# Conditional Normalizing flow for Monte Carlo sampling in lattice scalar
field theory

Ankur Singha  Department of Physics, Indian Institute of Technology Kanpur,
Kanpur-208016, India  Dipankar Chakrabarti  Department of Physics, Indian
Institute of Technology Kanpur, Kanpur-208016, India  Vipul Arora  Department
of Electrical Engineering, Indian Institute of Technology Kanpur,
Kanpur-208016, India

###### Abstract

The cost of Monte Carlo sampling of lattice configurations is very high in the
critical region of lattice field theory due to the high correlation between
the samples. This paper suggests a Conditional Normalizing Flow (C-NF) model
for sampling lattice configurations in the critical region to solve the
problem of critical slowing down. We train the C-NF model using samples
generated by Hybrid Monte Carlo (HMC) in non-critical regions with low
simulation costs. The trained C-NF model is employed in the critical region to
build a Markov chain of lattice samples with negligible autocorrelation. The
C-NF model is used for both interpolation and extrapolation to the critical
region of lattice theory. Our proposed method is assessed using the
1+1-dimensional scalar $\phi^{4}$ theory. This approach enables the
construction of lattice ensembles for many parameter values in the critical
region, which reduces simulation costs by avoiding the critical slowing down.

## 1 Introduction

In lattice field theory, Monte Carlo Simulation techniques are used to sample
lattice configurations based on a distribution defined by the action of the
lattice theory. The parameter value at which we generate the lattice samples
determines the cost of the simulation. The non-critical region of the lattice
theory has low simulation costs for algorithms like Hybrid Monte Carlo
(HMC))[1]. However, as we attempt to sample uncorrelated lattice
configurations from the critical region, the simulation cost increases
rapidly. In the critical region, the integrated autocorrelation time, which
gives the measure of correlation, increases rapidly and diverges at the
critical point. For a finite-size lattice critical point corresponds to the
peak point of the autocorrelation curve. This problem is known as the critical
slowing down[2, 3]. Many efforts have been made to lessen the impact of
critical slowing in statistical systems and lattice QFT [4, 5, 6]. But it
always remains a challenging task to simulate near the critical point of a
lattice QFT by overcoming the critical slowing down.

These days, ML-based solutions to this problem are becoming popular. Various
ML algorithms have been applied for statistical physics and condensed matter
problems[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] . Some generative
learning algorithms[21, 22, 23, 24, 25, 26, 27] have recently been developed
to avoid the difficulty in lattice field theory. Conditional GAN was used in a
2D lattice Gross Neveue model [20] and shown to be effective in mitigating the
influence of critical slowdown. However, explicit probability density
estimation is not accessible in GAN; thus, we cannot guarantee that the model
distribution is identical to the true lattice distribution. Flow based
generative leanings are found successful in avoiding the problem of critical
slowing down in scalar field theory[21], fermionic system[22], U(1) gauge
theory[24] and Schwinger models[26]. In the flow-based approach [21, 24] an NF
model is trained at a single value of the action parameter with reverse KL
divergence. Finally, the trained model can generate lattice samples at the
same parameter value. This model is initialized from scratch for a parameter
value and does not use any lattice samples. Since HMC simulation in the non-
critical region is not affected by the critical slowing down problem, we can
use samples from that region to train a generative model. We present a method
for sampling lattice configurations near the critical regions using
Conditional Normalizing Flow (C-NF) to reduce the problem of critical slowing
down. This method involves training a C-NF model in a non-critical region to
produce samples for several parameter values in the critical region. Our goal
is to generate lattice configurations from the distribution
$p(\phi|\lambda_{crit})=\frac{1}{Z}e^{-S(\phi|\lambda_{crit})}$, where $\phi$
denotes the lattice field, $\lambda_{crit}$ denotes the action parameter close
to critical point and Z is the partition function. We train a C-NF model
$\tilde{p}({\phi|\lambda})$ with HMC samples from $p(\phi|\lambda)$ for
various non-critical $\lambda$ values. We train the C-NF model to be a
generalized model over $\lambda$ parameters. The model is then interpolated or
extrapolated to the $\lambda$ values in the critical region to generate
lattice configurations. However, the interpolated or extrapolated model may
not directly provide samples from the true distribution. But the exactness can
be guaranteed by using the Metropolis-Hastings(MH) algorithm at the end. So,
after training, we use the interpolated/extrapolated model
$\tilde{p}(\phi|{\lambda_{crit}})$ at critical region as proposal for
constructing a Markov Chain via an independent MH algorithm[21]. This method
is useful when the probability distribution is known up to a normalizing
factor.

The primary contributions of this study are as follows:

  1. 1.

