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https://github.com/saforem2/lattice23
Slides for Lattice 2023
https://github.com/saforem2/lattice23
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Slides for Lattice 2023
- Host: GitHub
- URL: https://github.com/saforem2/lattice23
- Owner: saforem2
- Created: 2023-07-28T14:01:06.000Z (over 1 year ago)
- Default Branch: main
- Last Pushed: 2024-09-10T14:28:20.000Z (about 2 months ago)
- Last Synced: 2024-09-10T16:13:54.547Z (about 2 months ago)
- Language: TeX
- Homepage: https://saforem2.github.io/lattice23/
- Size: 24.3 MB
- Stars: 4
- Watchers: 1
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: docs/README.md
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README
# MLMC: Machine Learning Monte Carlo
Sam Foreman
2024-04-03
#
MLMC:
Machine Learning Monte Carlo
for
Lattice Gauge Theory
[](https://samforeman.me) Sam Foreman
Xiao-Yong Jin, James C.
Osborn
[
`saforem2/`](https://github.com/saforem2/)`{`[`lattice23`](https://github.com/saforem2/lattice23),
[`l2hmc-qcd`](https://github.com/saforem2/l2hmc-qcd)`}`
# Overview
1. [Background:
`{MCMC,HMC}`](#markov-chain-monte-carlo-mcmc-centeredslide)
- [Leapfrog Integrator](#leapfrog-integrator-hmc-centeredslide)
- [Issues with HMC](#sec-issues-with-hmc)
- [Can we do better?](#sec-can-we-do-better)
2. [L2HMC: Generalizing MD](#sec-l2hmc)
- [4D $SU(3)$ Model](#sec-su3)
- [Results](#sec-results)
3. [References](#sec-references)
4. [Extras](#sec-extras)
# Markov Chain Monte Carlo (MCMC)
> [!NOTE]
>
> ### Goal
>
> Generate **independent** samples $\{x_{i}\}$, such that[^1]
> $$\{x_{i}\} \sim p(x) \propto e^{-S(x)}$$ where $S(x)$ is the *action*
> (or potential energy)
- Want to calculate observables $\mathcal{O}$:
$\left\langle \mathcal{O}\right\rangle \propto \int \left[\mathcal{D}x\right]\hspace{4pt} {\mathcal{O}(x)\, p(x)}$
If these were independent, we could
approximate:
$\left\langle\mathcal{O}\right\rangle \simeq \frac{1}{N}\sum^{N}_{n=1}\mathcal{O}(x_{n})$
$$\sigma_{\mathcal{O}}^{2} = \frac{1}{N}\mathrm{Var}{\left[\mathcal{O} (x)
\right]}\Longrightarrow \sigma_{\mathcal{O}} \propto \frac{1}{\sqrt{N}}$$
# Markov Chain Monte Carlo (MCMC)
> [!NOTE]
>
> ### Goal
>
> Generate **independent** samples $\{x_{i}\}$, such that[^2]
> $$\{x_{i}\} \sim p(x) \propto e^{-S(x)}$$ where $S(x)$ is the *action*
> (or potential energy)
- Want to calculate observables $\mathcal{O}$:
$\left\langle \mathcal{O}\right\rangle \propto \int \left[\mathcal{D}x\right]\hspace{4pt} {\mathcal{O}(x)\, p(x)}$
Instead, nearby configs are correlated,
and we incur a factor of
$\textcolor{#FF5252}{\tau^{\mathcal{O}}_{\mathrm{int}}}$:
$$\sigma_{\mathcal{O}}^{2} =
\frac{\textcolor{#FF5252}{\tau^{\mathcal{O}}_{\mathrm{int}}}}{N}\mathrm{Var}{\left[\mathcal{O}
(x) \right]}$$
# Hamiltonian Monte Carlo (HMC)
- Want to (sequentially) construct a chain of states:
$$x_{0} \rightarrow x_{1} \rightarrow x_{i} \rightarrow \cdots \rightarrow x_{N}\hspace{10pt}$$
