i=0} M^{n-i} \lb \lp 2M+1 \rp^i + \mathbbm{1}_\N \lp i \rp \lp 2M+1 \rp ^{\max \left\{i-1,0\right\}} \rb 
	\end{align}
	We prove this by induction. The base case of $n=0$ already implies that $U_0 \les \alpha \les \max \left\{\alpha, \beta \right\}$. Next assume that $U_n \les \lp 2M+1 \rp^n \max \left\{ \alpha, \beta \right\}$ holds for all integers upto and including $n$, it is then the case that:
	\begin{align}
		U_{n+1} &\les \alpha M^{n+1} + \sum^n_{i=0} M^{n+1-i}\lp \max \left\{ \beta, U_i \right\} + \mathbbm{1}_\N \lp i \rp \max \left\{ \beta, U_{\max \left\{i-1,0 \right\}} \right\} \rp \nonumber \\
		&\les \alpha M^{n+1} + \sum^n_{i=0} M^{n+1-i} \lb \max \left\{ \beta, \lp 2M+1 \rp^k\max \left\{ \alpha,\beta \right\}\right\} \right. \nonumber\\&\left. + \mathbbm{1}_\N \lp i \rp \max\left\{ \beta, \lp 2M+1 \rp ^{\max \left\{ k-1,0 \right\}} \max \left\{ \alpha, \beta \right\}\right\} \rb \nonumber \\
		&\les \alpha M^{n+1} + \max \left\{ \alpha,\beta\right\} \sum^n_{i=0} M^{n+1-i} \lb \lp 2M+1 \rp^i + \mathbbm{1}_\N \lp i \rp \lp 2M+1\rp ^{\max\left\{i-1,0 \right\}} \rb \nonumber \\
		&\les \max \left\{\alpha,\beta \right\} S_{n+1}
 	\end{align}
 	Then (\ref{7.0.2}) and the assumption that $M\in \lb 0, \infty \rp$ tells us that:
 	\begin{align}
 		S_{n+1} &= M^{n+1} + \sum^n_{i=0} M^{n+1-i} \lb \lp 2M+1 \rp^i + \mathbbm{1}_\N \lp i \rp \lp 2M+1 \rp^{\max\left\{i-1,0 \right\}} \rb \nonumber \\
 		&= M^{n+1} \sum^n_{i=0} M^{n+1-i} \lp 2M+1\rp^k + \sum^n_{i=1} M^{n+1-i} \lp 2M+1 \rp ^{i-1} \nonumber \\
 		&=M^{n+1} + M \lb \frac{\lp 2M+1 \rp^{n+1} - M^{n+1}}{M+1} \rb + M \lb \frac{\lp 2M+1 \rp^n-M^n}{M+1} \rb \nonumber \\
 		&= M^{n+1} + \frac{M\lp 2M+1\rp^{n+1}}{M+1} + \frac{\lp 2M+1 \rp^n}{M+1} - M \lb \frac{M^{n+1}+M^n}{M+1} \rb \nonumber \\
 		&\les M^{n+1} + \frac{M \lp 2M+1 \rp ^{n+1}}{M+1} + \frac{\lp 2M+1\rp ^{n+1}}{M+1} - M^{n+1} \lb \frac{\cancel{M+1}}{\cancel{M+1}} \rb \nonumber \\
 		&= \lp 2M+1\rp ^{n+1}
 	\end{align}
 	This completes the induction step proving (\ref{7.0.1}).
\end{proof}
\begin{lemma}
	Let $\Theta = \lp \bigcup^{n\in \N} \Z^n \rp$, $d,M \in \N$, $T\in \lp 0,\infty \rp$, $f \in C \lp \R, \R \rp$, $g,\in C \lp \R^d, \R \rp$, $\mathsf{F}, \mathsf{G} \in \neu$ satisfy that $\real_{\rect} \lp \mathsf{F} \rp = f$ and $\real_{\rect} \lp \mathsf{G} \rp = g$, let $\mathfrak{u}^\theta \in \lb 0,1 \rb$, $\theta \in \Theta$, and $\mathcal{U}^\theta: \lb 0,T \rb \rightarrow \lb 0,T \rb$, $\theta \in \Theta$, satisfy for all $t \in \lb 0,T \rb$, $theta \in \Theta$ that $\mathcal{U}^\theta_t = t+(T-t)\mathfrak{u}^\theta$, let $\mathcal{W}^\theta: \lb 0,T \rb \rightarrow \R^d$, $\theta \in \Theta$, for every $\theta \in \Theta$, $t\in \lb 0,T\rb$, $s \in \lb t,T\rb$, let $\mathcal{Y}^\theta_{t,s} \in \R$ satisfy $\mathcal{Y}^\theta_{t,s} = \mathcal{W}^\theta_s - \mathcal{W}^\theta_t$ and let $\mathcal{U}^\theta_n: \lb 0,T\rb \times \R^d \rightarrow \R$, $n\in \N_0$, $\theta \in \Theta$, satisfy for all $\theta \in \Theta$, $n\in \N_0$, $t\in \lb 0,T\rb$, $x\in \R^d$ that:
	\begin{align}
		&U^\theta_n \lp t,x\rp = \frac{\mathbbm{1}_\N\lp n \rp}{M^n} \lb \sum^{M^n}_{k=1} g \lp x + \mathcal{Y}^{(\theta,0,-k)}_{t,T}\rp\rb \nonumber\\
		&+ \sum^{n-1}_{i=0} \frac{T-t}{M^{n-i}} \lb \sum^{M^{n-i}}_{k=1} \lp \lp f \circ U^{(\theta,i,k)}_i\rp - \mathbbm{1}_\N \lp i \rp \lp f \circ U^{(\theta,-i,k)}_{\max \{ i-1,0\}} \rp \rp \lp \mathcal{U}^{(\theta,i,k)}_t,x+ \mathcal{Y}^{(\theta,i,k)}_{t,\mathcal{U}_t^{(\theta,i,k)}}\rp\rb 
	\end{align}
	it is then the case that:
	\begin{enumerate}[label = (\roman*)]
		\item there exists unique $\mathsf{U}^\theta_{n,t} \in \neu$, $t \in \lb 0,T \rb$, $n\in \N_0$, $\theta \in \Theta$, which satisfy for all $\theta_1,\theta_2 \in \Theta$, $n\in \N_0$, $t_1, t_2 \in \lb 0,T \rb$ that $\lay \lp \mathsf{U}^{\theta_1}_{n,t_1} \rp = \mathcal{L} \lp \mathsf{U}^{\theta_2}_{n,t_2} \rp$.
