# Machine learning the deuteron: new architectures and uncertainty
quantification

J Rozalén Sarmiento1,2, J W T Keeble3, A Rios1,2,3 1 Departament de Física
Quàntica i Astrofísica, Universitat de Barcelona (UB), c. Martí i Franquès 1,
E08028 Barcelona, Spain  2 Institut de Ciències del Cosmos (ICCUB),
Universitat de Barcelona (UB), Barcelona, Spain  3 Department of Physics,
University of Surrey, Guildford GU2 7XH, United Kingdom<EMAIL_ADDRESS>
###### Abstract

We solve the ground state of the deuteron using a variational neural network
ansatz for the wave function in momentum space. This ansatz provides a
flexible representation of both the $S$ and the $D$ states, with relative
errors in the energy which are within fractions of a percent of a full
diagonalisation benchmark. We extend the previous work on this area in two
directions. First, we study new architectures by adding more layers to the
network and by exploring different connections between the states. Second, we
provide a better estimate of the numerical uncertainty by taking into account
the final oscillations at the end of the minimisation process. Overall, we
find that the best performing architecture is the simple one-layer, state-
independent network. Two-layer networks show indications of overfitting, in
regions that are not probed by the fixed momentum basis where calculations are
performed. In all cases, the error associated to the model oscillations around
the real minimum is larger than the stochastic initialisation uncertainties.
The conclusions that we draw can be generalised to other quantum mechanics
settings.

##  1 Introduction

Artificial Neural Networks (ANNs) have become routine computational tools in a
wide range of scientific domains [1, 2]. Nuclear physics is no exception, and
the number of machine learning (ML) tools is increasing steadily [3, 4, 5, 6,
7, 8, 9]. An interesting and relatively recent use of ANNs is the solution of
quantum mechanical problems, including in the many-body domain [10, 11]. These
methods use ANNs as an ansatz for the many-body wave function, and employ
ready-made ML tools to minimise the energy of the system [12, 13]. The first
attempts in condensed matter [14, 15] were quickly followed by advances both
within [16] and outside the field [5], including quantum chemistry [17, 18]. A
good example in this direction is FermiNet [19], which uses an inherently
antisymmetric wave function as an ansatz in a variational approach to solve
the many-electron problem.

The methodology that we use to solve the deuteron here is based on the same
variational philosophy and stems from Ref. [20]. In that case, we used a
minimal, one-layer variational ANN ansatz to describe the wave function of the
deuteron (the only $2-$body nuclear bound state) in momentum space. This
yields excellent results in comparison with benchmark solutions for the
energies and the wave functions, quantified through fidelity measures.
Moreover, we estimated the out-of-sample uncertainty of the model by using
several random initialisations in the minimisation problem.

The assessment of uncertainties in variational ANN (vANN) is particularly
relevant to pinpoint the fundamental limitations of this methodology. There
are several systematic errors that are important for many-body vANNs. For
instance, network architectures play a key role in determining the convergence
properties of the network. Similarly, for a fixed architecture, vANN widths
and depths change the expressivity of the network, and are thus fundamental
ingredients in providing faithful representations of continuous wave
functions. One should explore this systematic uncertainties extensively before
reaching conclusions about any intrinsic shortcomings of the vANN method
itself.

The deuteron is an excellent test bed for such studies. It is a relatively
simple, one-body-like problem, which already includes the complexity of the
strong force. Moreover, dealing with a wave function with two different
angular momentum states adds an additional level of sophistication. In this
manuscript, we extend the elementary approach of Ref. [20] in two different
directions. First, we look at somewhat more complex ANN architectures to
describe the wave function. This yields insight on the importance of the
network architecture and should ultimately provide information on the
limitations of the method beyond a specific architecture. Second, we introduce
a novel way to compute the out-of-sample error. In particular, we look at
oscillations in the energy, which is our cost function, around the real
minimum. These oscillations are an additional source of uncertainty,
associated to the minimisation process. The analysis of such errors provides a
new insight into uncertainty quantification for variational ANN solvers.

This paper is structured in the following manner. In Section 2 we briefly go
over the methods used to approach the deuteron problem with ANNs. Section 3 is
devoted to the analysis of the numerical results, including their
uncertainties. Finally, in Section 4, we draw a series of conclusions and
comment on ways in which our work could be expanded and improved.

##  2 Methodology

The approach that we use to solve the problem is variational. The ANN plays
the role of a wave function ansatz, which we denote
$\ket{\psi_{\mathrm{ANN}}^{\mathcal{W}}}$.
$\mathcal{W}=\\{\mathbf{W^{(1)}}$,$\mathbf{W^{(2)}}$,$\mathbf{W^{(3)}}$,$\mathbf{b}\\}$
is a set formed by network weights, $\mathbf{W^{(i)}}$, and biases
$\mathbf{b}$. In a variational setting, the total energy of the system is the
loss function which reads

$\displaystyle
E^{\mathcal{W}}=\frac{\braket{\psi_{\mathrm{ANN}}^{\mathcal{W}}}{\hat{H}}{\psi_{\mathrm{ANN}}^{\mathcal{W}}}}{\braket{\psi_{\mathrm{ANN}}^{\mathcal{W}}}{\psi_{\mathrm{ANN}}^{\mathcal{W}}}}.$
(1)

The deuteron is the simplest nuclear two-body problem. We solve its structure
in momentum space, where we can trivially separate the center of mass from the
relative coordinates [21]. Consequently, the wave function of the system
depends only on the relative momentum coordinate, $\vec{q}$. Working in
momentum space allows us to skip numerically costly derivatives on the wave
functions when computing the kinetic energy.

We can further reduce the dimensionality of the deuteron problem via a partial
wave expansion, thus separating the dependence on the absolute value of the
momentum $q$ from its dependence on angles. For the deuteron, the tensor
component of the strong interaction admixes the $S$ ($L=0$) and the $D$
($L=2$) components of the ground-state wave function. The network will
consequently have two different outputs, one for each state. The potential
energy term used to compute Eq. (1) mixes these two states in a non-trivial
way, so they are not completely independent from each other in the variational
setting.

To implement the problem computationally, we use a one-dimensional grid in $q$
with $N_{q}=64$ points. This grid is used to compute the energy integrals in
Eq. (1) via quadrature. The loss function is obtained from a global integral,
the energy, and is always computed with the same set of momentum grid points.
As we shall see later, this can create some issues in deep models. To
efficiently capture the low-momentum structure and the high-momentum tails of
the wave function, we use a mesh that densely covers the region of low momenta
but is sparse in regions of high momenta. We first distribute $N$ Gauss-
Legendre points $x_{i}$ between $0$ and $1$, and then extend them tangentially
using the transform
$q_{i}=\dfrac{q_{\mathrm{max}}}{\tan\dfrac{\pi}{2}x_{N}}\tan\dfrac{\pi}{2}x_{i}\,,$
with $i=1,\cdots,N_{q}$ and $q_{\mathrm{max}}=500$ fm-1. In this set-up, the
overlap $\braket{\psi_{\text{targ}}^{L}}{\psi_{\text{ANN}}^{L}}$ , as well as
all analogous integrated quantities, are discretised as follows:

$\displaystyle\braket{\psi_{\text{targ}}^{L}}{\psi_{\text{ANN}}^{L}}=4\pi\int_{0}^{\infty}\\!\\!\\!dq\,q^{2}\,\psi_{\text{targ}}^{L*}(q)\psi_{\text{ANN}}^{L}(q)\approx
4\pi\sum_{i=1}^{N_{q}}w_{i}q_{i}^{2}\psi_{\text{targ}}^{L*}(q_{i})\psi_{\text{ANN}}^{L}(q_{i})\,,$
(2)

where $w_{i}$ are the integration weights associated to the tangentially-
transformed Gauss-Legendre mesh. Note that because the angular dependence has
been removed via a partial wave expansion, all the integrals involve a radial
$q^{2}$ term. Note also that all the wave functions are real-valued.

In the following, we present results corresponding to different ANN
architectures for the two wave functions, $\psi_{\text{ANN}}^{L}(q)$, with
$L=0$ and $2$. Following Ref. [20], we take three identical steps to train all
the networks. These three steps are common to most variational Artificial
Neural Network (vANN) problems in quantum mechanics [15]. First, we initialise
network parameters to uniform random values in the ranges
$\mathbf{W^{(1)}}\in[-1,0)$, $\mathbf{W^{(2)}}\in[0,1)$ and
$\mathbf{b}\in[-1,1)$. We choose a sigmoid activation function, $\sigma(x)$,
which provided marginally better results than the softplus function in Ref.
[20].

Second, we train the ANN to mimic a target function that bears a certain
degree of physical meaning. We choose a functional form
$\psi^{L}_{\mathrm{targ}}(q)=q^{L}e^{-\frac{\xi^{2}q^{2}}{2}}$, which has the
correct low-momentum asymptotic values and a reasonable real-space width,
$\xi=1.5\,\mathrm{fm}$. In this pre-training step, we use the overlap

$\displaystyle
K^{L}=\dfrac{\braket{\psi_{\text{targ}}^{L}}{\psi_{\text{ANN}}^{L}}^{2}}{\braket{\psi_{\text{targ}}^{L}}{\psi_{\text{targ}}^{L}}\braket{\psi_{\text{ANN}}^{L}}{\psi_{\text{ANN}}^{L}}}$
(3)

to compute a total cost function, $C$, defined as

${}C=(K^{S}-1)^{2}+(K^{D}-1)^{2}\,.$ (4)

Equation (4) is zero if, and only if, the two overlaps $K^{S}=K^{D}=1$. We
choose the optimiser RMSProp [1], which dynamically adapts a global learning
locally for all the network parameters. Training a medium-sized ANN with
$N_{\mathrm{hid}}\approx 20$ hidden nodes usually takes about $10^{4}$
iterations. We point out that this pretraining step is not strictly necessary
to achieve a correct energy minimisation, but it helps in guaranteeing a
successful energy minimisation with fewer epochs.