such that, as $N \rightarrow \infty$:
$$\left\{x_{i}, x_{i+1}, x_{i+2}, \cdots, x_{N} \right\} \xrightarrow[]{N\rightarrow\infty} p(x)
\propto e^{-S(x)}$$
> [!TIP]
>
> ### Trick
>
> - Introduce fictitious momentum
> $v \sim \mathcal{N}(0, \mathbb{1})$
> - Normally distributed **independent** of $x$, i.e. $$\begin{align*}
> p(x, v) &\textcolor{#02b875}{=} p(x)\,p(v) \propto e^{-S{(x)}} e^{-\frac{1}{2} v^{T}v}
> = e^{-\left[S(x) + \frac{1}{2} v^{T}{v}\right]}
> \textcolor{#02b875}{=} e^{-H(x, v)}
> \end{align*}$$
## Hamiltonian Monte Carlo (HMC)
- **Idea**: Evolve the
$(\dot{x}, \dot{v})$ system to get new states $\{x_{i}\}$❗
- Write the **joint distribution** $p(x, v)$: $$
p(x, v) \propto e^{-S[x]} e^{-\frac{1}{2}v^{T} v} = e^{-H(x, v)}
$$
> [!TIP]
>
> ### Hamiltonian Dynamics
>
> $H = S[x] + \frac{1}{2} v^{T} v \Longrightarrow$
> $$\dot{x} = +\partial_{v} H,
> \,\,\dot{v} = -\partial_{x} H$$
## Leapfrog Integrator (HMC)
> [!TIP]
>
> ### Hamiltonian Dynamics
>
> $\left(\dot{x}, \dot{v}\right) = \left(\partial_{v} H, -\partial_{x} H\right)$
> [!NOTE]
>
> ### Leapfrog Step
>
> `input` $\,\left(x, v\right) \rightarrow \left(x', v'\right)\,$
> `output`
>
> $$\begin{align*}
> \tilde{v} &:= \textcolor{#F06292}{\Gamma}(x, v)\hspace{2.2pt} = v - \frac{\varepsilon}{2} \partial_{x} S(x) \\
> x' &:= \textcolor{#FD971F}{\Lambda}(x, \tilde{v}) \, = x + \varepsilon \, \tilde{v} \\
> v' &:= \textcolor{#F06292}{\Gamma}(x', \tilde{v}) = \tilde{v} - \frac{\varepsilon}{2} \partial_{x} S(x')
> \end{align*}$$
> [!WARNING]
>
> ### Warning!
>
> - Resample $v_{0} \sim \mathcal{N}(0, \mathbb{1})$
> at the beginning of each trajectory
**Note**: $\partial_{x} S(x)$ is the
*force*
## HMC Update
- We build a trajectory of $N_{\mathrm{LF}}$ **leapfrog steps**[^3]
$$\begin{equation*}
(x_{0}, v_{0})%
\rightarrow (x_{1}, v_{1})\rightarrow \cdots%
\rightarrow (x', v')
\end{equation*}$$
- And propose $x'$ as the next state in our chain
$$\begin{align*}
\textcolor{#F06292}{\Gamma}: (x, v) \textcolor{#F06292}{\rightarrow} v' &:= v - \frac{\varepsilon}{2} \partial_{x} S(x) \\
\textcolor{#FD971F}{\Lambda}: (x, v) \textcolor{#FD971F}{\rightarrow} x' &:= x + \varepsilon v
\end{align*}$$
- We then accept / reject $x'$ using Metropolis-Hastings criteria,
$A(x'|x) = \min\left\{1, \frac{p(x')}{p(x)}\left|\frac{\partial x'}{\partial x}\right|\right\}$
## HMC Demo
# Issues with HMC
- What do we want in a good sampler?
- **Fast mixing** (small autocorrelations)
- **Fast burn-in** (quick convergence)
- Problems with HMC:
- Energy levels selected randomly $\rightarrow$ **slow mixing**
- Cannot easily traverse low-density zones $\rightarrow$ **slow
convergence**
# Topological Freezing
**Topological Charge**:
$$Q = \frac{1}{2\pi}\sum_{P}\left\lfloor x_{P}\right\rfloor \in \mathbb{Z}$$
**note:**
$\left\lfloor x_{P} \right\rfloor = x_{P} - 2\pi
\left\lfloor\frac{x_{P} + \pi}{2\pi}\right\rfloor$
> [!IMPORTANT]
>
> ### Critical Slowing Down
>
> - $Q$ gets stuck!
