Identifying Physical Law of Hamiltonian Systems via Meta-Learning
Authors: Seungjun Lee
Haesang Yang
Woojae Seong
Venue: ICLR 2021 Poster (2021)
Keywords: Learning physical laws
meta-learning
Hamiltonian systems

Abstract: Hamiltonian mechanics is an effective tool to represent many physical processes with concise yet well-generalized mathematical expressions. A well-modeled Hamiltonian makes it easy for researchers to analyze and forecast many related phenomena that are governed by the same physical law. However, in general, identifying a functional or shared expression of the Hamiltonian is very difficult. It requires carefully designed experiments and the researcher's insight that comes from years of experience. We propose that meta-learning algorithms can be potentially powerful data-driven tools for identifying the physical law governing Hamiltonian systems without any mathematical assumptions on the representation, but with observations from a set of systems governed by the same physical law. We show that a well meta-trained learner can identify the shared representation of the Hamiltonian by evaluating our method on several types of physical systems with various experimental settings.

Published as a conference paper at ICLR 2021
IDENTIFYING PHYSICAL LAW OF HAMILTONIAN SYS- TEMS VIA META-
Seungjun Lee, Haesang Yang, Woojac Scong Scoul National University
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ABSTRACT
Hamiltonian mechanics is an effective tool to represent many physical processes with concise yet gell-gener mathematical expressions. well-modeled Hamiltonian makes it easy for researchers to analyze and forecast many related phenomena that are governed by the same physical law. However, in general, identifying functional or shared expression of the Hamiltonian is very difficult. It requires carefully designed experiments and the researcher's ' insight that comes from years of experience. We propose that meta-learning algorithms can be poten- tially powerful data-driven tools for identifying the physical law governing Hamil- tonian systems without any mathematical assumptions on the representation, but with observations from a set of systems governed by the same physical law. We show that a well meta-trained learner can identify the shared representation of the Hamiltonian by evaluating our method on several types of physical systems with various experimental settings.
INTRODUCTION
Hamiltonian mechanics, reformulation of Newtonian mechanics, can be used to describe classical systems by focusing on modeling continuous- time evolution of system dynamics with a conservative quantity called Hamiltonian (Goldstein et al., 2002). Interestingly, the formalism of the Hamiltonian provides both geometrically meaningful interpretation (Arnol ' et al.) (2001) and efficient numerical schemes (Feng & Qin 2010) representing the state of complex systems in phase space with sym- plectic structure. Although formalism was originally developed for classical mechanics, has been applied to various fields of physics, such as fluid mechanics (Salmon, 1988), statistical mechanics (Reichl 1999), and quantum mechanics (Sakurai Commins 1995).
While it has many useful mathematical properties, establishing appropriate Hamiltonian of the unknown phenomena is a challenging problem. A Hamiltonian for system can be modeled by a shared expression of the Hamiltonian and physical parameters. For instance, the Hamiltonian of an ideal pendulum is described as H = 2m1z + mgl(1 - cos q) (shared expression), with mass m, pendulum length 1, and gravity constant g (physical parameters), whereas q and p are the angle of the pendulum and the corresponding conjugate momentum (state of the system), respectively. Once an appropriate functional of the Hamiltonian is established from observing several pendulums, new pendulum-like system can be readily recognized by adapting new physical parameters on the expression. Therefore, identifying an appropriate expression of the Hamiltonian an important yet extremely difficult problem in most science and engineering areas where there still remain numerous unknown processes where it is even uncertain whether closed-form solution or mathematically clear expression exists.
y²
In the recent era of deep learning. we can consider the use of learning- based algorithms to identify an appropriate expression of the Hamiltonian with sufficient data. To determine the Hamiltonian un- derlying the unknown physical process, the Hamiltonian should satisfy two fundamental conditions: (1) it should fit well previously observed data motions, (2) it should generalize well on newly observed data from new systems if the systems share the same physical law with previous ones. The first condition has been mitigated by explicitly incorporating symplectic structure conservation laws on neural networks, called Hamiltonian neural networks (HNN) (Greydanus et al., (0199) for learning Hamiltonian dynamics. HNN and its variants have been shown to be effective in learning

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H
H1g
Meta-optimization Task-specific adaptation
New System
Hg Ho
Hu H glg
Figure/ 1: There is resemblance between meta-learning and identifying the physical laws of Hamil- tonian. A hypothesized governing equation of Hamiltonian, usually corrected and established by evaluating many related systems, could be learned using meta-learning (left). Then, a well-established Hamiltonian can be utilized to predict new system dynamics, which could be viewed meta-transfee process by a well-trained meta-learner (right).
