Can machine learning really solve the three-body problem?


 Machine learning is becoming one of most rapidly growing technical fields, benefiting tremendous areas in science and industry. Recently, Breen et al demonstrated a new study with a multi-layered deep artificial neural network (ANN) on a chaotic three-body system and claimed that it can bring “success in accurately reproducing the results of a chaotic system” with “up to 100 million times faster than numerical integrator”. Here, we use their trained ANN model to predict periodic orbits of the same three-body system, but the detailed comparisons are disappointing. It might be due to the butterfly-effect of chaotic systems, i.e. very sensitive to tiny disturbance, but in practice nearly all ML algorithms derive their solutions statistically and probabilistically and therefore are rarely possible to train them to 100% accuracy. We illustrate here that the current accuracy of the machine learning is not precise enough for correct prediction of periodic orbits of a chaotic three-body system in a long enough duration. Thus, it is still a great challenge for machine learning to solve chaotic systems, such as the famous three-body problem. Without doubt, studies in machine learning on chaotic systems might greatly deepen and enrich our understandings not only on chaos but also on machine learning itself.


I. INTRODUCTION
Machine learning has emerged as one of the most attractive technologies in widespread fields, such as airport facial recognition 1 , money laundering detecting 2 , computer vision 3 , natural language processing 4 , advertising 5 , and so on. Deep leaning is widely used to identify objects in images, transcribe speech into text, match news items, posts or products with users' interests, and select relevant results of search. Dominating in Go game even made a splash throughout the world by AlphaGo 6 . It has also become one of the most essential parts of addressing scientific and engineering questions [7][8][9][10][11][12] . For example, a good prediction of fluid dynamic fields has been reported, which has the ability of learning velocity and pressure information from flow snapshots by means of machine learning strategy 12 . The deep neural networks (DNNs) have driven error rates down from 8.4% to 4.9% in voice recognition in five years, and the DNNs for image recognition have helped improve error rates on ImageNet from more than 30% in 2010 to less than 3% today 11 . Note that the 5% threshold is roughly the error rate of humans for image and speech recognition using similar data. Despite a growing amount of literature on the success and efficiency of machine learning in different areas, one would ask one important question: what are the limitations/boundaries of this silver bullet?
The famous three-body problem might be a litmus test to somehow answer this question.
The evolution of many bodies in space can be described by Newton's equations of motion, and the gravitational force is the only reason to drive them 13 . This discovery has an essential role in many physical areas, including the famous three-body problem. Numerous distinguished scientists 14,15 dived into this important topic since Newton mentioned it in 1680s. Poincaré 16 discovered that three-body problem is very sensitive to initial conditions, say, it is a chaotic system with the so-called butterfly-effect. That is the reason why this problem remains impenetrable 17 and only three families of periodic solutions were discovered in three hundred years, including the Euler-Lagrange 14,15 , the Broucke-Hadjidemetriou-Hénon 18-22 and the figure-eight orbit 23,24 . These years, seeking new periodic solutions of the three-body problem has been paid much attention [25][26][27] . A big breakthrough was made currently by Li and Liao, who found more than six hundred new periodic solutions of three-body system with three equal masses 28 and 1223 new families of periodic solutions with two equal masses 29 .
Besides, they also found 313 new families of collisionless periodic orbits for the free-fall triple system with different ratios of masses 30 (there were only 4 trajectories reported before 31,32 ).
To achieve this accomplishment, they used a new numerical strategy, namely the clean numerical simulation (CNS) [33][34][35][36][37][38][39] , to overcome the failure due to round-off and truncation errors in calculating chaotic issues, because the CNS can give reliable/convergent trajectory of chaotic systems with high precision in a long enough duration.
Recently, Breen et al 40 demonstrated a new study with a multi-layered deep artificial neural network (ANN) 3 involved in and claimed that it can bring "success in accurately reproducing the results of a chaotic system" with "up to 100 million times faster than numerical integrator". Breen et al 40 showed a good performance of predicting 100 trajectories of free-fall triple system with equal masses by an ANN model trained with 9900 samples.
However, it is a pity that they did not illustrate its ability to track periodic orbits, a famous classical problem for three-body system. As mentioned above, seeking periodic solutions of a three-body system has been a challenging work in more than three hundreds year. It, therefore, would be a perfect benchmark to validate this ANN approach.
There are 30 equal-mass cases among the 313 collisionless free-fall periodic orbits found