> - as $\beta\longrightarrow \infty$:
> - $Q \longrightarrow \text{const.}$
> - $\delta Q = \left(Q^{\ast} - Q\right) \rightarrow 0 \textcolor{#FF5252}{\Longrightarrow}$
> - \# configs required to estimate errors
> **grows exponentially**:
> $\tau_{\mathrm{int}}^{Q} \longrightarrow \infty$
# Can we do better?
- Introduce two (**invertible NNs**) `vNet` and `xNet`[^4]:
- `vNet:`
$(x, F) \longrightarrow \left(s_{v},\, t_{v},\, q_{v}\right)$
- `xNet:`
$(x, v) \longrightarrow \left(s_{x},\, t_{x},\, q_{x}\right)$
- Use these $(s, t, q)$ in the *generalized* MD update:
- $\Gamma_{\theta}^{\pm}$
$: ({x}, \textcolor{#07B875}{v}) \xrightarrow[]{\textcolor{#F06292}{s_{v}, t_{v}, q_{v}}} (x, \textcolor{#07B875}{v'})$
- $\Lambda_{\theta}^{\pm}$
$: (\textcolor{#AE81FF}{x}, v) \xrightarrow[]{\textcolor{#FD971F}{s_{x}, t_{x}, q_{x}}} (\textcolor{#AE81FF}{x'}, v)$
# L2HMC: Generalizing the MD Update
> [!NONE]
>
> ### L2HMC Update
>
> - Introduce $d \sim \mathcal{U}(\pm)$ to determine the direction of
> our update
>
> 1. $\textcolor{#07B875}{v'} =$
> $\Gamma^{\pm}$$({x}, \textcolor{#07B875}{v})$
> $\hspace{46pt}$
> update $v$
>
> 2. $\textcolor{#AE81FF}{x'} =$
> $x_{B}$$\,+\,$$\Lambda^{\pm}$$($$x_{A}$$, {v'})$
> $\hspace{10pt}$
> update first **half**: $x_{A}$
>
> 3. $\textcolor{#AE81FF}{x''} =$
> $x'_{A}$$\,+\,$$\Lambda^{\pm}$$($$x'_{B}$$, {v'})$
> $\hspace{8pt}$
> update other half: $x_{B}$
>
> 4. $\textcolor{#07B875}{v''} =$
> $\Gamma^{\pm}$$({x''}, \textcolor{#07B875}{v'})$
> $\hspace{36pt}$
> update $v$
> [!NONE]
>
> ### 🎲 Re-Sampling
>
> - Resample both $v\sim \mathcal{N}(0, 1)$, and
> $d \sim \mathcal{U}(\pm)$ at the beginning of each trajectory
> - To ensure ergodicity + reversibility, we split the
> $x$ update into sequential
> (complementary) updates
> - Introduce directional variable $d \sim \mathcal{U}(\pm)$, resampled
> at the beginning of each trajectory:
> - Note that $\left(\Gamma^{+}\right)^{-1} = \Gamma^{-}$, i.e.