a data-driven method
many useful properties of the Hamiltonian Tooth et al., 2020 Chen et al. 2020, /Zhong et al. 2020a Sanchez-Gonzaler et al., 2019; Jin et al., 2020). In their experiments, it has been shown that HNN and its variants work well on learning conservation laws continuous-time translational symmetry, enable the learning of complex systems stably by incorporating numerical integrators and general- ize multiple initial conditions controls for the given system. However, there is limited work regarding a trained model that works well on totally new systems governed by the same physical law with novel physical parameters.
To consider the second condition, we propose that meta-learning, which aims to train a model well generalized on novel data from observing few examples, can be a potential key to learning a func- tional of Hamiltonian as a data-driven method. There have been several representative categories of meta-learning algorithms, such as the metric-based method (Snell al. 2017 [ung et al.) 2018), black-box method (Santoro et al. 2016 Bertinetto et al., 2019), and gradient- method (Rusu al., 2019; Flennerhag al., 2020). Among these methods, we especially focus on the gradient- based method, which is readily compatible with any differentiable model and flexibly applicable to
a wide variety of learning problems (Finn et al. 2017 Xu et al., 2018 (ooseedaals et al. 2020).
One of the most successful algorithms of the gradient-basee based method is Model-Agnostic Meta- Learning (MAML) (Finn et al. 2017), which consists of a task-specific adaptation process and a meta-optimization process. The key observations supporting its potential are the resemblance be- tween these processes and the identification of the physical laws of the Hamiltonian. The schematic is shown in Figure I The task-adaptation process, which adapts the initial model parameters to task-specific train set, resembles the process of adapting hypothesized governing equations to obser- vations of several physical systems. The meta-optimization process, which updates the initial model parameters by validating each task-specific adapted parameters to a task-specific test set, is similar to correcting the hypothesized governing equations by validating each gystem-specifee Hamiltonian on new data from the corresponding physical systems. In addition, (Raghu et al. 2020) proposed that the recent success behind these meta-learning algorithms was due to providing qualitative shared representation across tasks rather than learning initial model parameters that encourage rapid adap- tation (Finn et al. 2017). This hypothesis may support our suggestion that a meta-learner can be efficient in identifying the shared representation of a Hamiltonian. From this point of view, we ex- periment on several types of physical systems to verify whether these meta-learning algorithms are beneficial to our desired lcarning problems. Our contributions are summarized as follows:
We formulate the problem of identifying the shared representation of unknown Hamiltonian meta-learning problem.
For learning to identify the Hamiltonian representations, we incorporate the HNN architec- ture on meta-learning algorithms.
After meta vectattaining the meta-learner, we adapt the model on new systems by learning the data of partial observations and predict the dynamics of the systems as a vector field in phase space.
We evaluate our method on several types of physical systems to explore the efficiency of our methods with various experimental settings.
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2 PRELIMINARIES
2.1 HAMILTONIAN NEURAL NETWORKS
In Hamiltonian mechanics, the state of a system can be described by the vector of canonical coordi- nates, (q.p), which consist of position, q (q) 92.... and its conjugate momentum, p (Pi, P2, .... Pu)) in phase space, where n is the degree of freedom of the system. Then, the time evo- lution of the system is governed by Hamilton's s equations a: 0p 04
2V æ (ar), where
I
H (x): 12n + Ri is the Hamiltonian that is conservative during the process and = - 0
is
2n x 2n nkew-symmetric matrix. From the Hamiltonian equations, the Hamiltonian vector field in phase space, which is interpreted as the time evolution of the system æ is the symplectic gradient of the Hamiltonian ΩV m. (ar), which is determined by the Hamiltonian function and the state of the system itself. Then, the trajectory of the state can be computed by integrating the symplectic gradi- ent of the Hamiltonian. If the Hamiltonian does not depend on the time variable, the Hamiltonian remains constant during the time evolution, because moving along the direction of the symplectic gradient keeps the Hamiltonian constant (Arnol d 2013). In (Greydanus et al., 2019), the Hamilto- nian function can be approximated by neural networks, Hg, called HNN. To make the Hamiltonian function constant motion, the loss of HNN is defined by the distance between the true vector field and the symplectic gradient of Ho.