In the third and final step, we define a new cost function: the energy, as
given in Eq. (1), and we computed it with the momentum quadrature described
above. The network is then trained to minimise $E^{\mathcal{W}}$ for
$2.5\times 10^{5}$ epochs. We note that the kinetic energy has diagonal
contributions in the $L=0$ and $2$ states, but the potential term mixes both
states. We choose the N3LO Entem-Machleidt nucleon-nucleon force [22] which is
easily implemented in our code as a momentum-dependent potential. We use the
PyTorch library [23, 24], which is particularly useful due to its automatic
differentiation capabilities [25]. A full version of the code is available on
GitHub [26].

###  2.1 Architectures

The vANN procedure is relatively general, and can be applied to any feed-
forward ANN. In the initial exploration of deuteron wave functions of Ref.
[20], a minimal approach was chosen deliberately to explore the potential and
the limitations of the method in the simplest possible model. We used a
single-layer ANN with an increasing number of hidden nodes in order to assess
systematics for a fixed architecture. Here, we take a step further and assess
the impact of increasing the depth of the ANN which leads to different
architectures.

(a) State-connected network with a single hidden layer. For this network, the
number of parameters is $N=4N_{\mathrm{hid}}$, with $N_{\mathrm{hid}}$
denoting the number of hidden nodes. Note that the first layer includes a
bias.

(b) State-connected network with two hidden layers, with
$N=\frac{1}{8}N_{\text{hid}}(N_{\text{hid}}+12)$ parameters.

(c) State-disconnected network with a single hidden layer, with
$N=3N_{\mathrm{hid}}$ parameters.

(d) State-disconnected network with two hidden layers, with
$N=\frac{1}{4}N_{\mathrm{hid}}\left(N_{\mathrm{hid}}+8\right)$ parameters.

Figure 1: Neural network architectures used in this work.

To this end, we introduce four different ANN architectures, which we show in
the four panels of Fig. 1. All our networks have a single input, $q$, and two
outputs, $\ket{\Psi^{S}_{\text{ANN}}}$ and $\ket{\Psi^{D}_{\text{ANN}}}$. We
distinguish between different configurations depending on how they connect to
the two final states. On the one hand, in fully-connected networks, which we
dub “state-connected” (sc) networks, all the parameters contribute to both
outputs (see the top panels, Figs. 1(a) and 1(b)). On the other hand, “state-
disconnected” (sd) networks (bottom panels, Figs. 1(c) and 1(d)) provide
independent parameters for each state. One may naively expect sd networks to
provide more flexibility (and hence better variational properties) for the two
wave functions.

In addition to the state dependence of the network, we also explore
architecture systematics in terms of depth. We implement both sc and sd
networks with one (left panels) and two (right panels) layers. In the
following, we use the acronyms $n$sc and $n$sd ($n=1,2$), with $n$ the number
of layers, to refer to the 4 architectures explored in this work. We stress
that in Ref. [20] we only looked at the $1$sc architecture. The motivation to
explore other architectures and, in particular further depths, is the widely
held belief that deep ANNs outperform their shallow single-layered
counterparts [27].

In the initial layer of all four architectures, we introduce both weights
$\mathbf{W^{(1)}}$ and biases $\mathbf{b}$. In contrast, all the second
(and/or third) layers only have weights, $\mathbf{W^{(2)}}$
($\mathbf{W^{(3)}}$). For clarity and completeness, we provide the functional
form of the four networks. For the one-layer $1$sd configurations, the wave
function ansatz reads as

$\displaystyle{}\psi^{L}_{\mathrm{ANN}}(q)=\sum^{N_{\mathrm{hid}}/2}_{i=1}W^{(2)}_{i,L}\sigma\left(W^{(1)}_{i,L}q+B_{i,L}\right).$
(5)

The corresponding $1$sc contribution is obtained by setting
$N_{\mathrm{hid}}/2\rightarrow N_{\mathrm{hid}}$ and using $L-$independent
$\mathbf{W^{(1)}}$ and $\mathbf{B}$ parameters. When $2$ layers are included,
in contrast, the sd ansatz is more complex, and becomes the sum of sums of two
nested activation functions,

$\displaystyle{}\psi_{\mathrm{ANN}}^{L}(q)=\sum_{i=N_{\mathrm{hid}}/2+1}^{N_{\mathrm{hid}}}W^{(3)}_{i,L}\sigma\left(\sum_{j=1}^{N_{\mathrm{hid}}/2}W^{(2)}_{j,L}\sigma\left(W_{j,L}^{(1)}q+B_{j,L}\right)\right).$
(6)

One can again easily manipulate the previous expression to obtain a $2$sd wave
function.

The relation between the total number of hidden neurons, $N_{\mathrm{hid}}$,
and the total number of variational parameters, $N$, is different for each
architecture. The relations $N(N_{\mathrm{hid}})$ are shown in the captions of
Fig. 1 for each network configuration. One-layer networks have a linear
$N(N_{\mathrm{hid}})$ relation, whereas the relation for two-layer networks is
quadratic. In other words, for the same $N_{\text{hid}}$, two-layer networks
will usually involve a much larger number of parameters than one-layer models,
and one may expect overfitting to become an issue. Whereas for the $1$sc
architecture the number of hidden nodes is unrestricted so long as
$N_{\text{hid}}>2$, the $2$sc architecture must have $N_{\text{hid}}>4$ with
$N_{\text{hid}}$ even. Likewise, the $1$sd network must have an even
$N_{\text{hid}}$. In contrast, for the $2$sd network, $N_{\text{hid}}$ must
increase $4$ units at a time.

###  2.2 Learning process

The results we present are obtained with the four network configurations and a
sigmoid activation function. We minimise the energy cost function using
RMSProp with hyperparameters $\epsilon=10^{-8}$ and a momentum $\beta=0.9$. We
set the learning rate to $10^{-2}$ and explore the $N_{\mathrm{hid}}$
dependence using the values $N_{\mathrm{hid}}\in\\{20,30,40,60,80,100\\}$ (in
the 2sc architecture we change $N_{\mathrm{hid}}=30\rightarrow
N_{\mathrm{hid}}=32$). To explore the full flexibility of the wave function
ansätze, we optimise the networks by minimising the overlap loss function for
$2\times 10^{3}$ epochs of pre-training, followed by $2.5\times 10^{5}$ epochs
of energy minimisation. Rather than doing this a single time, we use $20$
different random initialisations and display mean values and standard
deviations obtained with all these runs111This is to be compared to the $50$
runs shown for the $1$sc architecture in Ref. [20].. With this, we explore the
out-of-sample bias of the network and we attempt to draw generic conclusions
for the architecture network, rather than for a single, specific network
model.

We typically perform more than $20$ initialisation runs, with the exact number
depending on the architecture and $N_{\text{hid}}$. Not all of these runs
converge or, if they do, they may not have converged to meaningful values,
close enough to the minimum. Figure 2 shows the rate of convergence of
different architectures as a function of $N_{\text{hid}}$. This is obtained as
the ratio of the number of converged models, $N_{\text{con}}$, to the total
set of model initialisations, $N_{\text{tot}}$,
$r=N_{\text{con}}/N_{\text{tot}}$. As a selection criterion to compute
$N_{\text{con}}$, we define converged models as those that provide a final
energy within the range $E\in(-2.220,-2.227)$ MeV. Whenever more than $20$
initialisations lead to converged results, we randomly pick $20$ states to
have representative, but also homogeneous, statistics across the
$N_{\mathrm{hid}}$ domain. We stress the fact that these rates are obtained
from initialisations with a pretraining step. With no pretraining, all models
show slower convergence rates222This includes the $1$sc configuration, which
leads to a perfect convergence rate if pretraining is included.. We take this
difference as an indication of the fact that pretraining is effective in
providing a physical representation of the ANN wave function. Initally non-
pretrained networks may occasionally lead to accurate results, but are likely
to require many more energy minimisation epochs.

Figure 2: Convergence rate of the four network architectures as a function of
the number of hidden neurons. The rate (vertical axis) is defined as the ratio
between the number of models that converge and the total number of trained
models, $N_{\text{con}}/N_{\text{tot}}$.

Going back to Fig. 2, we find that the one-layer configurations provide better
convergence rates than their two-layer counterparts in almost all the range of
$N_{\mathrm{hid}}$. The $1$sc network (solid line) has a perfect convergence
rate. The $1$sd architecture (dash-dotted line) provides a relatively good
convergence rate, with $r>70\%$ relatively independent of the value of
$N_{\text{hid}}$. We attribute these similarities to the idea that both
pretraining and training are more easily done with a single layer ANN.