Using Conditional Normalizing Flows, we present a new method for sampling
lattice configurations near the critical regions. This method eliminates the
critical slowdown problem.

  2. 2.

The model has the ability to learn about the lattice system across multiple
$\lambda$ values and use this knowledge to generate sample at any given
$\lambda$ values. As a result, our model can generate samples at multiple
$\lambda$ values values in the critical region, which is not possible for the
existing flow-based methods in lattice thoery.

  3. 3.

We also demonstrate that Conditional Normalizing flow can be used to do both
interpolation and extrapolation for lattice $\phi^{4}$ theory. The
extrapolation demonstrates the possibility of using our approach for sampling
lattice gauge theory.

## 2 Lattice scalar $\phi^{4}$ Theory

In $2d$ euclidean space, the action for $\phi^{4}$ theory can be written as:

$\displaystyle S(\phi,\lambda,m)=\int
dx^{2}[(\partial_{\mu}\phi(x))^{2}+m^{2}\phi(x)^{2}+\lambda\phi(x)^{4}]$ (1)

where $\lambda$ and $m$ are the two parameters of the theory.

On lattice the action become:

$\displaystyle S(\phi,\lambda,m)=$
$\displaystyle\sum\limits_{x}\Big{[}\sum\limits_{\mu={1,2}}\Big{[}2\phi(x)^{2}+\phi(x)\phi(x+\hat{\mu})$
$\displaystyle-\phi(x)\phi(x-\hat{\mu})\Big{]}+m^{2}\phi(x)^{2}+\lambda\phi(x)^{4}\Big{]}$
(2)

where $x$ is a $2d$ discrete vector and $\hat{\mu}$ represents two possible
directions on the lattice. $\phi(x)$ is defined on each lattice site, taking
only real values.

We choose this specific form of action for HMC simulation because it is suited
for creating datasets for training the C-NF model. Using this form, we do not
need to apply any further transformations to the samples during training. This
action possesses $\phi(x)\rightarrow\phi(-x)$ symmetry, however it is
spontaneously broken at a specific parameter region. We set $m^{2}=-4$ for our
numerical experiments and observed spontaneous symmetry breaking in the theory
by varying $\lambda$. If we begin in the broken phase of the theory, the order
parameter under consideration $\langle\phi^{2}\rangle$ is nonzero which
approaches zero at the critical point and remains zero in the symmetric phase,
as shown in Figure 1. In HMC simulation we choose a parameter $\lambda$ and
produce configurations based on the probability distribution:

$\displaystyle P(\phi|\lambda)=\frac{1}{Z}e^{-S(\phi|\lambda)}$ (3)
$\displaystyle\text{where, }Z=\sum\limits_{\phi}e^{-S(\phi|\lambda)}$

Figure 1:  HMC simulation: order parameter $\langle\tilde{\phi}^{2}\rangle$ vs
$\lambda$ for $m^{2}=-4$, which shows a 2nd order phase transition.

Each lattice configuration is a $2d$ matrix with the dimensions $L_{x}\times
L_{y}$. For our experiment, we use $L_{x}=L_{y}=8$ and add a periodic boundary
condition to the lattice in all directions. More information on lattice
$\phi^{4}$ theory can be found in ref. [28]. Some of the observables which we
calculate on the lattice ensembles are:

  1. 1.

$\langle\tilde{\phi}^{2}\rangle$: $\tilde{\phi}=\frac{1}{V}\sum_{x}\phi(x)$

  2. 2.

Correlation Function:

$\displaystyle
G_{c}(x)=\frac{1}{V}\sum_{y}[\langle\phi(y)\phi(x+y)\rangle-\langle\phi(y)\rangle\langle\phi(x+y)\rangle]$

Zero momentum Correlation Function:
$C(t)=\sum_{x_{1}}G_{c}(x_{1},t)$

  3. 3.

Two Point Susceptibility: $\chi=\sum_{x}G(x)$

## 3 Conditional Normalizing Flow

Normalizing flows[29] are a generative model for constructing complex
distributions by transforming a simple known distribution via a series of
invertible and smooth mapping $f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ with
inverse $f^{-1}=g$. If $p_{z}(z)$ is the prior distribution and $p_{t}(x)$ is
the complex target distribution, then the model distribution $p_{m}(x;\theta)$
can be written using the change of variable formula as

$\displaystyle p_{m}(x;\theta)=p_{z}(z)\big{|}det\frac{\partial
f_{\theta}(z)^{-1}}{\partial x}\big{|}$ (4) $\displaystyle\text{where,
}x=f_{\theta}(z)$

Fitting a flow-based model $p_{m}(x;\theta)$ to a target distribution
$p_{t}(x)$ can be accomplished by minimising their KL divergence. The most
crucial step is to build the flow so that we can calculate
$\big{|}det\frac{\partial f_{\theta}(z)^{-1}}{\partial x}\big{|}$. One such
method is the affine coupling block, which divides the input $z$ into two
halves and applies an affine transformation to produce upper or lower
triangular Jacobians. The transformation rules for such a building are as
follows[30]:

$\displaystyle x_{1}=z_{1}\odot exp(s_{1}(z_{2})+t_{1}(z_{2}))$ $\displaystyle
x_{2}=z_{2}\odot exp(s_{2}(x_{1}))+t_{2}(x_{1})$ (5)

where, $\odot$ represent element-wise product of two vectors.

The inverse of this coupling layer is simply computed as:

$\displaystyle z_{2}=(x_{2}-t_{2}(x_{1}))\odot exp(-s_{2}(x_{1}))$
$\displaystyle z_{1}=(x_{1}-t_{1}(z_{2}))\odot exp(-s_{1}(z_{2}))$ (6)

Figure 2: One affine block of Conditional Normalizing flow. Here, c is the
conditional parameter.

Because inverting an affine coupling layer does not require the inverse of
$s_{1},s_{2},t_{1},t_{2}$, they can be any non-linear complex function and can
thus be represented by neural networks. Introducing a conditioning parameter
in NF is not as simple as it is in GAN. However, since $s$ and $t$ are only
evaluated in the forward direction, we may concatenate the conditioning
parameter $c$ with the input $x$ to the coupling layer in order to invert the
model[31], as illustrated in Figure 2. As a result, the affine coupling
transformation rules become:

$\displaystyle x_{1}=z_{1}\odot exp(s_{1}(z_{2},c)+t_{1}(z_{2},c))$
$\displaystyle x_{2}=z_{2}\odot exp(s_{2}(x_{1},c))+t_{2}(x_{1},c))$ (7)

And its inverse become

$\displaystyle z_{2}=(x_{2}-t_{2}(x_{1},c))\odot exp(-s_{2}(x_{1},c))$
$\displaystyle z_{1}=(x_{1}-t_{1}(z_{2},c))\odot exp(-s_{1}(z_{2},c))$ (8)

Let us designate the C-NF model as $f(x;c,\theta)$ and its inverse as
$g(z;c,\theta)$. The invertibility for any fixed condition $c$ is given by

$\displaystyle f^{-1}(.;c,\theta)=g(.;,c,\theta)$ (9)

The change of variable formula become

$\displaystyle p_{m}(x;c,\theta)=p_{z}(f(x;c,\theta))|detJ_{f}(z)|^{-1}$ (10)

And the loss function is the KL divergence between the model distribution
$p_{m}(x;c,\theta)$ and target distribution $p_{t}(x;c)$ :

$\displaystyle\mathcal{L}=D_{KL}[{p_{t}(x;c)||p_{m}(x;c,\theta)}]$ (11)

We can find the maximum likelihood network parameter $\theta_{ML}$ using this
loss. Then, for a fixed $c$, we can execute conditional generation of $x$ by
sampling $z$ from $p_{z}(z)$ and employing the inverted network
$g(z;c,\theta_{ML})$.

## 4 Numerical Experiments

This section discusses dataset preparation and the model architecture utilized
in training the C-NF model. In addition, the training details of the C-NF
model and the sampling process in the critical region are discussed. We train
the C-NF model with different datasets and model architecture for
interpolation and extrapolation.

###  4.1 Dataset

In the non-critical region, we generate lattice configurations using HMC
simulation where autocorrelation is low. In HMC simulation we use Molecular
Dynamics(MD) step size=0.1 and MD trajectory length=1. For training purposes,
we generate 10,000 lattice configurations for interpolation and 15,000
configurations for extrapolation of size $8\times 8$ for each $\lambda$ value.
In the critical region, for the evaluation purpose of the C-NF model, we use
HMC to generate around $10^{5}$ lattice configurations per $\lambda$. This
serves as a baseline for comparing observables to the observable produced
through the proposed sampling strategy.

###  4.2 C-NF Model Architecture

The affine coupling block displayed in Figure 2 is the fundamental building
component of our C-NF model. For all neural networks $s$ and $t$, we use the
same architecture. Figure 3 depicts the neural networks used for $s$ and $t$.
In the neural networks, we solely employ Fully Convolutional Layers. For the
first two layers, we use 64 filter for interpolation and 32 for extrapolation,
all of which are $3\times 3$ in size. For each Convolutional layer, we employ
the Tanh activation function. We use 8 such affine blocks while training for
both interpolation and extrapolation. For each 8 affine block, the conditional
parameter $c$ is concatenated with the input to $s$ and $t$ as shown in Figure
2.