LHNN = || - - Ho
2.2 MODEL-AGNOSTIC META-LEARNING AND FEATURE REUSE HYPOTHESES
A key assumption behind MAMI is that separately trained models for each task share meta-initial parameters 0 that could be improved rapidly for any task (Finn et al. 2017). Suppose that each given task, T, composed of D. = {D(r, }, is drawn from a task distribution, T/ p(T). The learning algorithms usually consist of bi-level optimization processes; (inner-loop) the task-specific adaptation to each train set,
0, 0 a VoLn :D)
(2)
and (outer-loop) the meta-optimization on each test set,
0 - 2-
Σ (D)
(3)
TemP(T)
where 0 could be any differentiable model's parameters that are expected to learn the shared repre- sentations of various tasks, and a and B are the step sizes of the inner and outer loops, respectively.
Meanwhile, (Raghu et al. 2020) observed that during the inner-loop process, the task-specific dis- tinction of the model parameters 0 is mostly from the last layer of the networks, whereas the entire body of the model hardly changed. Therefore, they hypothesized that the body of the model be- haves as a shared representation across the different tasks, whereas the head of the model behaves as task-specific parameter, which is called the feature reuse hypothesis. From this hypothesis, they proposed gradient-based meta-learning algorithm called Almost No Inner Loop (A (ANIL) by slightly modifying MAML by freezing all but updating the last layer of the networks during the inner-loop process. They showed that ANIL performs on par or better than MAML on several benchmarks, and has computational benefit compared to its counterpart. For the algorithm, when the meta-learner consists of layers 0 (0(1), ..., ((1-1), ((l)), the inner-loop update is modified
0% = (0(1), 00-1), 000 - aV gon Cyz (D)";
(4)
As many physical processes could be expressed as an invariant shared expression of Hamiltonian and variable physical parameters, such meta-learning scheme. which encourages to separate the invariant part and varying part, can be expected to be more efficient to learn new systems by the relatively small number of parameter update.
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3 METHOD
3.1 IDENTIFYING SHARED REPRESENTATION OF HAMILTONIAN VIA META-LEARNER
The main goal of our study is to train model identify the shared representation of the Hamiltonian using observations of dynamics from several systems that are assumed to be governed by the same physical law with different physical parameters. From meta-learning point of view, each system is regarded as a task Tí. where the physical parameters of the system are drawn from the distribution of p()). The observations of the system T: can be split into D; = (D D J. where D and D denote the task-specific train and test sets, respectively. The observations of both D and D.e are given by a set of tuples of canonical coordinates x = (q.P) and their time-derivatives o (q. p) as the ground truth.
For each system, the task-specific model parameters are obtained from Equation 2 or Equation by computing the task-specific loss using Equation
| on each train set D.