II. METHOD
Li and Liao 30 directly solved the planar Newtonian three-body problem by means of the CNS 33-39 , where r ′ i = (x ′ i , y ′ i ) and m i are the position vector and mass of the ith body, G is the gravity coefficient, respectively. Since they are identical mass particles with zero initial velocity for the considered free-fall case, and the dimensionless units are adopted, both mass m j and gravity G equal to one. The initial positions of the three particles are as shown in Fig. 1, where two of the three particles are at points (0.5, 0) and (-0.5, 0), and the other one is in a shadow region which is constructed by the x ′ and y ′ axes and an arc of the unit radius with the dot (-0.5, 0) as the center. This initial configuration will cover all planar free-fall three-body system 42 . To guarantee the reliability of the solution, we use the CNS 33-39 to solve Eq. (1). To find the periodic solutions of Eq. (1) for the free-fall triple systems, one has to make sure that where T ′ is the period. For details, please refer to Li and Liao 43 .
Note that the training dataset of the machine learning given by Breen et al 40  with the zero initial velocityṙ i (0) = 0, where i = 1, 2, 3. The transformation between the two coordinates r i (t) and r ′ i (t ′ ) follows where and the rotation angle the CNS in the frame r ′ i (t ′ ) are transformed into the frame r i (t) by means of the above formulas. Then, using the same initial position of each periodic orbits in the frame r i (t), we can calculate the corresponding orbit by means of the trained ANN model given by Breen et al 40 . Both of them are compared with each other in the frame r i (t).
Breen et al 40 used a combination of feedforward deep ANN, which is generally stated here to simplify the following understanding. The input ingredients are time t and one of the three particles initial position r 2 (0) = (x 2 (0), y 2 (0)). There are 10 hidden layers, each having 128 nodes, interconnected between the input layer and the output layer. For every node in each layer, there is an activation function σ(x) to map the input into a number that is further used as the input of the next layer. Without loss of generality, let us take the ith node of the first hidden layer as an example. The output from this perceptron is where

III. COMPARISON OF RESULTS
All of our test data are the free-fall periodic solutions of the three-body problem with equal mass given by Li and Liao 43 using the CNS [33][34][35][36][37][38][39] . To precisely estimate the deviation, we define the maximum relative error where r i is the i-th periodic orbit given by CNS 43 and o i is obtained using the ANN model Breen et al 40 gave three ANN models with t ≤ 3.9, t ≤ 7.8 and t ≤ 10, respectively. A test using four random periodic orbits indicates that the ANN model uploaded by Breen  Figure 3: the maximum relative-error δ is less than 3% in 0 ≤ t < 2. half period (T /2 < t < T ), the deviations of the ANN prediction become rather obvious, as shown in Figure 4.
Serious comparison for the other 29 periodic orbits in one period, i.e. t ∈ [0, T ], is impossible, since their periods are much larger than 3.9. Note that in many cases there exist large derivations even in a short duration 0 ≤ t ≤ 3.9 that is much smaller than the corresponding periods of these orbits. One example is given in Figure 5 for comparison of a periodic orbit with T = 12.8337947509, with the initial condition r 2 (0) = (−0.3762596888, 0.1031140835).
It is found that the corresponding trajectories of the triple system given by the ANN model (using the same initial condition) have obvious derivations from the CNS results after t > 1.
More comparisons of the small parts (within t ∈ [0, 3.9]) of the periodic orbits are given in Figures 6 to 9: all of them show a distinct deviation sooner or later and thus can not predict the corresponding periodic orbits (in a whole period) given by the CNS 43 . Note that the ANN model we used here is the finest one within t ≤ 3.9 uploaded by Breen et al. 40 . If the training dataset in a longer duration is used, a worse performance can be expected.
Note that, in t ∈ [0, 3.9], cases with a relative-error lower than 10% account for 73.3% the longer the duration of time, the more the ANN predictions depart from the CNS periodic orbits. However, most of the equal-mass 30 periodic orbits reported by Li and Liao 43 have a period much longer than 3.9. Therefore, it seems that the ANN model 40 could not predict the periodic orbits of the free-fall three-body system, especially for a long period. solution at t ≈ 2.5 and has an obvious deviation at t = 3.9), but also many of the ANN predictions have a distinct departure even from a small part (within t ∈ [0, 3.9]) of CNS periodic orbits, as shown in Figures 5 to 9.

IV. DISCUSSION
Although the deep neural networks (DNNs) have driven error rates down to 4.9% in voice recognition and less than 3% in image recognition today, which are less than the threshold (5%) of human for similar data 11 , nearly all ML algorithms derive their solutions statistically and probabilistically and therefore are rarely possible to train them to 100% accuracy. Even the best computer systems of voice and image recognition make errors, as do the best humans. Therefore, we had to be tolerant to the imperfection of machine learning system. However, we illustrate here that the current accuracy of the machine learning is not precise enough for correct prediction of periodic orbits of the famous three-body problem in a long enough duration, which has chaotic characteristic. give much larger weight on non-periodic orbits than periodic ones of three-body systems.
Thus, it seems that there should be a quite long way to go for machine learning to accurately predict periodic orbits of three-body systems in a long enough duration.
Finally, we should emphasized that the aim of this paper is not to discourage further exploration of machine learning, since many successful applications in science and industry strongly support its strength and potential. However, it is better for us to know its boundary: