Nonseparable Symplectic Neural Networks
Authors: Shiying Xiong
Yunjin Tong
Xingzhe He
Shuqi Yang
Cheng Yang
Bo Zhu
Venue: ICLR 2021 Poster (2021)
Keywords: Data-driven modeling
nonseparable Hailtonian system
symplectic networks

Abstract: Predicting the behaviors of Hamiltonian systems has been drawing increasing attention in scientific machine learning. However, the vast majority of the literature was focused on predicting separable Hamiltonian systems with their kinematic and potential energy terms being explicitly decoupled, while building data-driven paradigms to predict nonseparable Hamiltonian systems that are ubiquitous in fluid dynamics and quantum mechanics were rarely explored. The main computational challenge lies in the effective embedding of symplectic priors to describe the inherently coupled evolution of position and momentum, which typically exhibits intricate dynamics. To solve the problem, we propose a novel neural network architecture, Nonseparable Symplectic Neural Networks (NSSNNs), to uncover and embed the symplectic structure of a nonseparable Hamiltonian system from limited observation data. The enabling mechanics of our approach is an augmented symplectic time integrator to decouple the position and momentum energy terms and facilitate their evolution. We demonstrated the efficacy and versatility of our method by predicting a wide range of Hamiltonian systems, both separable and nonseparable, including chaotic vortical flows. We showed the unique computational merits of our approach to yield long-term, accurate, and robust predictions for large-scale Hamiltonian systems by rigorously enforcing symplectomorphism.

Published as a conference paper at ICLR 2021
NONSEPARABLE SYMPLECTIC NEURAL NETWORKS
Shiying Xiong Yunjin Tong", Xingzhe Hee, Shuqi Yang", Cheng Yang", Bo Zhu Dartmouth College, Hanover, NH, United States b ByteDance Al Lab, Beijing, China
ABSTRACT
Predicting the behaviors of Hamiltonian systems has been drawing increasing attention in scientific machine learning. However, the vast majority of the literature was focused on predicting separable Hamiltonian systems with their kinematic and potential energy terms being explicitly decoupled while building data-driven paradigms to predict nonseparable Hamiltonian systems that are ubiquitous in fluid dynamics and quantum mechanics were rarely explored. The main computational challenge lies in the effective embedding of symplectic priors to describe the herently coupled evolution of position and momentum, which typically exhibits intricate dynamics. To solve the problem, we propose a novel neural network architecture, Nonseparable Symplectic Neural Networks (NSSNNs), uncover and embed the symplectic structure of a nonseparable Hamiltonian system from limited observation data. The enabling mechanics of our approach is augmented symplectic time integrator to decouple the position and momentum energy terms and facilitate their evolution. We demonstrated the efficacy and versatility of our method by predicting a wide range of Hamiltonian systems, both separable and nonseparable, including chaotic vortical flows. We showed the unique computa- tional merits of our approach to yield long-term, accurate, and robust predictions for acaee Hamiltonian systems by rigorously enforcing symplectomorphism.
INTRODUCTION
Hamiltonian dynamic system refers to a formalism for modeling physical system exhibiting some specific form of energy conservation during its temporal evolution. A typical example is pendulum whose total energy (referred to as the system's 's Hamiltonian) is conserved as a temporally invariant sum of its kinematic energy and potential energy. Mathematically, such energy conservation indicates a specific geometric structure underpinning its time integration, named as symplectic structure, which further spawns wide range of numerical time integrators model Hamiltonian systems. These symplectic time integrators have proven their effectiveness in simulating a variety of energy-conserviee dynamics when Hamiltonian expressions are known as a prior. Examples encompass applications in plasma physics (Morrison) 2005), electromagnetics (Li et al. 2019). fluid mechanies (Salmon) 1988). and celestial mechanics (Saari Xia 1996), to name few.
On another front, the emergence of the various machine learning paradigms with their particular focus on uncovering the hidden invariant quantities and their evolutionary structures enable faithful prediction of Hamiltonian dynamics without knowing its analytical energy expression beforehand. The key mechanics underpinning these learning models lie in proper embedding of the strong mathematical inductive priors to ensure Hamiltonian conservation in neural network data flow. Typically, such priors are realized in a variational way or a structured way. For example, in Greydanus al. 2019), the Hamiltonian conservation is encoded in the loss function. This category of methods does not assume any combinatorial pattern of the energy term and therefore relies on the inherent expressiveness of neural networks distill the Hamiltonian structure from abundant training datasets (Choudhary et al. 2019). Another category of Hamiltonian networks, which we refer to as structured approaches, implements the conservation law indirectly by embedding a symplectic time integrator
(DiPietro et al. 2020 Tong et al. 20200: Chen et al.) 2020) or composition of linear, activation, and
gradient modules (Jin 2020 into the network architecture.
sshiyingxiong@ddrrrmuubbceccc

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Conventionll Convertiene Dowe Aarrey
(NSSSNN))
Cradteet
Gradert coseeeoeeee.
Natworka
Noural Fastworks
Beckward
Copparable System
Nonsoparable System
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Figure 1:
Comparison between NSSNN and HNN regarding the network design and prediction results of a vortex flow example.
One of the main limitations of the current structured methods lies the separable assumption of the Hamiltonian expression. Examples of separable Hamiltonian systems include the pendulum, the (Zhu et al. 2016), the Kepler (Antohe & Gladwell] (004), and the Hénon-Heilld systems (Zotos 2015). However, beyond this scope, there exist various nonseparable systems whose Hamiltonian has no explicit expression to decouple the position and momentum energies. Examples include incompressible flows (Suzuki al., (007), quantum systems (Bonnabel et al.) 2009). rigid body dynamics (Chadaj et al. (011)) charged particle dynamics (Zhang et al.) (0166), and nonlinear Schrödinger equation (Brugnano et al. 2018). This nonseparability typically causes chaos and instability, which further complicates the systems' dynamics. Although SympNet in Jin al. (2020) can be used learn and predict nonseparable Hamiltonian systems, multiple matrices of the same order with system dimension are needed in the training process of SympNet, resulting in difficulties generalizing into high-dimensional large-scale N-body problems which are common in series of nonseparable Hamiltonian systems, such as quantum multibody problems and vortex- particle dynamics problems. Such chaotic and large-scale nature jointly adds shear difficulties for a conventional machine learning model to deliver faithful predictions.
In this paper, we propose an effective machine learning paradigm to predict nonseparable Hamiltonian systems. We build novel neural network architecture, named nonseparable symplectic neural networks (NSSNNs), to enable accurate and robust predictions long-term Hamiltonian dynamics based short-term observation data. Our proposed method belongs the category of structured network architectures: it intrinsically embeds the symplectomorphism into the network design strictly preserve the symplectic evolution and further conserves the unknown, nonseparable Hamiltonian energy. The enabling techniques we adopted in our learning framework consist of an augmented symplectic time integrator to asymptotically "decouple" the position and momentum quantities that were nonseparable in their original form. We also introduce the Lagrangian multiplier the augmented phase space to improve the system's 's numerical stability. Our network design motivated by ideas originated from physics (Tao) 2016) and optimization (Boyd et al. 2004). The combination of these mathematical observations and numerical paradigms enables novel neural network architecture that drastically enhance both the scale and scope of the current predictions.
We show a motivational example in Figure by comparing our approach with a traditional HNN method (Greydanus et al. 2019) regarding their structural designs and predicting abilities. We refer the readers to Section 6 for a detailed discussion. As shown in Figure I the vortices evolved using NSSNN are separated nicely the ground truth, while the vortices merge together using HNN due to the failure of conserving the symplectic structure a nonseparable system. The conservative capability of NSSNN springs from design of the auxiliary variables (red and y) which converts
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the original nonseparable system into higher dimensional quasi-separable system where we can adopt a symplectic integrator.
2 RELATED WORKS
Data-driven physical prediction. systems including fluid mechanics (Brunton et al., 2020), wave physics (Hughes et al. 2019). quantum physics (Sellier al. 2019), thermodynamics (Hernandez al. 2020), and material science (Teicherta et al. 2019). Among these different physical systems, data-driven fluid receives increasing attention. We refer the readers to Brumton et al. (2020) for thorough survey of the fundamental machine learning methodologies as well as their uses for understanding, modeling, optimizing, and controlling fluid flows experiments and simulations based on training data. One of the motivations of our work is to design versatile learning approach that can predict complex fluid motions. On another front, many pieces of research focus on incorporating physical priors into the learning framework, e.g., by enforcing incompressibility (Mohan et al. (2020), the Galilean invariance (Ling et al. 2016), quasistatic equilibrium (Geng et al. 2020), the Lagrangian invariance (Cranmer et al. 2020 and Hamiltonian conservation (Hernandez et al. 22200; Greydanus et al.) [0119] Jin et al. 22200] Zhong et al. 2020). Here, inspired by the idea of embedding physics priors into neural networks, we
Data-driven approaches have been widely applied in physical
aim accelerate the learning process and improve the accuracy of our model.
Neural networks for Hamiltonian systems. Greydanus et al. (2019) introduced Hamiltonian neural networks (HNNs) to conserve the Hamiltonian energy of the system by reformulatio latingg the loss function. Inspired by HNN, series of methods intrinsically embedding a symplectic integrator into the recurrent neural network was proposed, such as SRNN (Chen et al.) (020)) TaylorNet (Tong et al. 2020) and SSINN (DiPietro et al.) 2020). to solve separable Hamiltonian systems. Combined with graph networks (onnalzz-Gonzalez et al. 2019 BBttaglia et al. (016). these methods were further generalized
large-scale N-body problems induced by interaction force between the particle pairs. Jin et al. (2020) proposed SympNet by directly constructing the symplectic mapping of system variables within neighboring time steps to handle both separable and nonseparable Hamiltonian systems. However, the scale of parameters in SympNet for training N dimensional Hamiltonian system is O(N)). which makes it hard to be generalized to the high dimensional N-body problems. Our NSSNN overcomes these limitations by devising new Hamiltonian network architecture that is specifically suited for nonseparable systems (see details in Section 55) In addition, the Hamiltonian- based neural networks can be extended to further applications. Toth et al. (2020) developed the Hamiltonian Generative Network (HGN) to learn Hamiltonian dynamics from high-dimensional observations (such images). Moreover, Zhong al. (2020) introduced Symplectic ODE-Net (SymODEN), which adds an external control term to the standard Hamiltonian dynamics.
FRAMEWORK
3.1
AUGMENTED HAMILTONIAN EQUATION
We start by considering Hamiltonian system with N pairs of canonical coordinates (i.e. N generalized positions and N generalized momentum). The time evolution of canonical coordinates is governed by the symplectic gradient of the Hamiltonian (Hand & Finch 2008). Specifically, the time evolution of the system is governed by Hamilton's 's equations as
dq 0H dt 0p
dp 0H dt aq
(1)
with the initial condition ().p), (go. Po). In a general setting, == (91, 42. , .qN) represents
the positions and p = ...)) denotes their momentum. Function H H(q. p) is the Hamiltonian, which corresponds to the total energy of the system. An important feature of Hamilton's equations is its symplectomorphism (see Appendix B for a detailed overview).
The symplectic structure underpinning our proposed network architecture draws inspirations from the original research of Tao (2016) in computational physics. In TTo (2016), a generic, high-order, explicit and symplectic time integrator was proposed to solve (T) of an arbitrary separable and nonseparable
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Hamiltonian H. This is implemented by considering augmented Hamiltonian
H(q. p. x, y) = H. + HB + wHo
(2)
with
HA H y). Hg = H( p): He 2 (1lq - x 1p y
(3)
in an extended phase space with symplectic two form dq A dp + dx A dy. where w is a constant that controls the binding of the original system and the artificial restraint.
Notice that the Hamilton's 's equations for H.
dq 0H 0HL(x.)) p) + w(p
- yy),
di 0p
Op
dp Htl Mll., y) w(q x),
dt 0q
dq
(4)
dæ HH 0H(q. y)
w(p y)),
di
dy
dy
dy
OH
0H( p) dx
ww xx),
dt dx
with the initial condition (g. P. v, y) ltto (go. Po, Qo Po) Po) have the same exact solution
in the sense that (q.P. P. x. y) = (q. q. p). Hence, we can get the solution of [ by solving (4)
Furthermore, it is possible to construct high-order symplectic integrators for H. in (4) with explicit
updates. Our model aims to learn the dynamical evolution of (q. p) in (I) by embedding 4 into the
framework of NeuralODE (Chen et al. 2018). The coefficient w acts as regularizer, which stabilizes the numerical results (see Section
3.2 NONSEPARABLE HAMILTONIAN NEURAL NETWORK
We learn the nonseparable Hamiltonian dynam- ics (T) by constructing augmented system (4). from which we can obtain the energy func- tion H(g. p) by training the neural network Ho(q. P) with parameter 0 and calculate the gradient Voo (g. p) by taking the in-graph gra- dient. For the constructed network Ho (q. ().
Algorithm Integrate (4) by using the second- order symplectic integrator
Input: Po: to t, dt;
Φi. Φ and Φc in 5):
Output: (q. ,, a. ÿ)
(qn Pay nn Yn
(go- Po. wo yo)
(4o. Qo, Po)
n
floor td)/ for =i=1
do de
we integrate 44) by using the second-order sym- plectic integrator (Tao] 2016). Specifically, we will have an input layer (g.p, x; y) (go. Po. qo. Po) at t to and an output layer end (q. p. x, y) = (qn. Pn. Fn. yn.) at to + nntt.
(q, Pr, Fz yi) = 01 002
dt/2
oØ
dt/2
dt/2 Φ
(q-11, 1i-1, æ; 1:=1);
The recursive relations of [g., Ps: 3;. y:),
1,2, n, be expressed by the algorithm (also
see Figure in Appendix A). The input functions Φf (q. p, x. y), (2(4, p, x. y). and & qg x. y) in algorithm are
9 p 810Ha q. y) /0q x + 8j07L /Op
q + 8(HHo(H.
+
+ R
q x
p + y
p y
and
(5)
2
+ 2
Rd
q T
y
yy 51876 (. p9
p
p y
respectively, Here
cos (2o)) I sin (2w6)
sin (2w)) I
Rd =
cos (2w6) 6 where I is identity matrix.
(6)
We remark that and y are just auxiliary variables, which are theoretically equal to q and Therefore, we can use the data set of (q. p) construct the data set containing variables (q. ,, x. y). In addition, by constructing the network Hd, we show that theorem B. in Appendix B holds, so the networks of Φš, and Φ$ in (5) preserve the symplectic structure of the system. Suppose that $ and 22 two symplectomorphismm.. Then, it is easy to show that their composite map o o is also symplectonorphism due to the chain rule. Thus, the symplectomorphism of algorithm can be guaranteed by the theorems B.1
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(0)
0.3
(6)
Test, or 2000
Test, or 2000
Test, 0=200
00
Test, u-
0.2
Test, o=-20
Test, 0-2000
Trin,
0.3
Trinn 200
0.1
Train a00
0.1
Test, a=200)
Train, 20
Train, om
Train, 20-2000
Train, e--200
0-
Train, e>
150
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spoch
gpoch
spoch
Comparisons of training and validation losses with different in the training process. The solid and dashed lines represent training and validation losses respectively. The networks are trained by (a) the clean dataset, (b) the dataset with noise ~ 0.050 (-,, and (c) the dataset with noise ~ 01,,)-
Figure 2:
4 TRAINING SETTINGS AND ABLATION TESTS
We use 6 linear layers with hidden size 64 to model Hg. all of which are followed by Sigmoid activation function except the last one. The derivatives öHa/ 0Ha/Op. 0Ha 0'Ha/qq oHo /Ox ö'Ho/ jo are all obtained by automatic differentiation in Pytorch (Paszke et al. (2019). The weights of the linear layers are initialized by Xavier initializaiton (Glorot & Bengio 2010).
We generate the dataset for training and validation using high-precision numerical solver (Tao 2016), where the ratio of training and validation datasets is We set the dataset (4b.P)) ) as the start input
and (q' pp) as the target with j 1,2, .N.. and the time span between (40= pb) and (q², pp) is Ttrain- Feeding (9o. Po) = -(g pb). to = 0, = Tirain and time step dt in Algorithm to get the predicted variables (q', p¹, 2., ))) Accordingly, the loss function is defined as
No
CNSSNN = No Σ
100|| + +1(p() -b0) 11- (l +6011 + (10()) - 1001 (7) (0
where N₂, 512 is the batch size of the training samples. We use the Adam optimizer (Kingma & Ba 2015) with learning rate 0.05. The learning rate is multiplied by 0.8 for every 10 epoches.
Taking system H(g. p) 0.5 + 1) + 1) an example, carry out series of ablation tests based on our constructed networks. Normally, we set the time span, time step and dateset size as T 0.01, dt 0.01 and N, 1280.
The choice of w in (4) is largely flexible since NSSNN not sensitive to the parameter w when is larger than certain threshold. Figure shows the training and validation losses with different w in the network trained by clean and noise datasets. Though the convergence rates are slightly different in small scope, the examples with various w are able to converge to the same size of training and validation losses. Here, we set w 2000, but w can be smaller than 2000. The only requirement for picking w is that has be larger than (10), which detailed in Appendix C
We pick the L1 loss function to train our network due to its better performance. Figure 3 compares the validation losses with different training loss functions in the network trained by clean and noise datasets. Figure 3 a) shows that either the network trained by L1 or MSE with clean dataset can converge small validation loss, but the network trained by L1 loss converges relatively faster. Figures b)) cc) both show that the network trained by L1 with noise dataset can converge to smaller validation loss. In addition, we already introduced regularization term in the symplectic integrator embedded in the network; thus, there no need to add the regularization term in the loss function.