CT. (D,
Σ
| - QV Hg
(5)
and the oretaaoptiizztiin can be operated on the batch of systems as Equation by minimizing the loss over the batch of physical parameters sampled from p(T). Each loss is computed by evaluating each task-specific adapted model parameters 02 to each test set Dir,
Σ (D, TT-P(T)
Σ Σ
④ - DV-VV No
(6)
TT<p(T) (00
Depending on the inner-loop methods, we call the algorithms Hamiltonian Model-Agnostic Meta- Learning (HAMAML) when using Equation 2 when using Equation 4
and Hamiltonian Almost No Inner-Loop (HANIL)
3.2 LEARNING NEW SYSTEM DYNAMICS FROM PARTIAL OBSERVATIONS
It can be expected that if the learner is efficient in identifying the underlying physical nature of an unknown process, the model can appropriately predict the dynamics of novel systems from their partial observations. To evaluate whether learner is efficient in identifying the representation of an unknown Hamiltonian, we set up the following validating scheme on several types of physical systems. After meta- meta-training, novel systems are given meta-test sets Dnew {Dir Die " generated from the system's Hamiltonian with novel physical parameters Taer p(T). The train set Dir is used to adapt the learner the system dynamics (0 (2ew). The new observed systems are given as К trajectories, which are reminiscent settings of the K-shoo learning problem (Snell al.) 2017). Starting from the K initial states, each trajectory is obtained by integrating the time-derivatives of the systems from Os to T's, with sampling rate L which yields L rolled out T sequence of states. The test set Die new is for validating the performance of the adapted model 6
4 RELATED WORKS
4.1 LEARNING DYNAMICS WITH NEURAL NETWORKS
Several works learning dynamical systems usually use various forms of graph neural networks (Battaglia et al.; 2016, Santoro et al.) 2017; Sanzaez-Gonzalez et al. 2018 Kipf et al., 2018 Battaglia et al. 2018 Gonzalz et al. 2020), where each node usually represents the state of individual objects and each edge between two arbitrary nodes usually represents the relation of the nodes. (Chnn et al. 2018)) introduced new type of deep architecture, which is considered as the differentialle ordinary differential equations (ODE) solver parametrized by neural networks, called neural ODEs. The most related to our work, (Greydanus et al., 2019) introduced Hamilto- nian Neural Networks (HNNs) to learn the dynamics of Hamiltonian systems by parameterizing the Hamiltonian with neural networks. The HNNs have been developed through combining graph networks for learning interacting systems (Sanchez-
et al.) 2019) or generative networks for learning Hamiltonian from high- dinh-iiiensionll data (Toth et al. 2020). (Chen et al.] 2020) pro-
posed Symplectic recurrent neural networks to handle observed trajectories of complex Hamiltonian
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systems by incorporating leapfrog integrator to recurrent neural networks. (Zong et al. 2020a con- sidered additional control terms to learn the conservative Hamiltonian dynamics and the dissipative Hamiltonian dynamics (Zhong et al. 2020b). For learning Hamiltonian in an intrinsic way, (Jin et al., 2020) proposed Symplectic networks where the architecture consists of the compositional modules for preserving the symplectic structure. In most existing studies for learning Hamiltonian, one model is trained per one system, and the evaluations are restricted to the same physical sys- tem during training with new initial conditions. Our focus is learning the shared representation of Hamiltonian which can be reused to predict new related systems governed by the same physical law that appear throughout the previously observed systems.
4.2 IDENTIFYING PHYSICAL LAWS OR GOVERNING EQUATIONS FROM DATA
In earlier works, symbolic regression was utilized to search the mathematical expressions of the governing equations automatically from observed data (Bongard & Lipson] 2007; Schhmitt & Lip- son 2009). These methods have been developed by constructing dictionary of candidate nonlinear functions (Brunton et al. 2016) Mangan et al.) 2017) or partial differentiations (Rudy et al.) [2017] Schaeffer 2017) by incorporating a apparstt-promottine
algorithm to extract the governing equations. Recently, they have been refined through combining with neural networks (Udrescu & Tegmark 2020 Both et al. 2019; Atkinson et al. 2019; Sahoo et al. 2018) or graph structures (Cranmer work, the existing methods of identifying the governing repre- sentations used the symbolic representation underlying the assumptions that the unknown physical laws are expressed by combinations of known mathematical terms.
al. 2019 2020). In contrast
4.3
GRADIENT-BASEL META-LEARNING
The core learning method of our work is related to gradient-basee meta-learning algorithms (Finn) cet al. 2017). The learning method has been developed by training additional learning rate (Li et al.) 2017. , removing the second-order derivatives (Nichol et al. 2018) and stabilizing the training proce- dure (Antoniou et al. 2019). Mcanwhile, in another direction of the research, the model parameters are separated by the shared representation part which is mainly updated in the outer-loop and task- specific varying part which is mainly updated in the inner-loop (Lee & Choi 2018 Javed & White 2019 Flennerhag et al., 2020 Raghu et al.) 2020: Zhou et al., 2020). Our study mainly focuses on whether a meta-learning could identify the shared expression of the Hamiltonian from several observed systems. MAML is chosen as the representative of the meta-learning algorithm. Also, for verifying whether the separative learning scheme improves the model to identify the shared ex- pression of the Hamiltonian, we choose ANIL as the representative of the meta-learning with the separative learning scheme. Since, without any structural difference, it is sufficient to verify that the existence of the separative learning scheme would be beneficial.
5 EXPERIMENTS
5.1 TYPES OF PHYSICAL SYSTEMS
Spring-Mass. Hamiltonian of the system is described by H = e Ma--o)² where the physical parameters mi k and qu are the mass, spring constant, and equilíbrium position, respectively. g and p are the position and conjugate momentum of the system, respectively.
Pendulum. Hamiltonian of the system is described by = - + mgl(1 - cos(q 2o)). where the physical parameters m, I, and go are the mass, pendulum length, and equilibrium angle from the vertical, respectively. and p are the pendulum angle from the vertical and conjugate momentum of the system, respectively. denotes the gravitational accelcration.
Kepler problem. The system consists of two objects that are attracted to each other by gravitational force. Hamiltonian of the system is described by H
ip|²
GMm where the physical parameters
2m
qq-0ol"
M and m are the mass of the two bodies and we set the object of M to be stationary at qu = (q qy) of the coordinate system. The state is represented by q (g1, 92) and p (P1.P2) which denote the position and conjugate momentum of the object m in two-dimensional space, respectively. G denotes the gravitational constant.
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Table I: MSEs across the test sets of 10 new systems from adapting to the corresponding train sets after 10 gradient steps.
Observations
Systems
Learner
Point Dynamics
Trajectorics
25-shot
50-shot
5-shot
10-shot
HNN from Scratch
82.3.148.9
82.1
84.7=48.9
82.4.188.8
Pretrained HNN
8.2.113.3
4.9-16.6
11.9=14.1
6.64::0.0
Naive NN MAML
159.8=65.6
155.8164.4
195.7=87.1
162.3=66.5
Spring-Mass
Naive NN ANIL
16.6=15.7
16.2=13.6
36.0=25.4
19.5=17.4
HAMAML
29:22.4
2.4=144
4.2=5.2
1.620.66
HANIL
0.02-+0001
0.01++0.006
0.4-0.4
0.1+0.07
HNN from Scratch
6.522.6
6.1=2.1
6.7=2.1
6,611.9
Pretrained HNN
5,433.5
5.1=3.1
5.933.5
5,8+3.2
Naive NN MAMI.
5.8+3.7
5.3+3.3
10.3+8.1
8.3+-.5
Pendulum
Naive NN ANIL
2.3+1.8
2.0+1.4
3.3=2.4
2.8+188
HAMAML
1.5+0055
1.4+0.3
4.7-2.9
1.9
HANIL
0.02.10.04
0.003
0.6
0.04
HNN from Scratch
6.7+11.3
6.6
7.4+12.4
6.9411..6
Pretrained HNN
1.742.6
1.6+2.6
1.9-2.9
1.722.7
Naive NN MAML
1,51.5
0.9+0.4
2.8=4.6
1.3+0.7
Kepler
Navie NN ANIL
1.1
0.910.6
2.4
1,2.1.99
HAMAML
1.5.1.5
1.3.1.6
1.7-2.4
1.541.6
HANIL
0.33
0.33
0.33-0.19 0.19
0.38
5.2 EXPERIMENTAL SETTINGS
Datasets. During the meta-training, we generate 10,000 tasks for meta-train sets of all systems. Each meta-train set consists of task-specific train set and test set given by 50 randomly sampled point states and their time-derivatives è in phase space with task-specific physical parameters. The distributions of the sampled states and physical parameters for each physical process are de- scribed in Appendix A During the meta-testing, the distributions of sampled states and physical parameters are the same as in the meta-training stage. To evaluate learners on the efficacy of identi- fying Hamiltonian of the new systems, meta-test sets are constructed as the following ways.
(1) Observing the new systems as point dynamics phase space: Dir consists of randomly sam- pled points in phase space with K {25, 50} and L = and Die new consists of equally fine-spaced
points in phase space.
(2) Observing the new systems as trajectories in phase space: Drr new consists of randomly sampled trajectories and Die new consists of equally fine-spaced points in phase space. The number sampled trajectories with K (5, 10) and L sequences during T 1s.
The trajectories are obtained by integrating the symplectic gradient of the Hamiltonian from ini- tial states using the adaptive step-size Linge-Kutta method (Hairer al. [993). Fine-spaced test sets consist of equally spaced grids for each coordinate in the region of the phase space where we sampled the point states. More details about data
represented in the Appendix A.
Baselines. We took several learners as baselines to assess the efficacy of our proposed methods, such (1) training HNN DAr from scratch (random initialization), (2) pretrained HNN across all of the meta-train set, (3) meta-trained naive fully connected neural networks (Naive NN), which given the inputs and the outputs è with MAML, and (4) with ANIL. The architectures and training details of the baselines and our methods are represented in the Appendix A.2
The evaluation metric is the average of the mean squared errors (MSE) between true vector fields è and the symplectic gradients of predicted Hamiltonian _2 (x) across the test sets of new systems Die by adapting the learners to the corresponding train set of the new sys- new- For all types of systems, the averaged MSE of randomly sampled 10 new systems are tems Der
Evaluations.
averaged for evaluating the learners after 10 gradient steps adapting to each DAr_... We also predict the state trajectories and the corresponding energies by integrating the output vector fields of the learners adapted to a new system by observing samples of point dynamics. Following (Greydanus) et al. 2019), evaluate the MSEs of the predicted trajectories and energies from their correspond- ing ground truth at each time step. The predicted values are computed by the learners adapted to 50 randomly sampled point dynamics of new systems in phase space after 50 gradient steps.

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HNN from scratch
Pretrained HNN
Ne've NN MAML
Naive NN ANIL
HAMALL
HANIL
Ground Truth
Intill Outputs
(a) Before adaptation
25 POIN
50 POU
Trajectoriss
10 Trajectories
(b) After gradient step
590 POIN
Trojectoriie
10 Trajectories
(c) After 10 gradient steps
Figure 2: Predicted vector fields (gray streamlines) by adapting the learners observations of new pendulum systems given as point dynamics (red arrows) or trajectories (red dots) after the corre- sponding gradient steps. The r-axis and y-axis denote q and p, respectively.

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State irajectory
MSE between coortinates
MSE between Energy
Spring
10
time
dime
time
Pensulum
10"1
18--
10
1011
10'
time
time
time
13.0
12.
Kapler
2.0
33 MAML
30
G1
time
Line
Dinre
HNN from scretch
Naive
HAMAML HANIL
Ground Truth
Pretrained HNN
Naive ANIL
Ivttlal state
Figure 3: Predicted state trajectories (or position coordinates for Kepler problem) and the corre- sponding energies, and the corresponding MSEs at each time step. Note that the MSEs of spring- mass and pendulum are represented as log-scaled.
5.3 RESULTS
Quantitative results. The quantitative results of meta-testing performance are shown in Table is shown that HANIL outperforms the others for all experimental settings in all types of physical systems. When observing 25-shot point dynamics and 5-shot trajectories, the number of given samples in the phase space is the same, and the same is true for observing 50-shot point dynamics and 10-shot trajectories (L 5). Comparing the point dynamics and trajectories with the same number of given samples, learning from observing point dynamics is slightly more accurate than that from observing trajectories.
Predicted vector fields. In Figure predicted pendulum dynamics by adapting the learners from observing partial observations are represented as phase portraits by the corresponding gradient steps. shown in Figure 5 Appendix A.3 In Figure (a), the
mass Note that those of spring mass systems initial outputs vector fields from the learners are represented. During the adaptation to the given observations, the output vectors of each learner are evolved to fit on the observations based on their own prior belief or representation learned from the meta-train set. In detail, HNN from scratch fails predict the dynamics of new systems from partial observations. At least hundreds of states should be given as train set, and thousands of gradient steps are required for training HNN for learning system (Greydanus
al., 2019). However, in our meta-testing, up to 50 states are given to adapt to the new system with few gradient steps. Thus, the number of samples and gradient steps is too small train HNN without any inductive bias. Pretrained HNN also fails, even though it trained using the meta-train sets. A model simply pretrained across all tasks may output the averaged values of time-derivative at each state point. As the time-derivatives each state would varying sensitive to the physical parameters of the systems, the simple averaged values are likely to have very different patterns from the actual vector fields. Therefore, such pretrained model would not be efficient to learn appropriate shared representation across the systems. Naive NNs, with MAML and ANIL also fail to predict the dynamics because naive NNs are hard to grasp the continuous and conservative structure of the vector fields where the number of given data not sufficient adapt
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10
10
10a
MSE
MSE
10
1104
10
50
12
Gradient steps
Tasks
Figure 4: Ablation studies conducted on 10 new pendulum systems by comparing (a) Learning curves during aas-addatttiio, and (b) the effects of the size of the meta-train sets.