Figure 3: Architecture of the Neural Network s and t of the $i$-th affine
block.

###  4.3 Training and Sampling Procedure

The loss function used for training the C-NF model is the forward KL
divergence as in Section 3 between the model distribution
$p_{m}(\phi;\lambda,\theta)$ and target distribution $p_{t}(\phi;\lambda)$:

$\displaystyle\mathcal{L(\theta)}$ $\displaystyle=\int d\phi
p_{t}(\phi;\lambda)(log[p_{t}(\phi;\lambda)]-log[p_{m}(\phi;\lambda,\theta)])$
$\displaystyle=E_{\phi\sim
p_{t}}(log[p_{t}(\phi;\lambda)]-log[p_{m}(\phi;\lambda,\theta)])$ (12)
$\displaystyle\text{where},$ $\displaystyle
log[p_{t}(\phi;\lambda)]=-[S(\phi|\lambda)+Z]$

The expectation $E_{\phi\sim p_{t}}$ is evaluated using HMC samples from the
non-critical region. During training, we maintain the learning rate at 0.0003
and employ the Adam optimizer.

Once training is complete, we invert the C-NF model, which we call the
proposal model. The inputs to the proposal model are i) lattices from the
Normal distribution $\mathcal{N}(0,1)$ and ii) $\lambda$ as a conditional
parameter for sample generation. Outputs are the lattice configurations and
probability densities for each configuration for a given $\lambda$. For the
critical region we give critical $\lambda$ values as conditional parameter to
the propsal model. We look at two scenarios in which a C-NF model can be
either interpolated or extrapolated to the critical region. Both require
different training, but the sampling technique is the same. The samples from
the interpolated/extrapolated model may not exactly represent the true
distribution of lattice theory $p_{t}(\phi|\lambda)$. As a result, we use this
model as a proposal for the independent MH algorithm, which generates a Markov
Chain with asymptotic convergence to the true distribution.

###  4.4 Interpolation to the Critical Region

The $\lambda$ set used to train the C-NF model for interpolation purposes is:
$\\{3,3.2,3.5,3.6,3.7,3.8,5.8,6,6.5,7,8,9\\}$. During training, we bypass the
critical region [4.1-5.0] so that we can interpolate the model where the
autocorrelation time is large for HMC simulation. We interpolate the trained
model for multiple $\lambda$ values belonging to the critical region (4.1,
4.2, 4.25, 4.3, 4.35, 4.4, 4.45, 4.5, 4.55, 4.6, 4.65, 4.7, 4.8, 5.0). For
each $\lambda$ values we generates one ensemble of $10^{5}$ configurations
from the proposed method. On each ensemble we calculate different observables
using bootstrap re-sampling method.

(a) $\langle\tilde{\phi}^{2}\rangle$

(b) $\chi$

Figure 4: Interpolation to the critical
region:-$\langle\tilde{\phi}^{2}\rangle$ and $\chi$ are calculated on samples
generated from i)HMC, ii)C-NF followed by MH, and iii) Naive C-NF. The Error
bars indicate standard deviation calculated using bootstrapping re-sampling
with bin size 100.

We compare the observables from HMC simulation and our proposed method.
Observable from the Naive C-NF without MH is also shown to demonstrate the
C-NF model’s proximity to the true distribution. In Figure 4 we plot two
observables $\langle\tilde{\phi}^{2}\rangle$ and $\chi$ for the interpolated
$\lambda$ values. Although the naive C-NF model has biases, MH can eliminate
them, and both observables match pretty well within the statistical
uncertainty. In the Appendix, we present a table of numerical values of the
observables with errors.

(a) $\lambda=4.3$

(b) $\lambda=4.5$

(c) $\lambda=4.55$

(d) $\lambda=4.70$

Figure 5: Interpolation-:Zero momentum Correlation function calculated on
samples generated from i)HMC, ii)C-NF followed by MH, and iii) Naive C-NF. The
Error bars indicate $95\%$ confidence interval calculated using bootstrapping
re-sampling method with bin-size 100.

In Figure 5, the two point zero momentum correlation function ${C(t)}$ is
shown for four different $\lambda$ values. For these $\lambda$ values, we can
observe that the correlation function is non-vanishing, which is a unique
property of the critical region. The plots for other critical $\lambda$ values
are included in the Appendix.

In Figure 6 we have also shown the histogram of $\tilde{\phi}$ for a
particular critical $\lambda=4.6$. This demonstrates that MH can eliminate
this kind artefacts created by the C-NF model.

We displayed plots of various observables on both phases around the critical
point for the interpolation. We found that the observables estimated using our
technique and the HMC match quite well.