In contrast, two-layer networks start with a relatively low convergence rate.
In architecture $2$sc we have $r\approx 20\%$ for $N_{\text{hid}}=20$, and the
rate increases in a relatively steady and linear fashion up to $r\approx
80-95\%$ for $N_{\text{hid}}=100$. In architecture $2$sd we find even lower
convergence rates, $r\approx 2\%$ for $N_{\text{hid}}=20$, and an erratic
behaviour in terms of $N_{\text{hid}}$. The stark differences in convergence
rates between one and two-layer ANNs may be due to a variety of factors. On
the one hand, network parameter initialisation may be an issue. Network
parameters in different domains than the ones we prescribe at the moment may
provide better starting points. On the other hand, we fix the total number of
energy minimisation epochs and the current rate may just be an indication that
smaller two-layer networks simply take, with RMSprop, a larger number of
epochs to converge. Finally, two-layer networks may be problematic in terms of
the sigmoid activation function, which is prone to a vanishing gradients issue
[28, 29]. This should be accentuated in deeper (as opposed to shallow one-
layer) networks. Along these lines, having few neurons increases the
probability that all of them become frozen, which may explain why low
$N_{\text{hid}}$ models have a harder time converging. An activation function
like softplus or ReLU may work better with $2$-layer architectures. Initial
tests with the softplus function do in fact indicate an improvement in
convergence rates for $2-$layer networks.

###  2.3 Error analysis

In addition to different network architectures, we examine two types of
uncertainties. First, as we have just explained, we run $20$ minimisations for
different $N_{\text{hid}}$ values and network architectures. We take the
standard deviation of these $20$ results as a measurement of the uncertainty
related to the stochastic minimisation process. This out-of-sample error,
which was also explored in Ref. [20], is represented in terms of dark bands in
the figures of the following section. It tends to be a relatively small error,
of the order of a fraction of a keV in energy.

Second, even after a full minimisation, our results still show some residual
oscillations. These oscillations are typically within $1\%$ of the total
energy. If we assume that the oscillations happen around the minimum value of
$E^{\mathcal{W}}$, we can estimate the true energy minimum by averaging over
these oscillations. The oscillation amplitude can then be used to quantify an
additional source of uncertainty, associated to the minimiser. To determine
this error, we take a trained model and let it evolve for $300$ additional
epochs - a process which we call post-evolution. The number of post-evolution
epochs is small enough so that the mean value does not change during the
process, but also large enough to observe periodicity in the oscillations.

Figure 3: Top panel: D-state fidelity post-evolution of a trained model with
architecture $1$sd and $N_{\mathrm{hid}}=40$ hidden neurons. Bottom panel:
total energy evolution of the same model as in the top panel.

We illustrate this behaviour in Fig. 3. The top panel shows the evolution of
the $D-$state fidelity $F^{D}$ over the $300$ post-evolution epochs for a
$1$sd network with $N_{\text{hid}}=40$. Here, we define $F^{L}$ as the overlap
between the ANN and benchmark (obtained via exact diagonalisation) wave
functions in analogy to Eq. (3). The solid blue line, corresponding to the
fidelity at every epoch, displays clear oscillations that are asymmetric with
respect to a mean value of $F^{D}\approx 0.99998$. To account for the
asymmetry in the oscillating behaviour, we assign an upper and lower error
estimate. The upper value corresponds to the maximum value across the post-
evolution phase - $\delta F^{D}_{+}=0.00001$ (dash-dotted line), in the case
shown in the figure. In contrast, the lower value is significantly lower for
this specific example, leading to $\delta F^{D}_{-}=0.00002$ (dotted line).
The bottom panel of Fig. 3 shows the energy (solid line) as a function of the
post-evolution epoch. The energy oscillates around the mean value (dashed
line), and the top and bottom bounds of this oscillations are shown in dash-
dotted and dotted lines respectively. In this particular example, the mean
value of the energy is $E^{\mathcal{W}}\approx-2.225\,\mathrm{MeV}$, with
oscillation errors $\delta E^{\mathcal{W}}_{+}=0.4\,\mathrm{keV}$ and $\delta
E^{\mathcal{W}}_{-}=0.4\,\mathrm{keV}$. We note that these values are typical
for almost any number of neurons, and are extremely small, less than $0.1\%$
of the total energy value.

In order to take into account the stochastic out-of-sample uncertainty in
these oscillations, we repeat the process above $20$ times for each network
configuration ($N_{\mathrm{hid}}$). As explained earlier, we extract central
values for the different physical quantities by taking the average over the
$20$ mean values. We quote post-evolution oscillation errors that also include
this stochastic element. In other words, both upper and lower bounds are
calculated by averaging over 20 individual post-evolution runs. In a cost
function with no near multiple minimums of similar heights, all oscillations
should happen around the same minimum, and the mean values of
$E^{\mathcal{W}}$ should all be similar. This, in tandem with the fact that we
estimate the stochastic $\delta E^{\mathcal{W}}$ as the standard deviation,
leads us to expect small stochastic errors and comparatively bigger
oscillation errors. We confirm this expectation in the following section.

##  3 Results

###  3.1 Energy

Figure 4:  Top panels: energy as a function of the number of hidden nodes.
Different panels correspond to the network architectures of Fig. 1. Lines
represent central values obtained with $20$ stochastic initialisations. Dark
bands show the associated standard deviations (stochastic uncertainty),
whereas light bands show the post-evolution oscillation uncertainty. The
dashed red lines is the benchmark value. Bottom panels: the same for the
fidelity of the ANN wave function with respect to the benchmark for the $S$
(purple) and $D$ (green) states.

The two quantities that we use as indicators of the quality of our final
trained models are the energy and the fidelity, represented in the top and
bottom panels of Fig. 4, respectively. The four columns correspond to the the
different network architectures of Fig. 1. As explained earlier, the solid
dark lines in Fig. 4 indicate the central values. We show two different types
of uncertainties for each quantity. Firstly, stochastic uncertainties
associated to $20$ different initialisations are shown using dark colour
bands. Secondly, post-evolution oscillation uncertainties are displayed with
pale colours. A key finding of our work is that post-evolution uncertainties
are always larger than stochastic uncertainties.

Before discussing uncertainties, we want to stress the quality of the results.
As discussed in the context of Fig. 2, the ANNs used to generate these results
have been preselected to lie in the range $E\in(-2.220,-2.227)$ MeV.
Therefore, all these models already lie within $0.3\%$ of the final energy
result. For some architectures, the rate of convergence to this energy window
is not $100\%$. Yet, converged results provide energy values which, according
to the stochastic uncertainty, are well within $1$ keV (or $0.04\%$) of each
other. Post-evolution uncertainties are much larger than stochastic
uncertainties, but are typically smaller than $4$ keV (or $0.2\%$). We take
this as an indication that all these ANNs provide faithful, high-quality
representations of deuteron wave function. While none of the models shown here
are able to reach the real minimum even when the post-evolution lower bounds
are considered, the distance between the lower bound and the real minimum is
always less than $2$ keV. Our analysis indicates that this limitation is due
to the fact that the network cannot always distinguish between the $S-$ and
$D-$state contributions in high-momentum regions, where the two wave functions
are equally small. Ultimately, this limitation only has minor consequences, of
a fraction of a percent, in the total energy. As an important conclusion of
this analysis, we find that the simplest architecture, $1$sc, provides the
more stable and overall best performing results.

An interesting conclusion of Fig. 4 is that state-disconnected architectures
perform seemingly worse than state-connected networks. One could have expected
the opposite, on the basis that disconnected architectures may provide more
independent flexibility for the $\psi^{S}$ and $\psi^{D}$ wave functions. One
possible explanation for this behaviour is as follows. Within a single
minimisation epoch, the optimiser proposes steps based on the backpropagation
of gradients. Changes in parameters associated to the $S-$state are thus
immediately propagated to the $D$ state in a fully connected configuration. In
contrast, in the disconnected case, these changes do not necessarily affect
the $D$ state. In spite of having less parameters than connected
architectures, the training may take longer due to this effect.

We now proceed to discuss the $N_{\text{hid}}$ dependence of the results. We
find two very different trends depending on the depth of the network. One-
layer network results are relatively independent on the network width. In
other words, in terms of energy (but also in terms of fidelity), models with
$N_{\text{hid}}\approx 30$ are as good as models with $N_{\text{hid}}=100$. In
contrast, the two-layer models display a clear improvement in energy for small
values of $N_{\mathrm{hid}}$. This suggests that two-layer networks with small
values of $N_{\mathrm{hid}}$ have not managed to reach the full minimum after
$2.5\times 10^{5}$ epochs. Single-layer models have notably less parameters
and can thereby be trained faster than their two-layer counterparts. In fact,
snapshots of the minimisation process indicate that two-layer models with few
neurons are still learning at the end of the $2.5\times 10^{5}$ epochs. The
decrease in energy as $N_{\mathrm{hid}}$ increases extends to the whole range
of $N_{\mathrm{hid}}$ in architecture $2$sc and $2$sd. The post-evolution
uncertainties for these two-layer models are relatively big and almost
compatible with a constant value of energy.

In addition to central values, the dependence of uncertainties in
$N_{\text{hid}}$ is informative. The stochastic uncertainties are relatively
constant across all values, but are marginally larger for two-layer models.
Post-evolution errors for one-layer networks are on the order of $\approx 2$
keV, with a mild decreasing dependence on network width. In contrast, two-
layer network post-evolution uncertainties are as large as $6$ ($4$) keV for
low (high) $N_{\text{hid}}$ values. Following the same argument about the
energies of such models, this can be understood by realising that the updates
in $\mathcal{W}$ from RMSProp affect both $\psi^{S}$ and $\psi^{D}$ in the
state-connected case (as opposed to the state-disconnected case). One expects
this may lead to larger energy variations and, hence, bigger oscillations.

The difference in oscillation errors can also be understood in terms of frozen
neurons. Two-layer models that converge will not have many frozen neurons, and
given that such models have comparatively many more parameters than their one-
layer counterparts, the changes in $\mathcal{W}$ are expected to be
comparatively bigger. This, when the minimum of the oscillation is not deep
enough, can cast the model away from the minimum which is reflected in large,
irregular oscillations. The argument above also explains why the stochastic
errors are comparatively bigger in two-layer models: big, non-regular
oscillations are prone to yield different central values, which directly
translates to bigger stochastic errors.

###  3.2 Fidelity

The bottom panels of Fig. 4 also provide an insight on the model quality after
minimisation. Again, we highlight that overall the fidelities are extremely
close to benchmark values, within a fraction of a percent in all cases. We
find general conclusions in terms of network width that are in line with those
we found for the energy. Across all architectures and network depths, the
fidelity of the $S-$state is better than that associated to the $D$-state. Not
only are the central values of $F^{S}$ closer to one, but post-evolution
uncertainties are significantly smaller for this state. This is in contrast to
the much smaller stochastic uncertainty, which is of the same size for both
states333The only exception is the $D-$state fidelity of the $2$sd
architecture (bottom right panel), which has a larger oscillation uncertainty.
.

When it comes to the $N_{\text{hid}}$ dependence of the results, we find again
that one-layer networks are relatively independent of the network width.
Overall, networks perform better with $1$sc than with $2$sc architectures as
measured by overlaps that are closer to one, although they have relatively
similar uncertainties. The fidelities of the $1$sd architecture improve as the
width increases. Again, we interpret this result in terms of incomplete
learning for networks with fewer parameters. The behaviour of the $2$sd
networks (rightmost panels) is more erratic. The $S-$state fidelity is
relatively close to the benchmark and seemingly improves with
$N_{\text{hid}}$. In contrast, the $D-$state fidelity is closer to one for
$N_{\text{hid}}=80$, and subsequently its quality decreases. We take this as a
sign that such architectures with $N_{\text{hid}}$ values larger than $80$ may
be problematic and, in fact, we show in the following subsection that
overfitting is an issue in this region. Finally, we stress again that there is
no observable bias-variance trade-off in the fidelities.

###  3.3 Wave functions

Figure 5:  Top panels: $20$ instances of the $S-$state wave function as a
function of momentum, for each of the 4 architectures considered here for
$N_{\text{hid}}=100$. The dashed line represents the benchmark wave function.
The insets focus on the region around the origin, $q\approx 0$. Bottom panels:
the same for the $D-$state wave function. Note the difference in scales of the
momentum in both rows.

We now turn our attention to analysing directly the ANN outputs: the wave
functions. This is an instructive exercise that allows us to have clearer and
more local indicators of model quality, in addition to the information
provided by integrated quantities such as the energy or the fidelity. We show
$20$ different instances of the wave functions of the $S-$ (top panels) and
$D-$states (bottom panels) in Fig. 5. To show the “best” possible wave
functions, we focus on the number of hidden nodes $N_{\text{hid}}$ that
provide the lowest (central value of the) energies. For all four architectures
considered, this corresponds to $N_{\text{hid}}=100$. By looking at these
instances directly, one gets an idea of the possible spread of variationally
evolved models. Overall, we find that all networks provide very similar, high-
quality representations of the benchmark wave function which is shown in
dashed lines.

In most cases, the largest discrepancies occur towards the origin. As
explained in Ref. [20], the presence of a $q^{2}$ factor in the energy
integrals allows the networks to push a large amount of variance towards the
origin. In other words, changing the wave function near the origin has no
reflection on the global cost function. The large variance can be seen in the
insets of all panels which show the behaviour of the wave functions close to
the origin. For the $S-$state, one-layer models have relatively linear
behaviours as $q\to 0$, whereas $2-layer$ networks saturate close to the
origin. None of these behaviours matches the benchmark, showed in dashed
lines, in this limit. We note that the $20$ realisations provide a relatively
similar amount of deviations around a central value. This is in contrast to
the $D-$state wave function of the $1$sc architecture, which has a large
variation around the origin, in spite of providing the best overall energies.

Unlike in Ref. [20], we display the wave functions of Fig. 5 in a linear and
denser grid of points than the one used to compute energy quadratures. This
allows us to investigate whether the network is able to learn efficiently not
only the properties at the mesh points or the origin, but also the continuity
of the wave function across a set of points at finite momentum. Indeed, we
observe a step-like behaviour in the two-layer $D-$state wave functions
(bottom panels). These steps occur precisely in the vicinity of each
quadrature mesh point and they are more easily seen at high momenta where mesh
points are relatively spaced apart. Clearly, two-layer networks have enough
flexibility to generate horizontal steps around these points. In other words,
the networks learn the properties of the wave function only locally around the
meshpoints and interpolate in a stepwise fashion between them. We take this as
another, different sign of out-of-sample uncertainty. We stress that the step
do not occur in simpler, less flexible one-layer models. We speculate that the
lack of flexibility of such models forces them to behave in a more continuous
fashion. Further, these effects are not detected in integrated quantities,
since the quadrature meshpoint values are well reproduced by all the networks.
In other words, in our set-up, the regions between mesh points do not
contribute to the loss functions. Therefore, the network can push the variance
to these regions with virtually no observable effect.

Figure 6: Top panels: standard deviation associated to $20$ instances of the
$S-$state wave function as a function of momentum, for each of the 4
architectures considered here. Different line styles correspond to different
$N_{\mathrm{hid}}$ values. Bottom panels: the same for the $D-$state wave
function. Note the difference in scales of the momentum in both rows.

Figure 6 provides further insight into the structure of the wave function
variance in our models. The variance here is estimated as the standard
deviation associated to the $20$ model initialisations of our networks. The
different line styles in the figure correspond to different values of
$N_{\text{hid}}$. Top (bottom) panels show $S$ ($D$) state variances. It is
quite clear from these figures that the overall variance at finite momentum is
relatively independent of the network width. Moreover, most network models
have maximum variances towards the origin which is in line with Fig. 5.
$1-$layer networks have relatively flat declines with momentum, and saturate
at values of the order of $\sigma\approx 10^{-3}$.

In contrast, $2-$layer networks have pronounced oscillatory values. The
oscillation positions have a one-to-one correspondence with the plateaus in
Fig. 5. Variance minima occur at the quadrature mesh points, and maxima happen
in between. The oscillation minima (eg the smallest uncertainties) have
variances which are similar to the $1-$layer case. This is a clear indication
that the network has learned locally the wave functions of the system at the
quadrature mesh grids, but is overfitting the values in between which are not
penalised by the cost function. Note also that the variance for the $2$sd
models with low $N_{\text{hid}}$ differs from the general trend. This is due
to the poor adaptiveness of such architectures. In fact, some of these
converged wave functions bear little resemblance to the exact ones, despite
the fact that their energy is still within the range of Fig. 4. We discuss
potential mitigation strategies to these problems in the following sections.

Before discussing solutions to this problem, we want to quantify its
relevance. To this end, we generate wave function models in a new uniform
$10^{3}$-point mesh in the interval $[0,5]$ fm-1. We use this new linear mesh
to compute, in quadrature, the overlaps of the network models to the benchmark
wave function as well as the total energy. To summarise our findings, we show
in Table 1 the results of this exercise for 2 different models. On the one
hand, the $1$sd model with $N_{\mathrm{hid}}=100$ (top row) shows almost no
signs of overfitting. In contrast, the middle row shows results for the $2$sd
architecture with $N_{\mathrm{hid}}=100$, which has significant overfitting.
The bottom row quotes the values obtained with a benchmark wave function,
obtained in the very same uniform mesh.

| $F^{S}$ | $F^{D}$ |  $E$ (MeV)
---|---|---|---
$1$sd | $0.99997\pm 0.00001$ | $0.99994\pm 0.00003$ | $-2.2172\pm 0.0008$
$2$sd | $0.9992\pm 0.0004$ | $0.993^{+0.007}_{-0.012}$ | $-2.0\pm 0.3$
Benchmark | 1 | 1 | -2.22486
Table 1: Fidelity and energies of the $1$sd and $2$sd architectures in a
uniform mesh, where the overfitting is captured. Each value is the mean over
the set of $20$ initialisations, and the errors are the associated standard
deviations.

First, we discuss the fidelities reported in the first and second columns of
Table 1. These are computed in the uniform mesh, as opposed to the results
presented in Fig. 4 for the original quadrature. For the $1$sd model, the
fidelities are practically the same, close to $1$ up to the fourth or fifth
significant digit. This is due to the fact that the model has barely any
overfitting. The fidelities of the $2$sd model are, however, one and two
orders of magnitude further away from one than those reported in Fig. 4. We
take this as an early indication of potential overfitting.

A second, stronger indication comes from the energies in the final column.
These values should be compared to the benchmarks in the same uniform mesh,
reported in the bottom row. Two conclusions can be drawn here. First, the
energy values obtained in this mesh are significantly worse than the exact
ones. Whereas for the $1$sd value the energy lies within a fraction of a
percent of the benchmark, for the $2$sd model the model is only accurate up to
$10\%$ of the total energy. Second, the stochastic uncertainties are
significantly larger than those reported in Fig. 4. For the $1$sd network, the
values are relatively competitive, of the order of $1$ keV. The $2$sd model
has stochastic uncertainties of $300$ keV, several orders of magnitude higher.

This substantial increase in stochastic uncertainty, as well as the important
change in energy central values, indicate that overfitting is particularly
problematic for $2-$layer networks. Among other things, this indicates that
the results shown in Fig. 4 are not fully representative of the true values of
$2-$layer networks. The general behaviour of these quantities as a function of
$N_{\mathrm{hid}}$ is, however, expected to remain the same.

Having quantified the importance of overfitting, we now discuss its causes.
The differences between the exact and predicted wave functions appear almost
exclusively in two-layer models. These models have comparatively more
parameters, which are presumably redundant. Two-layer networks can “hide” this
redundancy in regions with little or no contributions to the cost functions,
either towards the origin or in regions in-between fixed quadrature mesh
points. We stress that having too many parameters does not restrict the ANN
outputs, but it may increase the difficulty of the task assigned to the
optimiser. The easy path out in this case is to overfit the data.

Moreover, as one increases the number of parameters, the network may find
multiple ways of fitting the data set evaluated on the same fixed $N_{q}$ mesh
points. In other words, larger networks can represent the same wave functions
in many different ways, and as the number of parameters grow, so do the number
of ways in which a network can represent the wave functions. This is usually
identified as a bias-variance trade-off.

We now speculate how to mitigate the overfitting effects on the wave
functions. An obvious alternative is to try and increase the mesh density. One
expects that in regions which are densely covered by the mesh, the networks
may not have enough freedom to develop the artificial plateaus observed in
Fig. 5. Just as in traditional variational models, improving the quality of
the (momentum-space) basis by, say, increasing $N_{q}$, may also be beneficial
in terms of the optimisation. Alternatively, one can envisage a set-up in
which meshes change from epoch to epoch. If such changes are entirely
stochastic, one could rephrase them as traditional variational quantum Monte
Carlo implementations, although these are not usually formulated with non-
local interactions like the Entem-Machleidt potential. Alternatively, one
could work with deterministic meshes but change the total number of meshpoints
or the positions at each epoch. In the most favourable scenario, one would
expect to find a network that does not memorise specific data mesh points, but
rather learns the wave function structure. In the end, any such strategy would
try to enforce a network learning process that is independent of the specific
quadrature that is used to compute the global cost energy function. In other
applications of supervised learning, the quality of the local cost function is
quantified via comparison to a validation dataset from which the network
doesn’t learn but merely compares.

##  4 Conclusions and future outlook

In this work, we use vANNs to compute the ground state properties of the
deuteron with a realistic hamiltonian. To this end, we discretise the problem
in a fixed mesh on the relative momentum coordinate. We use standard ML tools,
based on Pytorch, to pre-train our models and subsequently minimise the energy
for a fixed number of epochs.

High-quality solutions for the variational wave function were already obtained
in Ref. [20] with a similar set-up. We extend this work here in two
directions, aimed at identifying fundamental limitations of the vANN approach.
First, we look at different ANN architectures, increasing the number of layers
and treating the connection to the output states in a connected or
disconnected fashion. Second, we identify a new source of uncertainty
associated to the oscillations around the final energy minimum. Third, by
carefully analysing the wave function outputs, we identify conditions in which
finite-momentum overfitting arises.

All vANNs models provide excellent results for the energies and fidelities,
when compared to benchmark wave functions. We find that ANNs do not have to be
specially tailored to perform well at a problem.

By looking at the rate of model convergence for different architectures, we
find a first qualitative sign that two-layer networks have a harder time
minimising the energy than their one-layer counterparts. In terms of model
uncertainties, the post-evolution oscillation errors dominate over the
stochastic initialisation errors, but they remain relatively small (of the
order of $6$ to $8$ keV) across a wide range of network widths. When it comes
to the structure of the network, state-connected networks, with output nodes
that are connected to the internal hidden layer, provide marginally better
results than state-disconnected architectures.

Overall, we find that two-layer networks provide worse results than one-layer
models. This may have been expected since the input layer is based on a
single, positive-defined degree of freedom (the magnitude of the relative
momentum, $q$). Central values of the energy are less attractive, and the
stochastic and post-evolution uncertainties are larger. We also identify a
dependence on the number of hidden nodes which is absent in the one-layer
case. Representing the wave functions of these models at grid points that are
not used in the minimisation process, we find anomalous horizontal steps
between mesh points for the two-layer models. These are clear signs of network
overfitting. It seems that, during the minimisation process, the network is
able to push some of the redundant dependence of parameters into these
regions, which do not contribute to the energy cost function. This is
analogous to the observation of a large variance (in terms of wave function
values) around the origin, where the spherical $q^{2}$ factor allows the
network to change values arbitrarily around $q\approx 0$ without affecting the
total energy. Unlike the situation at the origin, however, a change of
integration mesh for the energy can easily detect the degradation of the model
associated to overfitting at finite momentum. We find that, in the new mesh,
the fidelities with respect to benchmark wave functions become worse. The
energy values in a different mesh are also substantially less attractive. The
associated stochastic uncertainty increases, reaching up to $300$ keV in some
cases.

While unveiling the internal behaviour of the ANN is hard, the comparison
between different architectures certainly sheds some light on the fundamental
limitations of these variational methods. When it comes to presenting wave
functions that depend on a single continuous variable, our results indicate
that one-layer networks provide a better starting point than the more complex
two-layer approaches. This effect may be due to the fixed (if long) total
number of epochs, but other observations indicate that overfitting arises much
more easily in deeper networks. Similarly, our results suggest that the
network can tackle states with different quantum numbers in a fully connected
configuration. In training ANNs to represent continuous quantum systems, non-
fixed grid methods may provide superior learning capabilities. Overall, this
experience provides useful ideas to build more sophisticated nuclear models
and tackle more difficult problems, including those of a many-body nature.

This work is supported by STFC, through Grants Nos ST/L005743/1 and
ST/P005314/1; by grant PID2020-118758GB-I00 funded by
MCIN/AEI/10.13039/501100011033; by the “Ramón y Cajal” grant RYC2018-026072
funded by MCIN/AEI /10.13039/501100011033 and FSE “El FSE invierte en tu
futuro”; and by the “Unit of Excellence María de Maeztu 2020-2023” award to
the Institute of Cosmos Sciences, funded by MCIN/AEI/10.13039/501100011033

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Boundedness of Fractional Integral operators and their commutators in
vanishing generalized weighted Morrey spaces

Bilal Çekiça,111Corresponding author. E-mail address<EMAIL_ADDRESS>and
Ayşegül Çelik Alabalıkb

Abstract

> In this article, we establish some conditions for the boundedness of
> fractional integral operators on the vanishing generalized weighted Morrey
> spaces. We also investigate corresponding commutators generated by BMO
> functions.
>
> Keywords : Vanishing generalized weighted Morrey spaces, fractional integral
> operator, Commutator
>
> MSC : 42B20; 42B35; 46E30

##  1 Introduction

Morrey spaces $L^{p,\lambda}$ that play important role in the theory of
partial differential equations an in harmonic analysis were introduced by C.
Morrey [9] in 1938. Since then these spaces and various generalizations of
Morrey spaces have been extensively studied by many authors. Mizuhara [8]
introduced the generalized Morrey space $L^{p,\varphi}$ and Komori and Shirai
[6] defined the weighted Morrey spaces $L^{p,\kappa}(w)$. Guliyev [3] have
given the notion of generalized weighted Morrey space
$L^{p,\varphi}\left(w\right)$ which can be accepted as an extension of
$L^{p,\varphi}$ and $L^{p,\kappa}(w)$. We refer readers to the survey [13] and
to the elegant book [2] for further generalizations about these spaces and
references on recent developments in this field.

Fractional maximal operator $M_{\alpha}$ and fractional integral operator
$I_{\alpha}$ play an important role in harmonic analysis. In recent years many
authors have studied the boundedness of these operators on Morrey type spaces.
The boundedness of fractional integral operator $I_{\alpha}$ was proved by
Adams [1] on classical Morrey spaces. In [5] authours find conditions on the
pair $\left(\varphi_{1},\varphi_{2}\right)$ which ensure Spanne type
boundedness of fractional maximal $M_{\alpha}$ and fractional integral
operator $I_{\alpha}$ from one generalized Morrey spaces to another
generalized Morrey space. The boundedness of fractional integral operator
$I_{\alpha}$ was proved by Komori and Shirai [6] on weighted Morrey spaces.
The boundedness of $I_{\alpha,b}$ was proved by Di Fazio and Ragusa [15] on
classical Morrey spaces, by Guliyev and Shukurov [5] on generalized Morrey
space and by Komori and Shirai [6] on weighted Morrey spaces. Recently,
Guliyev proved the boundedness of fractional integral operator $I_{\alpha}$
and fractional maximal operator $M_{\alpha}$ and their commutators on
generalized weighted Morrey space $L^{p,\varphi}\left(w\right)$ in the elegant
paper [3].