The integral time step in the sympletic integrator is vital parameter, and the choice of d' largely depends on the time span Ttrein- Figure 4 compares the validation losses generated by various integral time steps dt based fixed dataset time spans Taaaán 0.01, 0.1 and 0.2 respectively in the training process. The validation loss converges to a similar degree with various di based on fixed T Trrain 0.01 and T Trrain 0.1 in aa) and (b), while it increases significantly as di increases based
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(8)
0.3
(6)
Triined byLL Tested byLi
Trained byå1, Tested by
Trained by ses Tested by
Trained
A1l
Trained byL1 Tested by sis
0.2
Trained by Tested by Mss
lass
Trined by MSK Tested by sesB
Triied by MRR Tested MSE
Triined by Tested by L.1
0.3
0.1
0.1
Trained bySIs, Tested by L.l
0-
Trained byd1, Tested by aists Trained by wer. Testod bynasoo
150
200
50
100
200
150
200
spoch
epoch
spoch
Figure 3: Comparisons of validation losses with different training loss functions in the training process. The red solid and dashed lines represent the networks trained by L1 loss function and validated by L1 and MSE respectively; the blue solid and dashed lines represent the networks trained by MSE and validated by L1 and MSE respectively. The networks are trained by (a) the clean dataset, (b) the dataset with noise 0.05U 1), and (c) the dataset with noise (-4
(a) 0.2
(b)
2
(c)
0.01
0.1
0.005
1.5
di 0.05
d: 0.04
dt= 0.002
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lozs
di 0.001
loss
dt 0.01
loss
0.5
0
-0.5
50
100
130
200
50
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opoch
spock
spoch
Figure 4: Comparisons validation losses with different dt in the training process. (a), (b), and (c) are trained based on different time spans Train 0.01, 0.1, and 0.2, respectively.
on fixed Terain = 0.02 in C). Thus, we should take relatively small di for the dataset with larger time span T Ttrain
5 COMPARISONS WITH OTHER METHODS
5.1 METHODOLOGIES
We compare our method with other recently proposed methods, such as HNN (Greydanus et al. 2019 NeuralODE Chen et al. 2018), TaylorNet (Tong et al. 2020), SSINN (DiPietro et al. 2020). SRNN (Chen al. 2020), and SympNet (Jin et al.) 2020). There several features distinguishing method from others, as shown in Table [I] HNN first enforces conservative features of a Hamiltonian system by reformulating its loss function, which incurs two main shortcomings. On the hand, requires the temporal derivatives of the momentum and the position of the systems to calculate the loss function, which difficult to obtain from real-world systems. On the other hand, HNN doesn't strictly preserve the symplectic structure, because its symplectomorphism is realized by its loss function rather than its intrinsic network architecture. NeuralODE successfully bypasses the time derivatives of the datasets by incorporating integrator solver into the network architecture
Embedding the Hamiltonian prior into the NeuralODE, series of methods are proposed, such as SRNN, SSINN, and TaylorNet, to predict the continuous trajectory of system variables; however, presently these methods are only designed to solve separable Hamiltonian systems. Instead of updating the continuous dynamics by integrating the neural networks in NeuralODE, SympNet adopts a symplectomorphism composed of well-designed both linear and non-linear matrices to intrinsically map the system variables within neighboring time steps. However, the parameters scale in the matrix map for training N dimensional Hamiltonian system in SympNet is O(N2), which makes it hard to generalize the high dimensional N-body problems. For example, in Section 6 predict the dynamic evolution of 6000 vortex particles, which is challenging for the training process of the SympNet on the level of (66002).

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Table 1: Comparison between HNN
(Greydanus al. 2019), NeuralODE (Chen al. 2018).
TaylorNet (Tong et al., 2020). SSINN (DPiictro et al. 2020), SRNN (Chen et al., 2020). Sym SympNet (Jin et al. 2020), and NSSNN. V represents the method preserves such property.
Methods
NSSNN HNN NeuralODE TaylorNet SSINN SRNN SympNet
Solve nonseparable systems
Solve separable systems
Preserve symplectic structure
Partially
Partially
Utilize continuous dynamics
No need for derivatives in dataset
Long-term predictability
Partially
Extend to N-body system
NeuralODE
HNN
IHNN
HRK
NSSNN
Figure 5: Comparison of prediction results of (q, p) for the spring system 71 = 0.5(g² + p²) from t 200 with (go.Po) (0, 3).. The time span the datasets Terain 0.4 (first row) and Tirain = 1 (second row). The five columns are five different methods NeuralODE, HNN, IHNN, HRK, and NSSNN, respectively. The red line denotes the ground truth; the blue line denotes prediction, which perfectly overlapping in NSSNN. The prediction ability of HNN and IHNN improves significantly with the decreasing of Trroin of the dataset which however may be hard to obtain in the actual experimental measurements.
NSSNN overcomes the weaknesses mentioned above. Under the framework of NeuralODE, NSSNN utilizes continuously-defing dynamics in the neural networks, which gives it the capability to learn the continuous-ting time evolution of dynamical systems. Based on Tao (2016), NSSNN embeds the symplectic prior into the nonseparable symplectic integrator to ensure the strict symplectomorphism, thereby guaranteeing the property of long-term predictability. In addition, unlike SympNet, NSSNN highly flexible and can be generalized to high dimensional N-body problems by involving the interaction networks (Sanalez-Gonzalez al. 2019), which will be further discussed in Section 6
5.2 EXPERIMENTS
We compare five implementations that learn and predict Hamiltonian systems. The first is NeuralODE, which trains the system by embedding the network fo - (dg/dt. dp/dt) into the Runge-Kutee Kutta (RK) integrator. The other four, however, achieve the goal by fitting the Hamiltonian Ha + H based on [I]. Specifically, HNN trains the network with the constraints of the Hamiltonian symplectic gradient along with the time derivative of system variables and then embeds the well- trained H&s into the RK integrator for predicting the system. The third and fourth implementations ablation tests. One of them improved HNN (IHNN), which embeds the well-trained He into the nonseparable symplectic integrator (Tao's 's integrator) for predicting. The other is to directly embed Ho into the RK integrator for training, which we call HRK. The fifth method is NSSNN, which embeds Hp into the nonseparable symplectic integrator for training.
For fair comparison, we adopt the same network structure (except that the dimension of output layer in NeuralODE is two times larger than that in the other four), the same I.1 loss function and same
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NeuralODE
HNN
IHNN
HRK
NSSNN
-2
Figure 6: Comparison of prediction results of (q, p) for the Tao's system H 0.5 1) 1)
from t = 0 to 20000 with (qo.Po) = (0. -3). The network is trained by the dataset with noise 0.05U (-1, 1), and the time span of the dataset Terain 0.2. The red line denotes the ground truth; the blue line denotes the prediction, which are perfectly overlapping in NSSNN.
size of the dataset, and the precision of all integral schemes is second order, and the other parameters keep consistent with the one in Section 4 The time derivative in the dataset for training HNN and IHNN is obtained by the first difference method
dq (Terai))) (()) and
dp p(Tirain) q(0)
(8)
dt
T Ttrain
df
Ttrain
Figure demonstrates the differences between the five methods using spring system 7 0.5(g² p²) with different time span Tirain = 0.4, and same time step dt 0.2. We can see that by introducing the nonseparable symplectic integrator into the prediction of the Hamiltonian system, NSSNN has stronger long-term predicting ability than all the other methods. In addition, the prediction of HNN and IHNN lies the dataset with time derivative; consequently, it will lead to larger error when the given time span Tirain is large.