Figure 6: Histogram of $\tilde{\phi}$ for $\lambda=4.6$ from the a)naive C-NF
model and b) C-NF with MH is compared against HMC results with a bin-size=100.

###  4.5 Extrapolation to the Critical Region

The $\lambda$ set used to train the C-NF model for the extrapolation is
:$\\{3,3.1,3.2,3.3,3.4,3.5,3.6,3.7,3.8,3.9\\}$. After training the model in
the broken phase, we extrapolate it in the critical region around
$\lambda=4.6$, which we take to be the critical region’s midpoint. The model
is extrapolated for five distinct $\lambda$ values:[4.2,4.3,4.4,4.5,4.6]. For
each $\lambda$ values we generates one ensemble of $10^{5}$ configurations
from our method. Again, we calculate several observables for each ensemble
using the bootstrap re-sampling method and compare them to the HMC results. We
find that the size of the training dataset needs to be increased for
extrapolation in order to achieve a C-NF model that is close to the true
distribution in the broken phase. We plot the observables
$\langle\tilde{\phi}^{2}\rangle$ and $\chi$ in Figure 7 and the zero momentum
correlation function is plotted in Figure 8 for four different $\lambda$ in
the critical region .

(a) $\langle\tilde{\phi}^{2}\rangle$

(b) $\chi$

Figure 7:  Extrapolation to the critical region-:
$\langle\tilde{\phi}^{2}\rangle$ and $\chi$ are calculated on samples
generated from i)HMC, ii)C-NF followed by MH, and iii) Naive C-NF. The Error
bars indicate standard deviation calculated using bootstrapping re-sampling
with bin size 100.

(a) $\lambda=4.20$

(b) $\lambda=4.30$

(c) $\lambda=4.40$

(d) $\lambda=4.45$

Figure 8: Extrapolation:-Zero momentum Correlation function calculated on
samples generated from i)HMC, ii)C-NF followed by MH, and iii) Naive C-NF. The
Error bars indicate $95\%$ confidence interval calculated using bootstrapping
re-sampling method with bin-size 100.

The naive C-NF model produces configurations that are inherently uncorrelated.
Since we obtain a Markov Chain after applying MH, therefore we can’t guarantee
the same. With a $25-40\%$ acceptance rate from the model, we see negligible
correlation between the samples. The integrated autocorrelation time for
$\chi$ from our proposed approach and HMC is plotted in Figure 9. It shows
that we successfully reduce the Autocorrelation time for the Markov chain.The
comparison is not absolute because HMC is affected by algorithmic settings.
But we want to show that for the critical region, there is almost no
correlation between samples in the Markov Chains obtained from both
interpolation and extrapolation.

Figure 9: Integrated Auto-correlation time calculated on samples generated
from HMC and C-NF model with MH. The straight line represents the
autocorrelation time for uncorrelated samples.

## 5 Cost Analysis

The sampling algorithm for the baseline approach (HMC) and the suggested
method is vastly different; thus, a direct cost comparison is opaque.
Nonetheless, we separate the simulation cost for the proposed technique into
two components: Training time and sample generation time. On a Colab Tesla
P100 GPU, the training time for the C-NF model for interpolation or
extrapolation is roughly 5-6 hours. However, sample generation is very fast
for the C-NF model. Generating one Markov Chain of $10^{5}$ configuration
takes 5-7 minutes with a 25-40$\%$ acceptance rate. Due to the short
generation time, a low acceptance rate is acceptable until the autocorrelation
time increases.

Once the C-NF model has been trained, it can be employed repeatedly to
generate configurations for a wide range of $\lambda$ values. From a single
training of the C-NF(interpolated) model, we have generated configurations for
13 $\lambda$’s in the critical region. So, our approach outperforms HMC for
sampling at multiple $\lambda$ values in the critical region.

## 6 Conclusion

The critical slowing down problem prevents generating a large ensemble in the
critical region of a lattice theory. In order to resolve this, we employ a
Conditional normalizing flow trained on HMC samples with low autocorrelation
and generate samples in the critical region. In order to learn a general
distribution over parameter $\lambda$, we train the C-NF model away from the
critical point of lattice $\phi^{4}$ theory. This model is interpolated in the
critical region and serves as a proposal for the MH algorithm to generate a
Markov chain. The degree to which the extrapolated or interpolated model
resembles the actual distribution determines the acceptance rate in the
critical region. In order to achieve a high acceptance rate and prevent the
development of autocorrelation, the C-NF model must be trained adequately. The
C-NF model generates uncorrelated samples, and we trained well enough to get
25-45$\%$ acceptance rate. With this much acceptance rate, we found no
correlation between configuration in the Markov chain. As a result, our method
significantly mitigates the critical slowing down problem. Aside from that,
our method can be highly efficient when we need interpolation/extrapolation to
numerous $\lambda$ values in the critical region. Since lattice gauge theory
requires extrapolation to the critical region, we have likewise extrapolated
$\phi^{4}$ lattice theory to the critical region. We observe high agreement
between observables estimated using the suggested technique and HMC simulation
for both interpolation and extrapolation.

## References

  * [1] Simon Duane, Anthony D Kennedy, Brian J Pendleton, and Duncan Roweth.  Hybrid monte carlo.  Physics letters B, 195(2):216–222, 1987.
  * [2] Ulli Wolff.  CRITICAL SLOWING DOWN.  Nucl. Phys. B Proc. Suppl., 17:93–102, 1990.
  * [3] Stefan Schaefer, Rainer Sommer, and Francesco Virotta.  Critical slowing down and error analysis in lattice QCD simulations.  Nucl. Phys. B, 845:93–119, 2011.
  * [4] Alberto Ramos.  Playing with the kinetic term in the HMC.  PoS, LATTICE2012:193, 2012.
  * [5] Arjun Singh Gambhir and Kostas Orginos.  Improved sampling algorithms in lattice qcd, 2015.
  * [6] Michael G. Endres, Richard C. Brower, William Detmold, Kostas Orginos, and Andrew V. Pochinsky.  Multiscale monte carlo equilibration: Pure yang-mills theory.  Physical Review D, 92(11), Dec 2015.
  * [7] Kai Zhou, Gergely Endrődi, Long-Gang Pang, and Horst Stöcker.  Regressive and generative neural networks for scalar field theory.  Phys. Rev. D, 100:011501, Jul 2019.
  * [8] Dian Wu, Lei Wang, and Pan Zhang.  Solving statistical mechanics using variational autoregressive networks.  Phys. Rev. Lett., 122:080602, Feb 2019.
  * [9] Jan M Pawlowski and Julian M Urban.  Reducing autocorrelation times in lattice simulations with generative adversarial networks.  Machine Learning: Science and Technology, 1(4):045011, oct 2020\.
  * [10] Kim A. Nicoli, Shinichi Nakajima, Nils Strodthoff, Wojciech Samek, Klaus-Robert Müller, and Pan Kessel.  Asymptotically unbiased estimation of physical observables with neural samplers.  Physical Review E, 101(2), feb 2020.
  * [11] Juan Carrasquilla.  Machine learning for quantum matter.  Advances in Physics: X, 5(1):1797528, jan 2020.
  * [12] Junwei Liu, Huitao Shen, Yang Qi, Zi Yang Meng, and Liang Fu.  Self-learning monte carlo method and cumulative update in fermion systems.  Physical Review B, 95(24), jun 2017.
  * [13] Johanna Vielhaben and Nils Strodthoff.  Generative neural samplers for the quantum heisenberg chain.  Physical Review E, 103(6), jun 2021.
  * [14] Chuang Chen, Xiao Yan Xu, Junwei Liu, George Batrouni, Richard Scalettar, and Zi Yang Meng.  Symmetry-enforced self-learning monte carlo method applied to the holstein model.  Physical Review B, 98(4), jul 2018.
  * [15] Giacomo Torlai and Roger G. Melko.  Learning thermodynamics with boltzmann machines.  Phys. Rev. B, 94:165134, Oct 2016.
  * [16] G. Carleo and M. Troyer.  Solving the quantum many-body problem with artificial neural networks.  Science, 355:602, 2017.
  * [17] Lei Wang.  Discovering phase transitions with unsupervised learning.  Phys. Rev. B, 94:195105, Nov 2016.
  * [18] Pengfei Zhang, Huitao Shen, and Hui Zhai.  Machine learning topological invariants with neural networks.  Phys. Rev. Lett., 120:066401, Feb 2018.
  * [19] Japneet Singh, Mathias Scheurer, and Vipul Arora.  Conditional generative models for sampling and phase transition indication in spin systems.  SciPost Physics, 11(2), Aug 2021.
  * [20] Ankur Singha, Dipankar Chakrabarti, and Vipul Arora.  Generative learning for the problem of critical slowing down in lattice Gross Neveu model.  arXiv: 2111.00574, 2021.
  * [21] M. S. Albergo, G. Kanwar, and P. E. Shanahan.  Flow-based generative models for markov chain monte carlo in lattice field theory.  Phys. Rev. D, 100:034515, Aug 2019.
  * [22] Michael S. Albergo, Gurtej Kanwar, Sébastien Racanière, Danilo J. Rezende, Julian M. Urban, Denis Boyda, Kyle Cranmer, Daniel C. Hackett, and Phiala E. Shanahan.  Flow-based sampling for fermionic lattice field theories.  2021\.
  * [23] Phiala E Shanahan, Daniel Trewartha, and William Detmold.  Machine learning action parameters in lattice quantum chromodynamics.  Physical Review D, 97(9):094506, 2018.
  * [24] Gurtej Kanwar, Michael S. Albergo, Denis Boyda, Kyle Cranmer, Daniel C. Hackett, Sébastien Racanière, Danilo Jimenez Rezende, and Phiala E. Shanahan.  Equivariant flow-based sampling for lattice gauge theory.  Physical Review Letters, 125(12), Sep 2020.
  * [25] Michael S. Albergo, Denis Boyda, Daniel C. Hackett, Gurtej Kanwar, Kyle Cranmer, Sébastien Racanière, Danilo Jimenez Rezende, and Phiala E. Shanahan.  Introduction to normalizing flows for lattice field theory, 2021.
  * [26] Michael S. Albergo, Denis Boyda, Kyle Cranmer, Daniel C. Hackett, Gurtej Kanwar, Sébastien Racanière, Danilo J. Rezende, Fernando Romero-López, Phiala E. Shanahan, and Julian M. Urban.  Flow-based sampling in the lattice Schwinger model at criticality.  2 2022.
  * [27] Daniel C. Hackett, Chung-Chun Hsieh, Michael S. Albergo, Denis Boyda, Jiunn-Wei Chen, Kai-Feng Chen, Kyle Cranmer, Gurtej Kanwar, and Phiala E. Shanahan.  Flow-based sampling for multimodal distributions in lattice field theory.  7 2021.
  * [28] Ingmar Vierhaus.  Simulation of phi 4 theory in the strong coupling expansion beyond the ising limit.  Master’s thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2010.
  * [29] Danilo Jimenez Rezende and Shakir Mohamed.  Variational inference with normalizing flows, 2015.
  * [30] Lynton Ardizzone, Carsten Lüth, Jakob Kruse, Carsten Rother, and Ullrich Köthe.  Guided image generation with conditional invertible neural networks, 2019\.
  * [31] Lynton Ardizzone, Carsten Lüth, Jakob Kruse, Carsten Rother, and Ullrich Köthe.  Guided image generation with conditional invertible neural networks.  CoRR, abs/1907.02392, 2019.