		\item for all $\theta \in \Theta$, $t \in \lb 0,T\rb$ that $\mathsf{U}^\theta_{0,t} = \lb \lb 0 \quad 0 \quad \cdots \quad 0\rb,\lb 0 \rb \rp \in \R^{1 \times d}\times \R^1$
		\item for all $\theta \in \Theta$, $n\in \N$, $t \in \lb 0,T \rb$ that:
%		\begin{align}
%			\mathsf{U}^\theta_{n,t} = \lb \bigoplus^{M^n}_{k=1} \lp \frac{1}{M^n} \circledast \lp \mathsf{G} \bullet \aff_{\mathbb{I}_d, \mathcal{Y}^{(\theta,0,-k)}_{t,T}} \rp \rp \rb \nonumber\\
%			\boxplus_{\mathbb{I}} \lb \boxplus^{n-1}_{i=0,\mathbb{I}} \lb \lp \frac{T-t}{M^{n-i}} \rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathbb{I}} \lp \lp \mathsf{F} \bullet \mathsf{U}^{(\theta,i,k}_{i, \mathcal{U}_t^{(\theta,i,k)} \rp \rp \rp   
%		\end{align}
		\begin{align}
			&\mathsf{U}^\theta_{n,t} \\&= \lb \bigoplus^{M^n}_{k=1} \lp \frac{1}{M^n} \circledast \lp \mathsf{G}\bullet \aff_{\mathbb{I}_d, \mathcal{Y}^{(\theta,0,-k}_{t,T}} \rp \rp \rb \nonumber \\
			&\boxplus_{\mathbb{I}} \lb \boxplus^{n-1}_{i=0,\mathbb{I}} \lb \lp \frac{T-t}{M^{n-i}}\rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathbb{I}}\lp \lp \mathsf{F} \bullet \mathsf{U}^{(\theta,i,k)}_{i,\mathcal{U}_t^{(\theta,i,k)}} \rp \bullet \aff_{\mathbb{I}_d}, \mathcal{Y}^{(\theta,i,k)}_{t,\mathcal{U}_t^{(\theta,i,k)}} \rp\rp \rb\rb \nonumber\\
			&\boxplus_{\mathbb{I}} \lb \boxplus^{n-1}_{i=0,\mathbb{I}} \lb \lp \frac{(t-T)\mathbbm{1}_\N}{M^{n-i}}\rp \circledast\lp \boxplus^{M^{n-i}}_{k=1,\mathbb{I}} \lp \lp \mathsf{F} \bullet \mathsf{U}^{(\theta,-i,k)}_{\max \{i-1,0\}, \mathcal{U}_t^{(\theta,i,k)}}\rp \bullet \aff_{\mathbb{I}_d,\mathcal{Y}^{(\theta,i,k)}_{t,\mathcal{U}_t^{(\theta,i,k)}}} \rp \rp\rb \rb 
		\end{align}
		\item that for all $\theta \in \Theta$, $n\in \N_0$, $t\in \lb 0,T \rb$, that $\dep \lp \mathsf{U}^\theta_{n,t} \rp = n\cdot \hid \lp \mathsf{F} \rp + \max \left\{1,\mathbbm{1}_\N \lp n \rp \dep \lp \mathsf{G} \rp \right\}$ 
		\item that for all $\theta \in \Theta$, $n\in \N_0$, $t \in \lb 0,T \rb$, that $\left\| \lay \lp \mathsf{U^\theta_{n,t}} \rp\right\|_{\max} \les \lp 2M+1\rp ^n \max \left\{ 2, \left\| \lay \lp \mathsf{F} \rp\right\|_{\max}, \left\| \lay \lp \mathsf{G} \rp \right\|_F \right\}$ 
		\item it holds for all $\theta \in \Theta$, $n\in \N_0$, $t \in \lb 0,T \rb $, $x \in \R^d$ that $U^\theta_n \lp t,x \rp = \lp \real_{\rect} \lp \mathsf{U}^\theta_{n,t}\rp \rp \lp x \rp $, and 
		\item it holds for all $\theta \in \Theta$, $n \in \N_0$, $t\in \lb 0,T\rb$ that:
		\begin{align}
			\param \lp \mathsf{U}^\theta_{n,t} \rp \les 2n\hid \lp \mathsf{F} \rp + \max \left\{1,\mymathbb{1}_\N \lp n\rp \dep \lp \mathsf{G} \rp \right\} \lb \lp 2M+1\rp^n\max \left\{ 2, \left\| \lay\lp \mathsf{F}\rp\right\|_{\max}, \left\| \lay \lp \mathsf{G} \rp \right\|_{\max}\right\}\rb^2 
		\end{align}
	\end{enumerate}
\end{lemma}