Moreover, the datasets obtained by II in HNN and IHNN sensitive noise. Figure compares the predictions of (g.p) for the system 7- = 0.5 - () + 1). where the network is trained by the dataset with noise 0.05U 1). along with time span Teraiz 0.2 and time step dt 0.02. Under the condition with noise, NSSNN still performs well compared with other methods. Also, we compare the convergent error of a series of Hamiltonian systems with different H trained with noisy data in Appendix D which generally shows better robustness than HNN does.
6 MODELING VORTEX DYNAMICS OF MULTI-PARTICLE SYSTEM
For two-dimensional vortex particle systems, the dynamical equations of particle positions (Iz58) Mj), j 1,2, N, with particle strengths r can be written the generalized Hamilto- nian form as
r, dr dt
8HP oy,
dyj I dt Oxy""
8H"
with HI
Σ (oTlogg(22 xx()- =.k=1
(9)
By including the given particle strengths r in Algorithm I we still adopt the method mentioned above to learn the Hamiltonian in (9) when there are fewer particles. However, considering a system with N₂ 2 particles, the cost to collect training data from all N,, particles might be high, and the training process can be time-cons time-consuming. Thus, instead of collecting information from all N, particles train our model, only use data collected from two bodies training data make predictions of the dynamics of N, particles.
Specifically, assume the interactive models between particle pairs with unit particle strengths I = 1 are the same, and their corresponding Hamiltonian can be represented network H. (z) at.). based on which the corresponding Hamiltonian of N,, particles can be written as (Battaglia et al. 2016 Sanchez-Ginzalle al. 2019)
H Σ I,TNHo(z,,
(10)
We embed (10) into the symplectic integrator that includes 1 obtain the final network architecture.
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Talue votlox
Lexpleeg sceies
Touth
Gomnd
XSSNN
NESNN
JINN
HXX
122
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Trajectories worter porticlss
Figure 7: Taylor and Leapfrog vortex. We generate results of Taylor vortex and Leapfrop vortex using NSSNN and HNN, and compare them with the ground truth. 6000 vortex elements are used with corresponding initial vorticity conditions of Taylor vortex and Leapfrop vortex.
The setup of the multi-particle problem is similar to the previous problems. The training time span Tirain 0.01 while the prediction period can be up T predict 40. We use 2048 clean data samples to train our model. The training process takes about 100 epochs for the loss to converge. In F we use our trained model to predict the dynamics of 6000-particle systems, including Taylor and Leapfrog vortices. We generate results of Taylor vortex and Leapfrop vortex using NSSNN and HNN and compare them with the ground truth. Vortex elements are used with corresponding initial vorticity conditions of Taylor vortex and Leapfrop vortex (Qu al. 2019). The difficulty of the numerical modeling of these two systems lies the separation of different dynamical vortices instead of having them merging into bigger structure. In both cases, the vortices evolved using NSSNN are separated nicely as the ground truth shows, while the vortices merge together using HNN.
7 LIMITATIONS
The network with the embedded integrator is often more conne-innuming to train than the one based on the dataset with time derivative. For example, the ratio of training time of the methods HNN and NSSNN is 1 3 when df = Tirain, and the training time of the recurrent networks further increases with the decreasing of dt. Although smaller di often has higher discretization accuracy, there is tradeoff between training cost and predicting accuracy. Additionally, a smaller dt may potentially cause gradient explosion. In this case, may want use the adjoint method instead. Another limitation lies in the assumption that the symplectic structure is conserved. In real-world systems, there could be dissipation that makes this assumption unsatisfied.
8 CONCLUSIONS
We incorporate a classic ideal that maps a nonseparable system to higher dimensional space making it quasi-separable to construct symplectic networks. With the intrinsic symplectic structure, NSSNN possesses many benefits compared with other methods. In particular, NSSNN the first method that can learn the vortex dynamical system, and accurately predict the evolution of complex vortex structures, such as Taylor and Leapfrog vortices. NSSNN, based on the first principle of learning complex systems, has potential applications in fields of physics, astronomy, and weather forecast, etc. We will further explore the possibilities of neural networks with inherent structure- structure-preserving ability in fields like 3D vortex dynamics and quantum turbulence. In addition, applications of NSSNN with datasets based on images or other real scenes through automatically identifying coordinate variables of Hamiltonian systems based neural networks.
will also work general

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Published as a conference paper at ICLR 2021
ACKNOWLEDGMENTS
This project is supported in part by Neukom Institute CompX Faculty Grant, Burke Research Initiation Award, and ByteDance Gift Donation. Yunjin Tong is supported by the Dartmouth Women in Science Project (WISP), Undergraduate Advising and Research Program (UGAR), and Neukom Scholars Program.
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Published as a conference paper at ICLR 2021
Eessan ln. oll in Ml-M
6. +
Spapkeele ieAggrsto
JUSI
Figure 8: (a) The forward pass of an NSSNN is composed of forward pass through a differentiable symplectic integrator as well as a backpropagation step through the model. (b) The schematic diagram of NSSNN.
B. Zhu, R. Zhang, Y. Tang, X. Tu, and Y. Zhao. Splitting k-symplectic methods for non-canonical separable hamiltonian problems. J. Comput. Phys., 322:387-399 2016.
E. F. Zotos. Classifying orbits in the classical hénon- hamiltonian system. Nonlinear Dynam., 79:1665-1677, 2015.
A NETWORK ARCHITECTURE
Figure 8 () shows the forward pass of NSSNN is composed of forward pass through a differentiable symplectic integrator as well as a backpropagation step through the model. Figure 8 plots the schematic diagram of NSSNN. For the constructed network Ho q. p), integrate 44) by using the second-order symplectic integrator (Tao 2016). Specifically, The input layer of the integrator (g. P, æ. y) (go: Po; 90, po) / and the output layer is (g, p. v. y) (an Pr: wn; ya)
at = to + ndd.. The recursive relations (qt, Pi, y:), = 1, ... n, are expressed by the algorithm
B SYMPLECTOMORPHISK,
One of the most important features of the time evolution of Hamilton's equations is that it is symplectomorphism, representing transformation of phase space that is volume-press preserving. In the setting of canonical coordinates, symplectomorphism means the transformation of the phase flow of a Hamiltonian system conserves the symplectic two-form
dq dp Σ (dgy dp,),
(11)
where A denotes the wedge product of two differential forms. The rules of wedge products can be found in Lee (200)). In the two-dimensional case, [TT can be understood as the area element of the surface. In this case, the symplectomorphism can be interpreted as the area element of the surface is constant. As proved below, our constructed network structure intrinsically preserves Hamiltonian structure.