## 7 Appendix

$\lambda$ | $\langle\tilde{\phi}^{2}\rangle$ | $\chi_{2}$
---|---|---
HMC |  C-NF with MH |  Naive C-NF |  HMC |  C-NF with MH |  Naive C-NF
4.10 |  $0.2147\pm 0.0179$ |  $0.2169\pm 0.0169$ |  $0.1586\pm 0.0205$ |  $13.7446+\pm 0.8059$ |  $13.7446\pm 0.8059$ |  $10.1557\pm 0.9181$
4.20 |  $0.1925\pm 0.0181$ |  $0.1970\pm 0.0173$ |  $0.1463\pm 0.0199$ |  $12.3272\pm 0.8103$ |  $12.6235\pm 0.7822$ |  $9.3694\pm 0.8844$
4.25 |  $0.1819\pm 0.0090$ |  $0.1872\pm 0.0087$ |  $0.1411\pm 0.0097$ |  $11.6485\pm 0.5747$ |  $11.9992\pm 0.5566$ |  $9.0395\pm 0.6191$
4.30 |  $0.1724\pm 0.0181$ |  $0.1781\pm 0.0176$ |  $0.1348\pm 0.0190$ |  $11.0331\pm 0.8188$ |  $11.3775\pm 0.7762$ |  $8.6260\pm 0.8674$
4.35 |  $0.1622\pm 0.0177$ |  $0.1681\pm 0.0174$ |  $0.1289\pm 0.0184$ |  $10.3827\pm 0.5648$ |  $10.7440\pm 0.5640$ |  $8.2559\pm 0.5894$
4.40 |  $0.1530\pm 0.0178$ |  $0.1583\pm 0.0172$ |  $0.1225\pm 0.0182$ |  $9.7920\pm 0.5529$ |  $10.1284\pm 0.5541$ |  $7.8501\pm 0.5867$
4.45 |  $0.1446\pm 0.0173$ |  $0.1490\pm 0.0170$ |  $0.1166\pm 0.0176$ |  $9.2557\pm 0.5440$ |  $9.5575\pm 0.5516$ |  $7.4716\pm 0.5637$
4.50 |  $0.1357\pm 0.0172$ |  $0.1399\pm 0.0170$ |  $0.1115\pm 0.0175$ |  $8.7005\pm 0.5446$ |  $8.9528\pm 0.5442$ |  $7.1308\pm 0.5479$
4.60 |  $0.1197\pm 0.0163$ |  $0.1227\pm 0.0163$ |  $0.0996\pm 0.0165$ |  $7.6629\pm 0.5272$ |  $7.8601\pm 0.5188$ |  $6.3753\pm 0.5271$
4.65 |  $0.1127\pm 0.0157$ |  $0.1151\pm 0.0162$ |  $0.0939\pm 0.0159$ |  $7.1991\pm 0.5041$ |  $7.3486\pm 0.5126$ |  $6.0062\pm 0.5109$
4.70 |  $0.1054\pm 0.0153$ |  $0.1069\pm 0.0157$ |  $0.0879\pm 0.0154$ |  $6.7575\pm 0.4978$ |  $6.8422\pm 0.5075$ |  $5.6177\pm 0.4966$
4.8 |  $0.0931\pm 0.0148$ |  $0.0931\pm 0.0148$ |  $0.0769\pm 0.0140$ |  $5.9530\pm 0.4717$ |  $5.9569\pm 0.4671$ |  $4.9223\pm 0.4590$
5.0 |  $0.0733\pm 0.0128$ |  $0.0704\pm 0.0128$ |  $0.0593\pm 0.0121$ |  $4.6935\pm 0.4131$ |  $4.4963\pm 0.4180$ |  $3.7920\pm 0.3886$
Table 1:  Two observables calculated on samples generated from i) HMC ii) C-NF with MH and iii) Naive C-NF .The Error indicates standard deviation calculated using bootstrapping re-sampling with bin-size 100. $\lambda$ |  4.1 |  4.2 |  4.3 |  4.4 |  4.5 |  4.6 |  4.7 |  4.8 |  4.9 |  5.0