> $$\Gamma^{+}\left[\Gamma^{-}(x, v)\right] = \Gamma^{-}\left[\Gamma^{+}(x,
> v)\right] = (x, v)$$
## L2HMC: Leapfrog Layer
## L2HMC Update
> [!NONE]
>
> ### Algorithm
>
> 1. `input`: $x$
>
> - Resample:
> $\textcolor{#07B875}{v} \sim \mathcal{N}(0, \mathbb{1})$;
> $\,\,{d\sim\mathcal{U}(\pm)}$
> - Construct initial state:
> $\textcolor{#939393}{\xi} =(\textcolor{#AE81FF}{x}, \textcolor{#07B875}{v}, {\pm})$
>
> 2. `forward`: Generate proposal
> $\xi'$ by passing initial
> $\xi$ through $N_{\mathrm{LF}}$ leapfrog layers
> $$\textcolor{#939393} \xi \hspace{1pt}\xrightarrow[]{\tiny{\mathrm{LF} \text{ layer}}}\xi_{1} \longrightarrow\cdots \longrightarrow \xi_{N_{\mathrm{LF}}} = \textcolor{#f8f8f8}{\xi'} := (\textcolor{#AE81FF}{x''}, \textcolor{#07B875}{v''})$$
>
> - Accept / Reject: $$\begin{equation*}
> A({\textcolor{#f8f8f8}{\xi'}}|{\textcolor{#939393}{\xi}})=
> \mathrm{min}\left\{1,
> \frac{\pi(\textcolor{#f8f8f8}{\xi'})}{\pi(\textcolor{#939393}{\xi})} \left| \mathcal{J}\left(\textcolor{#f8f8f8}{\xi'},\textcolor{#939393}{\xi}\right)\right| \right\}
> \end{equation*}$$
>
> 3. `backward` (if training):
>
> - Evaluate the **loss function**[^5]
> $\mathcal{L}\gets \mathcal{L}_{\theta}(\textcolor{#f8f8f8}{\xi'}, \textcolor{#939393}{\xi})$
> and backprop
>
> 4. `return`: $\textcolor{#AE81FF}{x}_{i+1}$
> Evaluate MH criteria $(1)$ and return accepted config,
> $$\textcolor{#AE81FF}{{x}_{i+1}}\gets
> \begin{cases}
> \textcolor{#f8f8f8}{\textcolor{#AE81FF}{x''}} \small{\text{ w/ prob }} A(\textcolor{#f8f8f8}{\xi''}|\textcolor{#939393}{\xi}) \hspace{26pt} ✅ \\
> \textcolor{#939393}{\textcolor{#AE81FF}{x}} \hspace{5pt}\small{\text{ w/ prob }} 1 - A(\textcolor{#f8f8f8}{\xi''}|{\textcolor{#939393}{\xi}}) \hspace{10pt} 🚫
> \end{cases}$$
# 4D $SU(3)$ Model
> [!NOTE]
>
> ### Link Variables
>
> - Write link variables $U_{\mu}(x) \in SU(3)$:
>
> $$ \begin{align*}
> U_{\mu}(x) &= \mathrm{exp}\left[{i\, \textcolor{#AE81FF}{\omega^{k}_{\mu}(x)} \lambda^{k}}\right]\\
> &= e^{i \textcolor{#AE81FF}{Q}},\quad \text{with} \quad \textcolor{#AE81FF}{Q} \in \mathfrak{su}(3)
> \end{align*}$$
>
> where
> $\omega^{k}_{\mu}(x)$
> $\in \mathbb{R}$, and $\lambda^{k}$ are the generators of
> $SU(3)$
> [!TIP]
>
> ### Conjugate Momenta
>
> - Introduce
> $P_{\mu}(x) = P^{k}_{\mu}(x) \lambda^{k}$
> conjugate to $\omega^{k}_{\mu}(x)$
> [!IMPORTANT]
>
> ### Wilson Action
>
> $$ S_{G} = -\frac{\beta}{6} \sum
> \mathrm{Tr}\left[U_{\mu\nu}(x)
> + U^{\dagger}_{\mu\nu}(x)\right] $$
>
> where $U_{\mu\nu}(x) = U_{\mu}(x) U_{\nu}(x+\hat{\mu})
> U^{\dagger}_{\mu}(x+\hat{\nu}) U^{\dagger}_{\nu}(x)$
## HMC: 4D $SU(3)$
Hamiltonian: $H[P, U] = \frac{1}{2} P^{2} + S[U] \Longrightarrow$
> [!NONE]
>
> - $U$ update:
> style="font-size:1.5em;">$\frac{d\omega^{k}}{dt} = \frac{\partial H}{\partial P^{k}}$
> $$\frac{d\omega^{k}}{dt}\lambda^{k} = P^{k}\lambda^{k} \Longrightarrow \frac{dQ}{dt} = P$$
> $$\begin{align*}
> Q(\textcolor{#FFEE58}{\varepsilon}) &= Q(0) + \textcolor{#FFEE58}{\varepsilon} P(0)\Longrightarrow\\
> -i\, \log U(\textcolor{#FFEE58}{\varepsilon}) &= -i\, \log U(0) + \textcolor{#FFEE58}{\varepsilon} P(0) \\