The vanishing Morrey space $VL^{p,\lambda}$ which has been first introduced by
Vitanza in [17] is a subspace of functions in $L^{p,\lambda}$ satisfying the
following condition

$\underset{r\rightarrow
0}{\lim}\sup\limits_{\begin{subarray}{c}x\in\mathbb{R}^{n}\\\
0<t<r\end{subarray}}t^{-\frac{\lambda}{p}}\left\|f\right\|_{L^{p}\left(B\left(x,t\right)\right)}=0.$

Ragusa [14] proved the boundedness of $I_{\alpha,b}$ on vanishing Morrey
spaces. Persson et al. [12] proved the boundedness of commutators of Hardy
operators in vanishing Morrey spaces. Later N. Samko [16] introduced the
vanishing generalized Morrey spaces $VL_{\Pi}^{p,\varphi}\left(\Omega\right)$
and proved the boundedness of $I_{\alpha}$ and $M_{\alpha\text{ }}$ on these
spaces.

Inspired by the above papers, we introduce the vanishing generalized weighted
Morrey spaces which are suitable subspace of functions in
$L^{p,\varphi}\left(w\right)$ and prove the boundedness of $M_{\alpha\text{
}}$and $I_{\alpha}$ from the vanishing generalized weighted Morrey space
$VL_{\Pi}^{p,\varphi}\left(\mathbb{R}^{n};w^{p}\right)$ to another vanishing
generalized weighted Morrey space
$VL_{\Pi}^{q,\psi}\left(\mathbb{R}^{n};w^{q}\right).$ Furthermore we show that
the commutators of both $M_{\alpha\text{ }}$and $I_{\alpha}$ are bounded in
vanishing generalized weighted Morrey spaces. In all the cases the conditions
for the boundedness are given in terms of Zygmund-type integral inequalities
on $\left(\varphi,\psi\right)$, where there is no any assumption on
monotonicity of $\varphi$, $\psi$ in $r$.

Throughout this paper, $C,c,c_{i}$ etc. are used as positive constant that can
change from one line to another. $A\lesssim B$ means that $A\leq cB$ with some
positive constant $c$. If $A\lesssim B$ and $B\lesssim A$, then we say
$A\approx B$ which means $A\ $and $B$ are equivalent.

##  2 Preliminaries

Let $w$ be a weight function on $\mathbb{R}^{n},$ such that $w(x)>0$ for
almost every $x\in\mathbb{R}^{n}.$ For $1\leq p<\infty$ and
$\Omega\subset\mathbb{R}^{n}$ we denote the weighted Lebesgue space by
$L^{p,w}\left(\Omega\right)$ with the norm

$\left\|f\right\|_{L^{p,w}\left(\Omega\right)}=\left(\int\limits_{\Omega}\left|f(x)\right|^{p}w(x)dx\right)^{1/p}<\infty.$

Let $1\leq p<\infty$, $\varphi$ be a positive measurable function on
$\mathbb{R}^{n}\times\left(0,\infty\right)$ and $w$ be a weight function on
$\mathbb{R}^{n}$. We denote by $L^{p,\varphi}\left(\mathbb{R}^{n};w\right)=$
$L^{p,\varphi}\left(w\right)$ the generalized weighted Morrey space, the space
of all functions $f\in L_{loc}^{p,w}\left(\mathbb{R}^{n}\right)$ with finite
quasinorm

$\left\|f\right\|_{L^{p,\varphi}\left(w\right)}=\underset{x\in\mathbb{R}^{n},r>0}{\sup}\frac{1}{\varphi^{\frac{1}{p}}\left(x,r\right)}\left\|f\right\|_{L^{p,w}\left(B\left(x,r\right)\right)}$

where $B\left(x,r\right)$ denote the open ball centered at $x$ of radius $r.$

The fractional integral operator (Riesz potential) $I_{\alpha}$ and fractional
Maximal operator $M_{\alpha},$ which play important roles in real and harmonic
analysis, are defined by

$I_{\alpha}f(x)=\mathop{\displaystyle\int}\limits_{\mathbb{R}^{n}}\frac{f(y)}{\left|x-y\right|^{n-\alpha}}dy,\qquad
0<\alpha<n$

and

$M_{\alpha}f(x)=\underset{r>0}{\sup}\frac{1}{r^{n-\alpha}}\mathop{\displaystyle\int}\limits_{B(x,r)}\left|f(y)\right|dy,\qquad
0\leq\alpha<n$

where $f\in L_{loc}^{1}\left(\mathbb{R}^{n}\right)$. If $\alpha=0,$ then
$M\equiv M_{0}$ is the Hardy-Littlewood maximal operator.

The class of $A_{p}$ weights was introduced by B. Muckenhoupt in [10]. The
Muckenhoupt classes characterize the boundedness of the Hardy-Littlewood
maximal function $M$ on weighted Lebesgue spaces. It is known that, $M$ is
bounded on $L^{p,w}(\mathbb{R}^{n})$ if and only if $w\in A_{p}$,
$1<p<\infty.$

Now we define Muckenhoupt class $A_{p}.$ Let $1<p<\infty$ and $\left|B\right|$
be the Lebesgue measure of the ball $B.$ A weight $w$ is said to be an $A_{p}$
weight, if there exists a positive constant $c_{p}$ such that, for every ball
$B\subset\mathbb{R}^{n},$

$\left(\mathop{\displaystyle\int}\limits_{B}w\left(x\right)dx\right)\left(\mathop{\displaystyle\int}\limits_{B}w\left(x\right)^{1-p^{\prime}}dx\right)^{p-1}\leq
c_{p}\left|B\right|^{p},$

when $1<p<\infty$, and for $p=1$

$\left(\mathop{\displaystyle\int}\limits_{B}w\left(y\right)dy\right)\leq
c_{1}\left|B\right|w\left(x\right)$

for a.e. $x\in B.$

A weight $w$ belongs to $A_{p,q}$ for $1<p<q<\infty$ if there exists
$c_{p,q}>1$ such that

$\left\|w\right\|_{L^{q}(B)}\left\|w^{-1}\right\|_{L^{p^{\prime}}(B)}\leq
c_{p,q}\left|B\right|^{1+\frac{1}{q}-\frac{1}{p}},$

where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. $A_{p,q}$ class was introduce by
B. Muckenhoupt and R. Wheeden [11] to study weighted norm inequalities for
fractional integral operators.

Following lemma gives the relation between $A_{p,q}$ and $A_{p}$ class.

###### Lemma 2.1

[7] If $w^{q}\in A_{1+\frac{q}{p^{\prime}}}$ with $1<p<q,$ then $w\in
A_{p,q}.$

###### Lemma 2.2

[3] Let $0<\alpha<n,$ $1<p<\frac{n}{\alpha},$
$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$ and $w\in A_{p,q}$. Then the
inequality

$\left\|I_{\alpha}f\right\|_{L^{q,w^{q}}\left(B\left(x,r\right)\right)}\lesssim\left\|w\right\|_{L^{q}\left(B(x,r)\right)}\mathop{\displaystyle\int}\limits_{2r}^{\infty}\left\|f\right\|_{L^{p,w^{p}}\left(B\left(x,t\right)\right)}\left\|w\right\|_{L^{q}\left(B\left(x_{,}t\right)\right)}^{-1}\frac{dt}{t}$
($2.1$)

holds for any ball $B(x,r)$ and for all $f\in
L_{loc}^{p,w}\left(\mathbb{R}^{n}\right).$

###  2.1 Vanishing generalized weighted Morrey spaces

Let $\Omega$ be an open set in $\mathbb{R}^{n}$ and $\Pi$ be an arbitrary
subset of $\Omega$. Let also $w$ be a weight function on $\Omega$,
$\varphi\left(x,r\right)$ be a measurable non negative function on
$\Pi\times\left[0,l\right)$ $\left(l=diam~{}\Omega\right)$ and positive for
all $(x,t)\in\Pi\times(0,l)$. The vanishing weighted Morrey space
$VL_{\Pi}^{p,\varphi}\left(\Omega;w\right)=VL_{\Pi}^{p,\varphi}\left(w\right)$
is defined as the space of functions $f\in L_{loc}^{p,w}\left(\Omega\right)$
with finite quasi norm

$\left\|f\right\|_{VL_{\Pi}^{p,\varphi}\left(w\right)}=\underset{x\in\Pi,0<r<l}{\sup}\frac{1}{\varphi^{\frac{1}{p}}\left(x,r\right)}\left\|f\right\|_{L^{p,w}\left(\widetilde{B}\left(x,r\right)\right)}$

such that

$\underset{r\rightarrow 0}{\lim}\text{
}\underset{x\in\Pi}{\sup}\frac{1}{\varphi^{\frac{1}{p}}\left(x,r\right)}\left\|f\right\|_{L^{p,w}\left(\widetilde{B}\left(x,r\right)\right)}=0,$
($2.2$)

where $\widetilde{B}\left(x,r\right)=B\left(x,r\right)\cap\Omega$ and $1\leq
p<\infty.$

Naturally, it is suitable to impose the function $\varphi\left(x,r\right)$ on
the following conditions

$\underset{r\rightarrow 0}{\lim}\text{
}\underset{x\in\Pi}{\sup}\frac{\left\|w\right\|_{L^{1}\left(\widetilde{B}(x,r)\right)}}{\varphi\left(x,r\right)}=0$
($2.3$)

and

$\underset{r>1}{\inf}\text{ }\underset{x\in\Pi}{\sup}\text{
}\varphi\left(x,r\right)>0$ ($2.4$)

where the last condition must be imposed when $\Omega$ is unbounded. These
conditions makes $VM_{\Pi}^{p,\varphi}\left(w\right)$ non trivial, since
bounded functions which have compact support belong to these spaces.