---|---|---|---|---|---|---|---|---|---|---
$\tau_{int}$ |  1.033 |  1.084 |  1.071 |  1.059 |  1.048 |  1.039 |  1.041 |  1.018 |  1.030 |  1.037
Table 2:  Integrated Autocorrelation time for $\langle\tilde{\phi}^{2}\rangle$ from the interpolated C-NF model after applying MH.  $\lambda$ |  4.2 |  4.3 |  4.4 |  4.5 |  4.6
---|---|---|---|---|---
$\tau_{int}$ |  1.046 |  1.049 |  1.027 |  1.032 |  1.010
Table 3:  Integrated Autocorrelation time for $\langle\tilde{\phi}^{2}\rangle$ from the extrapolated C-NF model after applying MH.  $\lambda$ | $\langle\tilde{\phi}^{2}\rangle$ | $\chi_{2}$
---|---|---
HMC |  C-NF with MH |  Naive C-NF |  HMC |  C-NF with MH |  Naive C-NF
4.20 |  $0.1925\pm 0.0181$ |  $0.1966\pm 0.0170$ |  $0.1676\pm 0.0189$ |  $12.3272\pm 0.8103$ |  $12.5670\pm 0.5490$ |  $10.7348\pm 0.5989$
4.30 |  $0.1724\pm 0.0181$ |  $0.1756\pm 0.0171$ |  $0.1509\pm 0.0177$ |  $11.0331\pm 0.8188$ |  $11.2388\pm 0.5571$ |  $9.6593\pm 0.5868$
4.40 |  $0.1530\pm 0.0178$ |  $0.1562\pm 0.0171$ |  $0.1348\pm 0.0173$ |  $9.7920\pm 0.5529$ |  $9.9943\pm 0.5506$ |  $8.6316\pm 0.5510$
4.50 |  $0.1357\pm 0.0172$ |  $0.1393\pm 0.0167$ |  $0.1200\pm 0.0164$ |  $8.7005\pm 0.5446$ |  $8.9094\pm 0.5293$ |  $7.6643\pm 0.5342$
4.60 |  $0.1197\pm 0.0163$ |  $0.1233\pm 0.0157$ |  $0.1075\pm 0.0158$ |  $7.6629\pm 0.5272$ |  $7.8881\pm 0.5143$ |  $6.8783\pm 0.4992$
Table 4:  Two observables calculated on samples generated from i)HMC ii)C-NF
with MH and iii) Naive C-NF .The Error bars indicates standard deviation
calculated using bootstrapping re-sampling with binsize 100.

(a) $\lambda=4.2$

(b) $\lambda=4.4$

(c) $\lambda=4.5$

(d) $\lambda=4.6$

(e) $\lambda=4.8$

(f) $\lambda=5.0$

Figure 10: Interpolation-:Zero momentum Correlation function calculated on
samples generated from i)HMC, ii)C-NF followed by MH, and iii) Naive C-NF. The
Error bars indicate $95\%$ confidence interval calculated using bootstrapping
re-sampling method with bin-size 100.