> U(\textcolor{#FFEE58}{\varepsilon}) &= e^{i\,\textcolor{#FFEE58}{\varepsilon} P(0)} U(0)\Longrightarrow \\
> &\hspace{1pt}\\
> \textcolor{#FD971F}{\Lambda}:\,\, U \longrightarrow U' &:= e^{i\varepsilon P'} U
> \end{align*}$$
$\textcolor{#FFEE58}{\varepsilon}$ is the step
size
> [!NONE]
>
> - $P$ update:
> style="font-size:1.5em;">$\frac{dP^{k}}{dt} = - \frac{\partial H}{\partial \omega^{k}}$
> $$\frac{dP^{k}}{dt} = - \frac{\partial H}{\partial \omega^{k}}
> = -\frac{\partial H}{\partial Q} = -\frac{dS}{dQ}\Longrightarrow$$
> $$\begin{align*}
> P(\textcolor{#FFEE58}{\varepsilon}) &= P(0) - \textcolor{#FFEE58}{\varepsilon} \left.\frac{dS}{dQ}\right|_{t=0} \\
> &= P(0) - \textcolor{#FFEE58}{\varepsilon} \,\textcolor{#E599F7}{F[U]} \\
> &\hspace{1pt}\\
> \textcolor{#F06292}{\Gamma}:\,\, P \longrightarrow P' &:= P - \frac{\varepsilon}{2} F[U]
> \end{align*}$$
$\textcolor{#E599F7}{F[U]}$ is the force
term
## HMC: 4D $SU(3)$
- Momentum
Update:
$$\textcolor{#F06292}{\Gamma}: P \longrightarrow P' := P - \frac{\varepsilon}{2} F[U]$$
- Link Update:
$$\textcolor{#FD971F}{\Lambda}: U \longrightarrow U' := e^{i\varepsilon P'} U\quad\quad$$
- We maintain a batch of `Nb` lattices, all updated in parallel
- $U$`.dtype = complex128`
- $U$`.shape`
`= [Nb, 4, Nt, Nx, Ny, Nz, 3, 3]`
# Networks 4D $SU(3)$
$U$-Network:
`UNet:`
$(U, P) \longrightarrow \left(s_{U},\, t_{U},\, q_{U}\right)$
$P$-Network:
`PNet:`
$(U, P) \longrightarrow \left(s_{P},\, t_{P},\, q_{P}\right)$
# Networks 4D $SU(3)$
$U$-Network:
`UNet:`
$(U, P) \longrightarrow \left(s_{U},\, t_{U},\, q_{U}\right)$
$P$-Network:
`PNet:`
$(U, P) \longrightarrow \left(s_{P},\, t_{P},\, q_{P}\right)$
$\uparrow$
let’s look
at this
## $P$-`Network` (pt. 1)
- `input`[^6]:
$\hspace{7pt}\left(U, F\right) := (e^{i Q}, F)$
$$\begin{align*}
h_{0} &= \sigma\left( w_{Q} Q + w_{F} F + b \right) \\
h_{1} &= \sigma\left( w_{1} h_{0} + b_{1} \right) \\
&\vdots \\
h_{n} &= \sigma\left(w_{n-1} h_{n-2} + b_{n}\right) \\
\textcolor{#FF5252}{z} & := \sigma\left(w_{n} h_{n-1} + b_{n}\right) \longrightarrow \\
\end{align*}$$
- `output`[^7]:
$\hspace{7pt} (s_{P}, t_{P}, q_{P})$
- $s_{P} = \lambda_{s} \tanh(w_s \textcolor{#FF5252}{z} + b_s)$
- $t_{P} = w_{t} \textcolor{#FF5252}{z} + b_{t}$
- $q_{P} = \lambda_{q} \tanh(w_{q} \textcolor{#FF5252}{z} + b_{q})$
## $P$-`Network` (pt. 2)
- Use $(s_{P}, t_{P}, q_{P})$ to update
$\Gamma^{\pm}: (U, P) \rightarrow
\left(U, P_{\pm}\right)$[^8]:
- forward
$(d = \textcolor{#FF5252}{+})$:
$$\Gamma^{\textcolor{#FF5252}{+}}(U, P) := P_{\textcolor{#FF5252}{+}} = P \cdot e^{\frac{\varepsilon}{2} s_{P}} - \frac{\varepsilon}{2}\left[ F \cdot e^{\varepsilon q_{P}} + t_{P} \right]$$
- backward
$(d = \textcolor{#1A8FFF}{-})$:
$$\Gamma^{\textcolor{#1A8FFF}{-}}(U, P) := P_{\textcolor{#1A8FFF}{-}} = e^{-\frac{\varepsilon}{2} s_{P}} \left\{P + \frac{\varepsilon}{2}\left[ F \cdot e^{\varepsilon q_{P}} + t_{P} \right]\right\}$$
# Results: 2D $U(1)$
> [!IMPORTANT]
>
> ### Improvement
>
> We can measure the performance by comparing $\tau_{\mathrm{int}}$ for
> the **trained model** vs.