Theorem B.1. For a given 6, the mapping &f, Φb and oi in are symplectomorphissms.
Proof. Let
(7,,5,,5,,1) H3 9), j= 1,2,3.
(12)
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Published as a conference paper at ICLR 2021
(8)
(8)
(0)
(d)
2
-2
22
-2
(0)
(
(b)
500H0
50000
500K0
100000
50000
100000
Figure 9: Comparison of different (from left right columns) with (a) and (e) 0, (b) and (f) w = 0.8, (c) and (g) = 0.9, (d) and (h) w 10 in the symplectic integrator for nonseparable
Hamiltonian H = (q + 1) 1)/2. The upper row: projection of a trace (())), <x)), v(O) with (g()), a( s(0 (-3, 0, 3. 0) onto the q p plane. The bottom row: deviation
< = ()) (ar, 22 between (q. p) and (2r, v).
From the first equation of ( we have
dt; A de + de; dt"
dg Ad p 070o(q.0) y)
HHo (.:((q:)) y)
d
x
^ dy
dq
3p
(13)
(Holo. Oqdy
oHo y)
=dq A dp + dx A dy +
dq A dy
Qydq
=qq A dp dr ^ dy.
Similarly, we can prove that dt,; A dt" dt; dt" = dq ^ dp equation of (5), we can directly deduce that dty dt dtz A dt dq dp + dæ dy.
+ dæ A dy. In addition, from the third
Suppose that @ and 42 two symplectomorphissm. Then, it is easy show that their composite map 22 @ is also symplectomorphismm due to the chain rule. Thus, the symplectomorphism of algorithm can be guaranteed by the theorem B.I.
C DETERMINING COEFFICIENT w
To further elucidation, the Hamiltonian H. H without the binding, i.e., H. with w = 0, in extended phase space (q. p, x, 3/ may not be integrable, even if H(q.p) p) integrable the original phase space (). However, Hc is integrable. Thus, as w increases, larger proportion in the phase space for H. corresponds to regular behaviors (Kolmogorov 1954. For H(g.p) = 1)) 1)/2. shown in Fig. 9 we compare the trajectories starting from (0), p()) (0), =0) -,,00.3,,00 3) calculated by the symplectic integrator (Tao) (016)) with different where the calculation accuracy second order accuracy and the time interval is 0.001. As Figs. aa). (b), (c), and (d) shown, the chaotic region in phase space is significantly decreasing until forming stable limit cycle. We define == (l( p) (x. 2/2 as the calculation error of this system, shown in Figs. (c), (f), (g), and (h) that the error is decreasing with increasing, which fits the quantitative results of phase trajectory well.
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Published as a conference paper at ICLR 2021
Table 2: Comparison of Prediciton error and Hamiltonian deviation between NeuralODF, HNN and
NSSNN
Prediciton error
Hamiltonian deviation
Problems
NeuralODE
HNN
NSSNN
NeuralODE
HNN
NSSNN
Pendulum
3.4 10-2 3.1 x 10-2 2.6 X 10-2
1.5
10-2
1.3 x 10
7.4 X 10-3
Lotka-Volterra
2.2 x 10 3.9 x 10 2.7 x 10-2
7.4
10
8.8 x 10
7.5 X 10
Spring
2.1 x 10-2
x.1 1.6 10-2
x
6.7 x 10
6.7 X 10-3
Hénon-Heiles
1.0 10
9.4
8.4 10-2
3.7x
X 10 x.0 10-2
Tao's example
3.7 10-2
2
2.2
10-2
1.4 10-2
1.1 1
8.2 X 10
Schrödinger
8.7 X 10
5.9
5.7
10-2
3.8 x 10-2
2.3 10
2.0 X 10
Vortex (2 particles)
2.1 X 10
7.7
3.4X
10-3
1.5 10
2.8x
2.1 X 10-3
Vortex (4 particles)
3.4 10-2
x
6.9
10-3
8.2x 10
1.4
X 10-3
D OTHER EXPERIMENTS
We consider the pendulum, the Lotka-Volterra, the Spring, the Hénon-Heiles, the Tao's example TTao 2016), the Fourier form of nonlinear Schrödinger and the vortex particle systems in our implementation. The Hamiltonian energies of these systems (except vortex particle system) are summarized as follows:
Pendulum system: H(g. p) 3(1 3(1-cos()) +p².
system: H(g.p) p- 2- ea
Spring system: H(q.p) q
22 p2. Héilns system: H(gn, (2. P2,P1, P2) (p} +22) (of
) + (gi92 92 - /2/3) Tao's example (Tao 2016): H(g. p) (4 + +1 + +1/2.. Fourier form of
nonlinear Schrödinger equation: H(g 42. P1. P2) (4f + pi)² + (gz + p2) /4 +
qip3 p 49192P1P2).
The network is trained by the dataset with noise ~ (-1,(-1,1) The training time span, integral time step, and validation time span are 0.01, 0.01, and 0.1, respectively. quuth Proth 2|2 and the prediction error deviation = || Pawt)) - H Apqpecii.. Ppredic) //2/ (//uu
the Hamiltonian
ep = 1ruth Qpredier || + P Ppredier|| It is clearly from the Table 2 that NSSNN either outperforms or has similar performances as NeuralODE and HNN do.