Henceforth we denote by $\varphi\in\mathfrak{B}\left(w\right)$ if
$\varphi\left(x,r\right)$ is a measurable non negative function on
$\Pi\times\left[0,l\right)$ and positive for all $(x,t)\in\Pi\times(0,l)$ and
satisfies the condition (2.3) and (2.4).

###  2.2 Commutator of Fractional Integral Operator

Let $f\in L_{loc}^{1}\left(\mathbb{R}^{n}\right)$ we define

$\left\|f\right\|_{\ast}=\underset{x\in\mathbb{R}^{n},r>0}{\sup}\frac{1}{\left|B\left(x,r\right)\right|}\mathop{\displaystyle\int}\limits_{B\left(x,r\right)}\left|f\left(y\right)-f_{B\left(x,r\right)}\right|dy$

where

$f_{B\left(x,r\right)}=\frac{1}{\left|B\left(x,r\right)\right|}\mathop{\displaystyle\int}\limits_{B\left(x,r\right)}f\left(y\right)dy.$

The space of functions of bounded mean oscillation (BMO) are the class of
functions whose deviation from their means over cubes is bounded. The set

$BMO\left(\mathbb{R}^{n}\right)=\left\\{f\in
L_{loc}^{1}\left(\mathbb{R}^{n}\right):\left\|f\right\|_{\ast}<\infty\right\\}$

is called the function space of BMO space.

Next we shall introduce the commutators of maximal operator $M_{\alpha,b}$ and
the commutator of fractional integral operator $I_{\alpha,b}$.

Let $b$ be a locally integrable function, $M_{\alpha,b}$ and $I_{\alpha,b}$
which are formed by $b$ are defined by

$M_{\alpha,b}f\left(x\right)=b\left(x\right)M_{\alpha}f\left(x\right)-M_{\alpha}\left(bf\right)\left(x\right)$

and

$I_{\alpha,b}f\left(x\right)=b\left(x\right)I_{\alpha}f\left(x\right)-I_{\alpha}\left(bf\right)\left(x\right).$

###### Lemma 2.3

[3]

Let $0<\alpha<n,$ $1<p<\frac{n}{\alpha},$
$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n},b\in BMO\left(\mathbb{R}^{n}\right)$
and $w\in A_{p,q}$. Then the inequality

$\left\|I_{\alpha,b}f\right\|_{L^{q,w^{q}}\left(B(x,r)\right)}\lesssim\left\|b\right\|_{\ast}\left\|w\right\|_{L^{q}\left(B(x,r)\right)}\mathop{\displaystyle\int}\limits_{2r}^{\infty}\ln\left(e+\frac{t}{r}\right)\left\|f\right\|_{L^{p,w^{p}}\left(B\left(x,t\right)\right)}\left\|w\right\|_{L^{q}\left(B\left(x,t\right)\right)}^{-1}\frac{dt}{t}$
($2.5$)

holds for any ball $B(x,r)$ and for all $f\in
L_{loc}^{p,w^{p}}\left(\mathbb{R}^{n}\right).$

##  3 Main Results

###  3.1 Boundedness of Fractional Integral operators and their commutators on
vanishing generalized weighted Morrey space

###### Theorem 3.1

Let $0<\alpha<n,$ $1<p<\frac{n}{\alpha},$
$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n},$ $w\in A_{p,q}$,
$\varphi\in\mathcal{B}\left(w^{p}\right)$ and $\psi\in$
$\mathcal{B}\left(w^{q}\right)$. Then the operators $M_{\alpha}$ and
$I_{\alpha}$ are bounded from the vanishing space
$VL_{\Pi}^{p,\varphi}\left(\mathbb{R}^{n};w^{p}\right)$ to another vanishing
space $VL_{\Pi}^{q,\psi}\left(\mathbb{R}^{n};w^{q}\right),$ if

$c_{\delta}:=\mathop{\displaystyle\int}\limits_{\delta}^{\infty}\sup_{x\in\Pi}\frac{\varphi^{\frac{1}{p}}\left(x,t\right)}{\left\|w\right\|_{L^{q}\left(B\left(x,t\right)\right)}}\frac{dt}{t}<\infty$
($3.1$)

for every $\delta>0,$ and

$\mathop{\displaystyle\int}\limits_{r}^{\infty}\frac{\varphi^{\frac{1}{p}}\left(x,t\right)}{\left\|w\right\|_{L^{q}\left(B\left(x,t\right)\right)}}\frac{dt}{t}\leq
c_{0}\frac{\psi^{\frac{1}{q}}\left(x,r\right)}{\left\|w\right\|_{L^{q}\left(B\left(x,r\right)\right)}}$
($3.2$)

where $c_{0}$ does not depend on $x\in\Pi$ and $r>0.$

Proof. By the pointwise inequality $M_{\alpha}f(x)\leq
I_{\alpha}\left(\left|f\right|\right)(x)$ for $0<\alpha<n$ it is sufficient to
prove the theorem only for fractional integral operator. For every $f\in
VL_{\Pi}^{p,\varphi}\left(\mathbb{R}^{n};w^{p}\right)$ we have to show that

$\left\|I_{\alpha}f\right\|_{VL_{\Pi}^{q,\psi}\left(w^{q}\right)}\leq
c\left\|f\right\|_{VL_{\Pi}^{p,\varphi}\left(w^{p}\right)}$

and

$\underset{r\rightarrow
0}{\lim}\sup\limits_{x\in\Pi}\frac{1}{\psi^{\frac{1}{q}}\left(x,r\right)}\left\|I_{\alpha}f\right\|_{L^{q,w^{q}}\left(B\left(x,r\right)\right)}=0.$

Firstly, we prove that $I_{\alpha}f\in VL_{\Pi}^{q,\psi}\left(w^{q}\right)$
when $f\in VL_{\Pi}^{p,\varphi}\left(w^{p}\right)$. Namely,

$\underset{r\rightarrow
0}{\lim}\sup\limits_{x\in\Pi}\frac{1}{\varphi^{\frac{1}{p}}\left(x,r\right)}\left\|f\right\|_{L^{p,w^{p}}\left(B\left(x,r\right)\right)}=0\Rightarrow\underset{r\rightarrow
0}{\lim}\sup\limits_{x\in\Pi}\frac{1}{\psi^{\frac{1}{q}}\left(x,r\right)}\left\|I_{\alpha}f\right\|_{L^{q,w^{q}}\left(B\left(x,r\right)\right)}=0.$

Let $0<r<\delta_{0}$. To show that
$\sup\limits_{x\in\Pi}\frac{1}{\psi^{\frac{1}{q}}\left(x,r\right)}\left\|I_{\alpha}f\right\|_{L^{q,w^{q}}\left(B\left(x,r\right)\right)}<\varepsilon$
for $0<r<\delta_{0}$, by using the inequality $\left(2.1\right)$ we can write

$\frac{1}{\psi^{\frac{1}{q}}\left(x,r\right)}\left\|I_{\alpha}f\right\|_{L^{q,w^{q}}\left(B\left(x,r\right)\right)}\leq
C\left[A_{\delta_{0}}\left(x,r\right)+B_{\delta_{0}}\left(x,r\right)\right]$
($3.3$)

where

$A_{\delta_{0}}\left(x,r\right):=\frac{\left\|w\right\|_{L^{q}\left(B(x,r)\right)}}{\psi^{\frac{1}{q}}\left(x,r\right)}\left(\mathop{\displaystyle\int}\limits_{r}^{\delta_{0}}\frac{\varphi^{\frac{1}{p}}\left(x,t\right)}{t\text{
}\left\|w\right\|_{L^{q}(B(x,t))}}\underset{0<r<t}{\sup}\frac{1}{\varphi^{\frac{1}{p}}\left(x,r\right)}\left\|f\right\|_{L^{p,w^{p}}\left(B\left(x,r\right)\right)}dt\right)$

and

$B_{\delta_{0}}\left(x,r\right):=\frac{\left\|w\right\|_{L^{q}\left(B(x,r)\right)}}{\psi^{\frac{1}{q}}\left(x,r\right)}\left(\mathop{\displaystyle\int}\limits_{\delta_{0}}^{\infty}\frac{\varphi^{\frac{1}{p}}\left(x,t\right)}{t\text{
}\left\|w\right\|_{L^{q}(B(x,t))}}\underset{0<r<t}{\sup}\frac{1}{\varphi^{\frac{1}{p}}\left(x,r\right)}\left\|f\right\|_{L^{p,w^{p}}\left(B\left(x,r\right)\right)}dt\right).$

Now we fix $\delta_{0}>0$ such that
$\sup\limits_{x\in\Pi}\underset{0<r<t}{\sup}\frac{1}{\varphi^{\frac{1}{p}}\left(x,r\right)}\left\|f\right\|_{L^{p,w^{p}}\left(B\left(x,r\right)\right)}<\frac{\varepsilon}{2Cc_{0}}$,
where $c_{0}$ and $C$ are constants from $\left(3.2\right)$ and
$\left(3.3\right),$ respectively. This allows to estimate uniformly for
$0<r<\delta_{0}$

$\sup\limits_{x\in\Pi}CA_{\delta_{0}}\left(x,r\right)<\frac{\varepsilon}{2}.$

The estimation of the second term, in view the condition (3.1), we get

$B_{\delta_{0}}\left(x,r\right)\leq
c_{\delta_{0}}\frac{\left\|w\right\|_{L^{q}\left(B(x,r)\right)}}{\psi^{\frac{1}{q}}\left(x,r\right)}\left\|f\right\|_{VL_{\Pi}^{p,\varphi}\left(w^{p}\right)}$

where $c_{\delta_{0}}$ is the constant from $\left(3.1\right).$ Since
$\psi\in\mathcal{B}\left(w^{q}\right)$ it suffices to choose $r$ small enough
such that

$\sup\limits_{x\in\Pi}\frac{\left\|w^{q}\right\|_{L^{1}\left(B(x,r)\right)}}{\psi\left(x,r\right)}<\left(\frac{\varepsilon}{2Cc_{\delta_{0}}\left\|f\right\|_{VL_{\Pi}^{p,\varphi}\left(w^{p}\right)}}\right)^{q}$

which gives required result.