> **HMC**.
>
> **Note**: lower is better
## Interpretation
Deviation in $x_{P}$
Topological charge
mixing
Artificial influx
of energy
## Interpretation
# 4D $SU(3)$ Results
- Distribution of $\log|\mathcal{J}|$ over all chains, at each *leapfrog
step*, $N_{\mathrm{LF}}$ ($= 0, 1, \ldots, 8$) during training:
## 4D $SU(3)$ Results: $\delta U_{\mu\nu}$
## 4D $SU(3)$ Results: $\delta U_{\mu\nu}$
# Next Steps
- Further code development
- [`saforem2/l2hmc-qcd`](https://github.com/saforem2/l2hmc-qcd)
- Continue to use / test different network architectures
- Gauge equivariant NNs for $U_{\mu}(x)$ update
- Continue to test different loss functions for training
- Scaling:
- Lattice volume
- Network size
- Batch size
- \# of GPUs
## Thank you!
> [!NOTE]
>
> ### Acknowledgements
>
> This research used resources of the Argonne Leadership Computing
> Facility,
> which is a DOE Office of Science User Facility supported under
> Contract DE-AC02-06CH11357.
##
[](https://github.com/saforem2/l2hmc-qcd)
[
](https://wandb.ai/l2hmc-qcd/l2hmc-qcd)
## Acknowledgements
- **Links**:
- [ Link to github](https://github.com/saforem2/l2hmc-qcd)
- [ reach out!](mailto:[email protected])
- **References**:
- [Link to slides](https://saforem2.github.io/lattice23/)
- [ link to github with
slides](https://github.com/saforem2/lattice23)
- (Foreman et al. 2022; Foreman, Jin, and Osborn 2022, 2021)
- (Boyda et al. 2022; Shanahan et al. 2022)
- Huge thank you to:
- Yannick Meurice
- Norman Christ
- Akio Tomiya
- Nobuyuki Matsumoto
- Richard Brower
- Luchang Jin
- Chulwoo Jung
- Peter Boyle
- Taku Izubuchi
- Denis Boyda
- Dan Hackett
- ECP-CSD group
- **ALCF Staff + Datascience Group**
##
### Links
- [ `saforem2/l2hmc-qcd`](https://github.com/saforem2/l2hmc-qcd)
- [📊 slides](https://saforem2.github.io/lattice23) (Github: [
`saforem2/lattice23`](https://github.com/saforem2/lattice23))
### References
- [Title Slide Background (worms)
animation](https://saforem2.github.io/grid-worms-animation/)
- Github: [
`saforem2/grid-worms-animation`](https://github.com/saforem2/grid-worms-animation)
- [Link to HMC demo](https://chi-feng.github.io/mcmc-demo/app.html)
## References
(I don’t know why this is broken 🤷🏻♂️ )
Boyda, Denis et al. 2022. “Applications of Machine
Learning to Lattice Quantum Field Theory.” In *Snowmass 2021*.
.
Foreman, Sam, Taku Izubuchi, Luchang Jin, Xiao-Yong Jin, James C.
Osborn, and Akio Tomiya. 2022. “HMC with
Normalizing Flows.” *PoS* LATTICE2021: 073.
.
Foreman, Sam, Xiao-Yong Jin, and James C. Osborn. 2021. “Deep Learning
Hamiltonian Monte Carlo.” In *9th International
Conference on Learning Representations*.