14


Reviews:
[{"review_id": "xP5e35Dk4pN", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The work proposes a novel method for solving non-separable Hamiltonian systems, using Tao's approach in which two copies of the phase space are tied together by an additional Hamiltonian. This appears to be a novel proposal, and certainly of interest. \n\n# Positioning w.r.t. related work.\nThe proposed method specifically focuses on non-separable systems. The work by Jin et al. (2020) is mentioned in passing, but this work appears to address the problem of non-separability as well. It would thus make sense to dive deeper into how the proposed method compares to Jin's, both theoretically and empirically. Some arguments require citations, such as \"variational methods are not well suited for tackling such challenges due to heir inherent weaknesses\": which weaknesses? \n\n# Discussion of the method.\nAlthough the method appears to perform favourably in all provided benchmarks, it remains unclear how the method compares on computational and memory complexity. This should be added to the discussion, with a theoretical and/or empirical analysis. It would also be beneficial if areas would be highlighted where the method falls short. \n\n# Experiment section.\nAlthough there is a section called 'Ablation test', an actual empirical ablation study is missing. As the proposed method introduces various moving parts including a specific loss, phase space parametrisation, and specific neural architecture fo H. It is not clear what the relative contribution is of these parts to the improved performance over the baselines. A closer study on this, as well as the influence of the hyperparameters such as t, would shine more light on the characteristics of the proposed method. Moreover, the experimentation section does not convince that the baselines' hyperparameters have been fairly tuned, and if a potential increase of parameters in H might contribute to the stronger performance.\n\n# Clarity of writing.\nI found the paper difficult to follow. The meaning of some phrases is unclear, e.g. 'which contributes robust for wider datasets'. \n\n# Conclusion.\nThe work is interesting, but the analysis of the method is incomplete due to the lack of comparison with a recent competing method, and a missing discussion of the potential tradeoffs involved in choosing this method over others. \n\nCouple nitpicks outside of the review:\n- \"Nonsep_e_rable\" appears a couple of times\n- Line 197: 'systmes.\n\nUpdate: The authors have addressed the most pressing issues with the manuscript. I've increased my score and vote in favour of accept. The section of limitations was difficult to follow and would benefit from a more structured comparison with competing methods. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "tylnNV4SPlq", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors propose a variation of Hamiltonian Neural Networks (HNNs) with a built-in symplectic integrator. While built-in symplectic integrators have already been used in HNNs, the most popular symplectic integrators only work for separable Hamiltonians. The authors use Tao’s integrator (2016) at the core of their model (HSSNN) to achieve better conservation of energy in systems with non-separable Hamiltonians. Many works have experimented with training with different integrators, and in that sense this is a very incremental contribution, however, Tao’s integrator is a state of the art integrator, and I think the community will benefit from reading about it, and seeing it implemented.\n\nThe introduction and model description part of the paper is clear in general and reasonably easy to follow, and does a very decent job at introducing Tao’s integrators in a very compact way. Many details for reproducibility are missing from the experimental sections of the paper (specially with respect to the implementation of the baselines), but this should be alleviated by the source code provided by the authors. The final section on the multi-particle vortices is probably the most unclear, and should be improved.\n\nConsidering that this is a very empirical paper, with an incremental contributions that only make use of toy environments, it should probably do a bit more of a thorough job at really trying to ablate the model in more sophisticated ways.\n\nI think there are two important baselines missing:\n* Performance of their model e.g. using a RK4 integrator, instead of Tao's. This would really tease apart the the effect of the symplecticity. Note this is an important ablation that is between this model and HNN (which is trained by supervising the gradients, rather than through an integrator).\n* Training HNN using the gradients, but then integrate the learned Hamiltonian using Tao’s integrator at test time (or even better: a time adaptive version of Tao’s integrator). I would be surprised if this does not perform as well as HSSNN (Although as the authors say, requiring the gradients to train HNN is a very rigid constraint, so this would not take away from the importance of their work).\n\nAlso in the multi-particle system, I am surprised the authors did not choose a Hamiltonian Graph Network (https://arxiv.org/pdf/1909.12790.pdf). I do not think at this stage I would need to see this included, but probably a mention to Hamiltonian Graph Networks in the context of multi-particle systems would be appropriate.\n\nAlso, Figure 3 b) and c), seem to have weird patterns that would be interesting to try to gain more insights into because I cannot tell if, for example, the variance in Fig 3c is just noise, or something more intrinsic to the experiment. So I really don’t know what to make of plots Figure 3 b) and c). Maybe plotting them for more environments or adding some form of errors bars would help understand them better.  \n\nSimilarly, in Table 3, I would have expected larger differences between HNN, and HSSNN for the Hamiltonian Deviation column. Maybe it would be possible to use longer trajectories and amplify the differences more? In fact the differences in terms of conservation of energy for Tao’s environment seem to be much smaller quantitatively in Table 3 than qualitatively in Figure 4, was the example from Figure 4 cherrypicked?\n\nI think some additional comparison ablations, like some of those provided in https://arxiv.org/pdf/1909.12790.pdf, would really help gain more insight on the model:\n* Try different integrators at train and test time\n* Train on a range of time steps during training, instead of a single values.\n\nI think the paper is a bit borderline, and in its current form a lean on the side of \"Weak Reject\", but would be happy to raise the score, if the authors can make improvements in the axis mentioned above.\n\nSome questions I am curious about (have not affected my decision):\n* Any particular reason to use an l1 loss, instead of l2? I wonder if this can also cause differences with the baselines which use l2. The prediction error seems to be defined also using l1. Is it not unfair to compare l1 to the baselines, when baselines are trained with l2 loss (or at least they were in their original papers)?\n* Table 1 says that HNN requires the gradients, but if I remember correctly the HNN paper model does mention the possibility of estimating the derivatives with finite differences, which is equivalent to training through an Euler integrator.\n\nSome minor comments (have not affected my decision):\n\nThe text in the experimental and results sections feels a bit rough at times, would recommend rewriting most of it, making sure that the message of each sentence is clear and unambiguous.\n\n> these nonseparable systems exhibit plateau of degrees of freedom, demonstrating complexities that are orders-of-magnitude higher than separable systems (whose degree of Freedoms are typically below 10).