Speeding up Quantum Few-Body Calculation with Active Learning
In large-scale scientific calculations, it often encounters the problem of determining a multi-dimensional function, which can be time-consuming when computing each point in this multi-dimensional space is already demanding. In the letter, we will use a quantum three-boson problem to demonstrate that active learning algorithm can speed up this calculation. The basic idea is to fit this multi-dimensional function by neural networks, and the key point is to make the query of labeled date economically by using a stratagem called “query by committee”. We will present the general protocol of this fitting scheme, as well as the procedure of how to further compute physical observable with the fitted results. In the three-boson example we present, the multi-dimensional function is a scattering kernel, and the physical observable is the atom-dimer scattering length, which can exhibit the Efimov resonances. We show that this algorithm can capture the Efimov resonance and predict the atom-dimer scattering length with an error of a few thousandths, with only a few percents of total data points required.
Neural network (NN) based supervised learning methods have nowadays found broad applications in studying quantum physics of condensed matter materials and atomic, molecular and optical systems review1 ; review2 . On the theory side, applications include, for example, finding orders and topological invariants in quantum phases Phase1 ; Phase2 ; Phase3 ; Phase4 ; Phase5 ; Phase6 ; Phase7 , generating variational wave functions for quantum many-body states VariationalWF ; WFRBM ; WF ; WFFewBody ; TheoryTomography and speeding up quantum Monte Carlo sampling MC1 ; MC2 . On the experimental side, these methods can help both optimizing experimental protocols State ; Tomography and analyzing experimental data STM ; ExpPhase ; ExpFHM . Usually the supervised learning scheme requires a huge number of labelled dataset. However, in many of quantum physics applications, labelling data can be quite expensive, for instance, performing computation or experiments can be computational demanding. Therefore, in many cases labelled data are not abundant. This is a challenge that have prevented many applications.
The active learning is a scheme to solve this problem ALbook . It starts from training a NN with a small initial dataset, and then actively queries the labelled data based on the predication of the NN and iteratively improves the performance of the NN until the goal of the task is reached. With this approach, sampling the large parameter space can be more efficient, and the request of labelled data is usually much more economically than normal supervised learning methods. Recently a few works have applied the active learning algorithm to determine the inter-atomic potentials in quantum materials PotentialAL1 ; PotentialAL2 ; PotentialAL3 and to optimize control in quantum experiments ExpControl1 ; ExpControl2 , where labelled data have to be obtained either by ab initio calculation or by repeating experiments, and are both time consuming.
Few-body calculation of a quantum system can determine the collisional properties between particles, which plays a fundamental role in physics branches such as nuclear physics, cold atoms physics and quantum chemistry. However, such calculations are usually heavily numerical, and the computational cost increases rapidly as the particle number increases. Part of the difficulty often originates from determining a multi-dimensional function, say, . For instances, can be an energy landscape where are spatial coordinates, or as in the example we will discussed below, is an interaction kernel function and are momenta. Normally one needs to discretize each dimension into points, and it requires to calculate totally number of data points in order to determine this function. In many cases, calculation of each point already takes quite a bit time, and thus the total computational cost is massive.
In this letter we apply the active learning protocol to speed up numerical calculation of a quantum few-body problem. The basic idea is to use NN to fit the multi-dimensional function . The advantage is that, by utilizing the great expressive power of NN, we do not need to assume a specific function form of . Here the key point is how to obtain a reliable fitting with less data points. Below we will first introduce the general protocol and then discuss a concrete example.
General Protocol. The main procedure is summarized in Fig. 1 and explained as follows:
1) We start with an initial dataset with the number of data , whose values of have been computed exactly. We use this dataset to train a number of different NN.
2) We ask all NN to make predictions of on all number of data points, and for each point we compute the variance between predications made by different NN.
3) We select number of data point with the largest variance, again with , and we query the accurate value of of these points by numerical calculation.
4) We add the number of new data into the training set to train all NN again, and then repeat from step 2). We repeat the procedure for -steps until the results from all NN converge.
5) We use NN to calculate the value of on all data points.
In this protocol, one only needs to query the value of on number of data points, and we should keep . The train-off is that we need to train a number of NN and keep using NN to make predications on all data points. It will save computational cost if the computational cost for training NN and making predication with NN is much less than the computational cost of , which is often the case in many applications.
In this protocol, we implement the idea of “ query by committee ”. Here committee means different NN. For fitting, there are parameters regimes where the function is smooth enough and it does not require a lot of data to reach a reliable fitting. These are regimes where different NN can quickly reach consensus, which is mathematically described by a small variance between predications made by different NN. Therefore we only query data in the regimes with large variance, which makes the query more efficient.
Example. As a proof-of-principle demonstration, below we apply this active learning protocol to a quantum three-body problem, where we consider three bosons interacting with an -wave interaction near the vicinity of a resonance. We use this example because it is a well-studied few-body problem and there are known results to breachmark our new approach. From this calculation, one can determine the important physical observable such as the atom-dimer scattering length . In particular, there exists the famous Efimov effect in this system, where diverges, and the places it diverges exhibit a geometric scaling sequence. We will show that our active learning algorithm can predict with a relative error of the order of and can well reproduce the Efimov scaling, and remarkably, in this calculation only of total data points need to be calculated, i.e. .