.
———. 2022. “LeapfrogLayers: A Trainable Framework
for Effective Topological Sampling.” *PoS* LATTICE2021 (May):
508. .
Shanahan, Phiala et al. 2022. “Snowmass 2021 Computational Frontier
CompF03 Topical Group Report: Machine Learning,” September.
.
# Extras
## Integrated Autocorrelation Time
## Comparison
## Plaquette analysis: $x_{P}$
Deviation from
$V\rightarrow\infty$ limit, $x_{P}^{\ast}$
Average
$\langle x_{P}\rangle$, with $x_{P}^{\ast}$ (dotted-lines)
## Loss Function
- Want to maximize the *expected* squared charge difference[^9]:
$$\begin{equation*}
\mathcal{L}_{\theta}\left(\xi^{\ast}, \xi\right) =
{\mathbb{E}_{p(\xi)}}\big[-\textcolor{#FA5252}{{\delta Q}}^{2}
\left(\xi^{\ast}, \xi \right)\cdot A(\xi^{\ast}|\xi)\big]
\end{equation*}$$
- Where:
- $\delta Q$ is the *tunneling rate*: $$\begin{equation*}
\textcolor{#FA5252}{\delta Q}(\xi^{\ast},\xi)=\left|Q^{\ast} - Q\right|
\end{equation*}$$
- $A(\xi^{\ast}|\xi)$ is the probability[^10] of accepting the
proposal $\xi^{\ast}$: $$\begin{equation*}
A(\xi^{\ast}|\xi) = \mathrm{min}\left( 1,
\frac{p(\xi^{\ast})}{p(\xi)}\left|\frac{\partial \xi^{\ast}}{\partial
\xi^{T}}\right|\right)
\end{equation*}$$
## Networks 2D $U(1)$
- Stack gauge links as `shape`$\left(U_{\mu}\right)$`=[Nb, 2, Nt, Nx]`
$\in \mathbb{C}$
$$ x_{\mu}(n) ≔ \left[\cos(x), \sin(x)\right]$$
with `shape`$\left(x_{\mu}\right)$`= [Nb, 2, Nt, Nx, 2]`
$\in \mathbb{R}$
- $x$-Network:
- $\psi_{\theta}: (x, v) \longrightarrow \left(s_{x},\, t_{x},\, q_{x}\right)$
- $v$-Network:
- $\varphi_{\theta}: (x, v) \longrightarrow \left(s_{v},\, t_{v},\, q_{v}\right)$
$\hspace{2pt}\longleftarrow$ lets look at
this
## $v$-Update[^11]
- forward
$(d = \textcolor{#FF5252}{+})$:
$$\Gamma^{\textcolor{#FF5252}{+}}: (x, v) \rightarrow v' := v \cdot e^{\frac{\varepsilon}{2} s_{v}} - \frac{\varepsilon}{2}\left[ F \cdot e^{\varepsilon q_{v}} + t_{v} \right]$$
- backward
$(d = \textcolor{#1A8FFF}{-})$:
$$\Gamma^{\textcolor{#1A8FFF}{-}}: (x, v) \rightarrow v' := e^{-\frac{\varepsilon}{2} s_{v}} \left\{v + \frac{\varepsilon}{2}\left[ F \cdot e^{\varepsilon q_{v}} + t_{v} \right]\right\}$$
## $x$-Update
- forward
$(d = \textcolor{#FF5252}{+})$:
$$\Lambda^{\textcolor{#FF5252}{+}}(x, v) = x \cdot e^{\frac{\varepsilon}{2} s_{x}} - \frac{\varepsilon}{2}\left[ v \cdot e^{\varepsilon q_{x}} + t_{x} \right]$$
- backward
$(d = \textcolor{#1A8FFF}{-})$:
$$\Lambda^{\textcolor{#1A8FFF}{-}}(x, v) = e^{-\frac{\varepsilon}{2} s_{x}} \left\{x + \frac{\varepsilon}{2}\left[ v \cdot e^{\varepsilon q_{x}} + t_{x} \right]\right\}$$
## Lattice Gauge Theory (2D $U(1)$)
> [!NOTE]
>
> ### Link Variables
>
> $$U_{\mu}(n) = e^{i x_{\mu}(n)}\in \mathbb{C},\quad \text{where}\quad$$