\n\nAre not many n-body systems highly separable, yet they have much much higher numbers of degrees of freedom?\n\nFigure 1 could be clearer, the intended message of the crossed-out notation in red is not obvious.\n\nFigure 2b bit unclear. The plot implies n iteration steps, but this is not really conveyed really well.\n\nMissing “/” in line 169\nMisspelled “systmes” (197)\n\nNotation in equation (7):\nVectors (bold) or scalars (italics)\nWhat about the subindex, should I assume you train on one step data, and not on sequences?\nIf the formula refers to the training loss, the limit of the sum should probably be the batch size, and not the number of training samples?\n\nThe term “Strong stability” in Table 1 seems not very scientific. Maybe something like “symplectic stability” or something like that would be more specific.\n\nTable 2: Hamiltonian Deviation (Row “spring”) HNN and NSSNN seem to have the same error, so maybe both should be in bold.\n\nEquation 2016 wrong left side, H(p, q)_predict, should probably be H(p_predict, q_predict) since I assume the analytical Hamiltonian formula is used in all cases, to estimate the energy of the state.\n\nMODELING VORTEX DYNAMICS OF MULTI-PARTICLE SYSTEM section needs more clarity. For example notation in equations 8 and 9 seems inconsistent. Also lines 229 to 232 lack sufficient detail of how the generalization from 2 particles to N particles is achieved. My guess is that the authors just add up all possible pairwise interactions when moving towards systems with higher number of particles.\n\nEDIT: Updated rating after author revisions.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "OAsKl71G4Vk", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper extends the symplectic family of network architectures towards modeling nonseparable Hamiltonian dynamic systems. More specifically, the paper implements a  symplectic integration schema (from Tao (2016)) for solving arbitrary nonseparable (and separable) Hamiltonian systems within a symplectic neural network architecture. The results from several modeling tasks show that the proposed Nonseparable Symplectic NNs (NSSNNs) are more robust and accurate than vanilla HNNs and NeuralODEs when applied to nonseparable Hamiltonian systems. \n\nAlthough the idea of modeling nonseparable Hamiltonian systems with Symplectic NNs was already briefly outlined in the SRNN paper (Chen et al 2020), this paper implements it and further analyses various properties of this approach. Overall, the paper is well structured and well written, however there are still some inconsistencies that need to be addressed and clarified. \n\nNamely, the related work discussion is somewhat handled poorly: For instance, the authors state in only one sentence that NSSNNs are closely related to SympNets (Jin et al 2020), without discussing any further details on how are they related and, more importantly, how they differ. Moreover, from that point on, SympNets are never considered (in the experiments) nor mentioned, even though SympNets are indeed able to model nonseparable Hamiltonian systems. In Table 1, that compares the properties of NSSNNs w.r.t some benchmarks, the authors discus \"TaylorNet\" and \"SSINN\" - these two are never introduced before. I assume the former refers to Thong et al. 2020, but I have no idea about the latter.  \n\nRegarding the choice of \\omega, the authors provide some evidence that the choice of \\omega plays a role as a regularization, where larger values tend to restrain the system. The analyses given in Appendix B show that with \\omega 10 the system already is stable (which also supports the experiments presented in Tao 2016). But then the \\omega is set to 2000 in the experiments, which is orders of magnitudes larger than the analyses. How and why was this value chosen? \n\nLines 206-207 state that from the results in Fig4, it is \"clear that\" NSSNNs can perform long-term predictions but HNNs and NeuralODEs (in the legend they are listed as ODE-nets, are these the same method?)  fail. It is not clear how was this determined, since the results show that NSSNNs are more robust to noise than the other two, NeuralODEs are still able to perform long-term predictions (in a noiseless setting), and HNNs in a both scenarios w/o noise and w/ moderate amount of noise.\n\nSome typos and minor comments:\n\tL1: Hamiltonian systems are not a \"special\" category of physical systems, but is a formalism for modeling certain physical systems (eg. a pendulum, besides within Hamiltonian mechanics, can be modeled within classical (Newtonian) mechanics and Lagrangian mechanics).\n\tL42: \"e.g. see Tong et al. 2020\" -> \"Tong et al. 2020\"\n\tL56: \"degree of Freedoms\" -> \"degrees of freedom\"\n\tL206: \"figure 4\" -> \"Figure 4\"\n\n#Update \n\nI thank the authors for addressing my questions and revising the manuscript, which clarified many of my concerns regarding this work. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "O1SGytxdyuQ", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary:\nThis paper describes a deep learning approach for predicting Hamiltonian systems. The original paper enforces conservation in the loss function. Several of the follow-up papers embed a symplectic integrator instead, but these couldn't handle non-separable systems. This paper can both handle non-separable systems and use a symplectic integrator to enforce conservation. They demonstrate their system on quite a few examples and show lower error (both in prediction and the deviation in the Hamiltonian) than a NeuralODE or the original HNN paper. The final example, in Figure 5, shows a compelling visual improvement.\n\nStrong points:\n\nThe Greydanus, et al. NeurIPS 2019 paper on Hamiltonian Neural Networks was very successful and has already inspired many follow-up papers. This paper improves it in several ways (Table 1). Since the ICLR community is similar to the NeurIPS community, I think that this paper would be of interest.\n\nI like that the extension for non-separable systems directly builds off an approach for extending integrators to non-separable systems (Tao, 2016). Further, the Tao integrator is built into the network training. This suggests that it's a robust path to take. \n\nI appreciate that results are reported on quite a few examples to compare NeuralODE, HNN, and the new method (NSSNN). \n\nThe results in Figure 5 are quite impressive! I also think it's cool that the training data only needed to be from two particles. \n\nWeak points/clarification questions:\n\nIn the examples in this paper, the canonical coordinates need to be known ahead of time. The original HNN paper has an example where the coordinates can be learned from data (Pixel Pendulum), and I believe some of the follow-up papers have covered this case as well. Do you have any thoughts on if your method could learn the coordinates?\n\nThe font size in the figures is sometimes hard to read.\n\nIn Section 4.2 & Table 2, is there a held-out test set, or are these all training errors? I would like to see both training & test errors to see if there is overfitting. \n\nMinor points:\n\nI found it confusing that Figure 3 & Section 4.1 refer to an \"ablation test.\" It seems like a test to choose a suitable training set. \n\nDisclaimer:\n\nI should mention that I can't vouch for the proof in the appendix. ", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a method for learning non-separable Hamiltonian dynamics by including a state of the art symplectic integrator (Tao, 2016) into the model training pipeline. This is a nice improvement on the past work that primarily addressed separable Hamiltonians. The reviewers agree that the paper is well written and that the empirical evaluation is solid. The paper was further improved during the discussion period by incorporating the reviewers' feedback. For this reason I am happy to recommend this paper for acceptance.
Decision: Accept (Poster)