By the definition of the norm and Lemma 2.2 we have

$\displaystyle\left\|I_{\alpha}f\right\|_{VL_{\Pi}^{q,\psi}\left(w^{q}\right)}$
$\displaystyle=$
$\displaystyle\underset{x\in\Pi,r>0}{\sup}\frac{1}{\psi^{\frac{1}{q}}\left(x,r\right)}\left\|I_{\alpha}f\right\|_{L^{q,w^{q}}\left(B\left(x,r\right)\right)}$
$\displaystyle\leq$ $\displaystyle
c\underset{x\in\Pi,r>0}{\sup}\frac{1}{\psi^{\frac{1}{q}}\left(x,r\right)}\left\|w\right\|_{L^{q}\left(B(x,r)\right)}\mathop{\displaystyle\int}\limits_{r}^{\infty}\left\|f\right\|_{{}_{L^{p,w^{p}}\left(B\left(x,t\right)\right)}}\left\|w\right\|_{L^{q}\left(B\left(x,t\right)\right)}^{-1}\frac{dt}{t}.$

Thus by the condition (3.2) we get

$\left\|I_{\alpha}f\right\|_{VL_{\Pi}^{q,\psi}\left(w^{q}\right)}\leq
c\left\|f\right\|_{VL_{\Pi}^{p,\varphi}\left(w^{p}\right)}$

which completes the proof of Theorem 3.2.

###### Remark 3.2

Note that in the case $w\equiv 1$ Theorem 3.1 was proved in [16].

###### Theorem 3.3

Let $0<\alpha<n,$ $1<p<\frac{n}{\alpha},$
$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n},b\in
BMO\left(\mathbb{R}^{n}\right)$, $w\in A_{p,q}$ and $\varphi\in B(w^{p})$ and
$\psi\in B(w^{q})$. If

$c_{\delta}:=\underset{0<r<\delta}{\sup}\text{
}\mathop{\displaystyle\int}\limits_{\delta}^{\infty}\left(e+\ln\frac{t}{r}\right)\sup_{x\in\Pi}\frac{\varphi^{\frac{1}{p}}\left(x,t\right)}{\left\|w\right\|_{L^{q}\left(B(x,t)\right)}}\frac{dt}{t}<\infty$
($3.4$)

for every $\delta>0,$ and

$\mathop{\displaystyle\int}\limits_{r}^{\infty}\ln\left(e+\frac{t}{r}\right)\frac{\varphi^{\frac{1}{p}}\left(x,t\right)}{\left\|w\right\|_{L^{q}\left(B\left(x,t\right)\right)}}\frac{dt}{t}\leq
c_{0}\frac{\psi^{\frac{1}{q}}\left(x,r\right)}{\left\|w\right\|_{L^{q}\left(B\left(x,r\right)\right)}}$
($3.5$)

where $c_{0}$ does not depend on $x\in\Pi$ and $r>0,$ then $I_{\alpha,b}$ is
bounded from $VL_{\Pi}^{p,\varphi}\left(w^{p}\right)$ to
$VL_{\Pi}^{q,\psi}\left(w^{q}\right).$

Proof. Proof of Theorem 3.3 follows by Lemma 2.3 and the same procedure argued
in the proof of Theorem 3.1.

###### Corollary 3.4

Let $0<\alpha<n,$ $1<p<\frac{n}{\alpha},$
$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n},b\in
BMO\left(\mathbb{R}^{n}\right)$, $w\in A_{p,q}$ and $\varphi\in B(w^{p})$ and
$\psi\in B(w^{q})$. If $\varphi$ and $\psi$ satisfy conditions (3.4) and (3.5)
then $M_{\alpha,b}$ is bounded from $VL_{\Pi}^{p,\varphi}\left(w^{p}\right)$
to $VL_{\Pi}^{q,\psi}\left(w^{q}\right).$

###### Example 3.5

If $w$ is a weight and there exists two positive constants $C$ and $D$ such
that $C\leq w(x)\leq D$ for a.e. $x\in\mathbb{R}^{n}$, then obviously $w\in
A_{p,q}$ for $1<p<q<\infty.$ Let $0<\alpha<n,$ $1<p<\frac{n}{\alpha}$ and
$0<\lambda<n-\alpha p.$ Moreover, let
$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n},$ $\varphi(x,r)=r^{\lambda}$ and
$\psi(x,r)=r^{\frac{\lambda q}{p}}.$ Then $M_{\alpha}$ and $I_{\alpha}$ are
bounded from the vanishing space
$VL_{\Pi}^{p,\varphi}\left(\mathbb{R}^{n};w^{p}\right)$ to another vanishing
space $VL_{\Pi}^{q,\psi}\left(\mathbb{R}^{n};w^{q}\right).$

###### Example 3.6

Let $0<\alpha<n,$ $1<p<\frac{n}{\alpha}$ and $0<\lambda<n-\alpha p.$ Moreover,
let $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n},$ $\varphi(x,r)=r^{\lambda}$ and
$\psi(x,r)=r^{\frac{\lambda q}{p}}.$ Suppose that
$v(y)=\left|y\right|^{\beta}.$ Then $v\in A_{p}$, $1<p<\infty,$ if
$0<\beta<n(p-1).$ From Lemma 2.1 we have
$w(y)=\left|y\right|^{\frac{\beta}{q}}\in A_{p,q}$, $1<p<q<\infty$ if
$0<\beta<n\frac{q}{p^{\prime}}$. Thus we can obtain the following:

$\left\|w\right\|_{L^{q}(B(x,r))}\approx\left\\{\begin{array}[]{ccc}r^{\frac{n+\beta}{q}}&\text{when}&\left|x\right|<3r\\\
r^{\frac{n}{q}}(\left|x\right|+r)^{\frac{\beta}{q}}&\text{when}&\left|x\right|\geq
3r\end{array}\right..$

If $\left|x\right|<3r,$ then we get for the left hand side of (3.2)

$\mathop{\displaystyle\int}\limits_{r}^{\infty}\frac{\varphi^{\frac{1}{p}}\left(x,t\right)}{\left\|w\right\|_{L^{q}\left(B\left(x,t\right)\right)}}\frac{dt}{t}\lesssim
r^{\frac{\lambda}{p}-\frac{n+\beta}{q}}$

since $\frac{\lambda}{p}-\frac{n+\beta}{q}<0.$ On the other hand, for the
right hand side of the inequality (3.2), we have

$\frac{\psi^{\frac{1}{q}}\left(x,r\right)}{\left\|w\right\|_{L^{q}\left(B\left(x,r\right)\right)}}\approx
r^{\frac{\lambda}{p}-\frac{n+\beta}{q}}.$

If $\left|x\right|\geq 3r,$ then

$\mathop{\displaystyle\int}\limits_{r}^{\infty}\frac{\varphi^{\frac{1}{p}}\left(x,t\right)}{\left\|w\right\|_{L^{q}\left(B\left(x,t\right)\right)}}\frac{dt}{t}\lesssim\frac{r^{\frac{\lambda}{p}-\frac{n}{q}}+\left|x\right|^{\frac{\lambda}{p}-\frac{n}{q}}}{\left|x\right|^{\frac{\beta}{q}}}$

and

$\frac{\psi^{\frac{1}{q}}\left(x,r\right)}{\left\|w\right\|_{L^{q}\left(B\left(x,r\right)\right)}}\approx\frac{r^{\frac{\lambda}{p}-\frac{n}{q}}}{(\left|x\right|+r)^{\frac{\beta}{q}}}.$

Obviously

$\frac{r^{\frac{\lambda}{p}-\frac{n}{q}}+\left|x\right|^{\frac{\lambda}{p}-\frac{n}{q}}}{\left|x\right|^{\frac{\beta}{q}}}\lesssim\frac{r^{\frac{\lambda}{p}-\frac{n}{q}}}{(\left|x\right|+r)^{\frac{\beta}{q}}},\qquad\text{when
}\left|x\right|\geq 3r.$

Furthermore we have

$c_{\delta}\approx\delta^{\frac{\lambda}{p}-\frac{n+\beta}{q}}.$

Therefore $M_{\alpha}$ and $I_{\alpha}$ are bounded from the vanishing space
$VL_{\Pi}^{p,\varphi}\left(\mathbb{R}^{n};w^{p}\right)$ to another vanishing
space $VL_{\Pi}^{q,\psi}\left(\mathbb{R}^{n};w^{q}\right).$

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aDepartment of Mathematics,

Faculty of Science, Dicle University,

21280 Diyarbakir, Turkey

bDepartment of Mathematics,

Institute of Natural and Applied Sciences,

Dicle University,

21280 Diyarbakir, Turkey