> $$x_{\mu}(n) \in [-\pi,\pi)$$
> [!IMPORTANT]
>
> ### Wilson Action
>
> $$S_{\beta}(x) = \beta\sum_{P} \cos \textcolor{#00CCFF}{x_{P}},$$
>
> $$\textcolor{#00CCFF}{x_{P}} = \left[x_{\mu}(n) + x_{\nu}(n+\hat{\mu})
> - x_{\mu}(n+\hat{\nu})-x_{\nu}(n)\right]$$
**Note**:
$\textcolor{#00CCFF}{x_{P}}$ is the product of links around $1\times 1$
square, called a “plaquette”
##
## Annealing Schedule
- Introduce an *annealing schedule* during the training phase:
$$\left\{ \gamma_{t} \right\}_{t=0}^{N} = \left\{\gamma_{0}, \gamma_{1},
\ldots, \gamma_{N-1}, \gamma_{N} \right\}$$
where $\gamma_{0} < \gamma_{1} < \cdots < \gamma_{N} \equiv 1$, and
$\left|\gamma_{t+1} - \gamma_{t}\right| \ll 1$
- **Note**:
- for $\left|\gamma_{t}\right| < 1$, this rescaling helps to reduce
the height of the energy barriers $\Longrightarrow$
- easier for our sampler to explore previously inaccessible regions of
the phase space
## Networks 2D $U(1)$
- Stack gauge links as `shape`$\left(U_{\mu}\right)$`=[Nb, 2, Nt, Nx]`
$\in \mathbb{C}$
$$ x_{\mu}(n) ≔ \left[\cos(x), \sin(x)\right]$$
with `shape`$\left(x_{\mu}\right)$`= [Nb, 2, Nt, Nx, 2]`
$\in \mathbb{R}$
- $x$-Network:
- $\psi_{\theta}: (x, v) \longrightarrow \left(s_{x},\, t_{x},\, q_{x}\right)$
- $v$-Network:
- $\varphi_{\theta}: (x, v) \longrightarrow \left(s_{v},\, t_{v},\, q_{v}\right)$
## Toy Example: GMM $\in \mathbb{R}^{2}$
## Physical Quantities
- To estimate physical quantities, we:
- Calculate physical observables at **increasing** spatial resolution
- Perform extrapolation to continuum limit
# Extra
[^1]: Here, $\sim$ means “is distributed according to”
[^2]: Here, $\sim$ means “is distributed according to”
[^3]: We **always** start by resampling the momentum, $v_{0} \sim
\mathcal{N}(0, \mathbb{1})$
[^4]: [L2HMC:](https://github.com/saforem2/l2hmc-qcd) (Foreman, Jin, and
Osborn 2021, 2022)
[^5]: For simple $\mathbf{x} \in \mathbb{R}^{2}$ example,
$\mathcal{L}_{\theta} =
A(\xi^{\ast}|\xi)\cdot \left(\mathbf{x}^{\ast} - \mathbf{x}\right)^{2}$
[^6]: $\sigma(\cdot)$ denotes an activation function
[^7]: $\lambda_{s},\, \lambda_{q} \in \mathbb{R}$ are trainable
parameters
[^8]: Note that $\left(\Gamma^{+}\right)^{-1} = \Gamma^{-}$,
i.e. $\Gamma^{+}\left[\Gamma^{-}(U, P)\right] = \Gamma^{-}\left[\Gamma^{+}(U, P)\right] = (U, P)$
[^9]: Where $\xi^{\ast}$ is the *proposed* configuration (prior to
Accept / Reject)
[^10]: And $\left|\frac{\partial \xi^{\ast}}{\partial \xi^{T}}\right|$
is the Jacobian of the transformation from
$\xi \rightarrow \xi^{\ast}$
[^11]: Note that
$\left(\Gamma^{+}\right)^{-1} = \Gamma^{-}$,
i.e. $\Gamma^{+}\left[\Gamma^{-}(x, v)\right] = \Gamma^{-}\left[\Gamma^{+}(x, v)\right] = (x, v)$