To be more concrete about this problem, we write down the Hamiltonian of this three boson system as
where () is the spatial coordinate of the th particle, and is an isotropic short-range potential. Normally we focus on the -wave scattering for ultracold low-energy atoms, and can be described by the -wave scattering length and a high-momentum cut-off . Most features of this quantum three-body problem only depend on the dimensionless parameter , which we will use in Fig. 4 and Fig. 5 Braaten . We consider the case that is large and positive, where supports a low-energy bound state called a dimer. A key process to this three-body problem is the atom-dimer scattering illustrated in Fig. 2(a), and the quantity to describe this process is a scattering kernel, which is usually denoted by , and it can be computed diagrammatically with the Feynman diagram shown in Fig. 2(b) Braaten . When focusing on the -wave scattering only, only depends on the amplitude of and , and we can simplify as .
Knowing , one can calculate the few-body quantity, such as , through the celebrated Skorniakov-Ter-Martirosion (STM) equation, which can be written as
In practices, suppose that we discretize both and into points between zero and , Eq. 2 becomes a matrix equation after the discretization, which can be solved by inverting the matrix. Let denotes the set of total data points, and the number of data points in is . In the conventional method, we need to compute for all points, with which we solve Eq. 2 to obtain . This is referred as the exact results as reference in the Fig. 4 (dashed lines) and Fig. 5 (black dots). Here we will follow the general procedure described above to fit using NN, and we will show that at most only number of data points are needed in order to obtain a reliable fitting. Then we use the trained NN to generate on all data points in to solve Eq. 2, and we find that the error is insignificant when comparing to the exact result.
We start with an initial dataset with points. Here we design five different fully connected NN, whose structure is schematically shown in Fig. 3. The input of all NN are two numbers and , and the output is . Each layer of a NN is characterized by the number of nodes and an activation function , and we describe each NN with by . The five different NN used in this works are
where denotes the logistic sigmoid activation function with . For each NN, the training results also depend on the initialization, which is particularly so when the number of data points are not enough. Therefore, for each NN, we also consider different initialization.
Therefore, at each iteration, we totally have trained NN to predict on all points in the set . For each point , we can compute variance among all predications. We will select points with the largest variance to compute at these points, and add them into the training set. At each iteration, we can also compute the mean variance
In Fig. 4(a), we plot the mean variance as a function of the number of data in the training set. The tendency is clear that the variance decreases as increases, and the predications from different NN converge. Fig. 4(a) also shows that the convergence for the case with is slower than the case with . This is because for larger , the function becomes more steep in certain regime, which makes fitting more difficult.
Now when the variance becomes small, we will start to use the NN to calculate on all data points in and to solve Eq. 2 with these values to determine physical quantity . In this step, there are two considerations. First, since solving Eq. 2 is also computational demanding when the dimension of the matrix becomes large, we certain want to reduce the working load of solving Eq. 2. Secondly, although the variance becomes smaller, there are still fluctuations between different NN and between different iterations, we want to average out these fluctuations. With these two considerations in mind, we figure out following procedure to determine .
1) For each NN, since there are copies depending on different initializations, we first average over the predications over these copies as
where denotes five different NN, and denotes different initializations. We then choose one of that is the closest to , where the distance is measured by
and we use this (ignoring the upper index) as the representative of the -th NN.
2) We solve Eq. 2 with and obtain . Among all five , we discard the largest and the smallest one, and take an average over the rest three, which is taken as predicted by the active learning approach. The results are shown in Fig. 4(b). One can see that the results fluctuate around the exact value, and the fluctuation decreases as increases. For the case with , the fluctuation suppresses more quickly than the case with . This is because the case with is more close to an Efimov resonance, and small errors in can be amplified during solving Eq. 2.
3) To further suppress the fluctuation, we can further take a self-average of , that is to average over for five successive iterations. This result is denoted by and shown in Fig. 4(c). Indeed, one can see that the fluctuation is already suppressed strongly at .
In Fig. 5, we plot as a function of . We stop the iteration at , a number much smaller than total number of data points in the conventional approach, and we take a self-average over the last five iterations to obtain . It shows that this calculation can obtain the atom-dimer resonances at the right locations. By comparing to the exact result, we define a relative error as ‰. The around the first resonance is shown in the inset of Fig. 5, which shows that the relative error does increases nearby the Efimov resonance, but overall it is kept within a few thousandths.
Conclusion and Outlook. In summary, we have developed a method to determine a multi-dimensional function efficiently. The method combines the great expressivity of NN and the advantage of the active learning scheme to reduce the demand for labeled data to a minimum. Note that the prediction of NN more or less varies depending on different structures of NN, different initializations and the different number of training data, which is taken as a double-edged sword in this work. On the one hand, we make use of the discrepancy between NN with different structures and different initializations to guide the query processes more efficient; and on the other hand, we need to properly average out these variances when calculating physical observables using the fitted results. Since determining a multi-dimensional function is a task often encountered in scientific computation, though here we show it can work well using quantum three-body calculation as an example, we believe our method can find broad applications in many areas of computational physics.
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