systems. Thus, the phase portraits of them are discontinuous or dissipative- shape. HANIL can accurately predict the dynamics of new systems from partial observations with few gradient steps, while HAMAMI. is slower than HANIL to adapt the true vector fields because of the larger number of parameters update in the adaptation process.
Trajectory predictions. In Figure 3 we also evaluate the learners adapted to the new systems through their predictions of state and the corresponding energies starting from the initial states during 20s. HANIL (blue lines) adapted to new systems predicts the states and energy trajectories with relatively small errors from the ground truth (black dashed lines), whereas the others fail to predict the right trajectories and energies of the system at each time step.
Ablation study. We conduct ablation studies verify the efficacy of the separative learning schemes by comparing the learning curves during task-adaptation, and the effects of the size of the meta-train sets. We evaluate the same evaluation of MSEs as described Section 5.2]but but varying gradient steps from 0 to 50 to see the learning curves and varying the number of tasks from 10 to 10,000 which are observed during meta-training to verify the effects of the size of the meta-train set. In addition, in order to see how the number of parameters updated during the inner-loop affects the performance, we add another comparison meta-leamed learner called HANIL-Inverse (HANIL-INV), The method updates all but except the first layer of the network during the inner-loop, while HANIL only updates the last layer of the network. Therefore, during the inner-loop. the number of updated parameters of the HANIL-INV is between those of the HAMAMI and HANII. In Figure 4 the results of ablation studies on pendulum systems are represented where the dynamics of new systems are given as 50 point dynamics. HANIL-Inv converges the error between HAMAML and HANIL, while the HNN from scratch and the pretrained HNN are hardly improved in both studies. As comparing the meta- trained learners with the others, meta-learning improves the performance to predict new systems through their ability to learn the shared representation. As comparing HAMAML and meta- traine learners with the separative learning scheme such as HANIL-INV and HANIL, seems beneficial to separately learn the shared representation and physical parameters through the separative learning scheme. In addition, since the updated parameters are limited to the last layer during the inner-loop, HANIL converges with the lowest errors, which would be most efficient to generalize the related systems with the same physical law, as fewer updated parameters are required for the new system. Meanwhile, in Figure/ (b), the point where the effect of the size of tasks to the meta-trained learners begins to be noticeable is between 200 and 500 and slowly converges at around 10,000.
6 CONCLUSIONS
By observing the resemblance between seemingly unrelated problems, identifying the Hamiltonian and meta-learning, formulate the problem of identifying the Hamiltonian meta-learning problem. We incorporate HNN, which is an efficient architecture for learning Hamiltonian, with meta-learning algorithms order to discover the shared representation of unknown Hamiltonian across observed physical systems. Comparing the bascline models with various experiments, we show that our proposed methods, especially HANIL, is efficient to learn totally new systems dy- namics governed by the same underlying physical laws. The results state that our proposed methods have the ability to extract the meta-transferable knowledge, which can be considered as physical nature across the observed physical systems during meta-training.

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REFERENCES
Antreas Antoniou, Harrison Edwards, and Amos Storkey. How to train your MAMI.. Interna- tional Conference on Learning Representations, 2019.
Vladimir Igorevich Amol'd. Mathematical methods of classical mechanics, volume 60. Springer Science & Business Media, 2013.
Vladimir Aroo'd, Sergei Petrovich Novikov, et al. Symplectic geometry. Springer, 2001.
Steven Arkinson, Waad Subber, Liping Wang, Genghis Khan, Philippe Hawi, and Roger Ghanem. Data-driven discovery of free-form governing differential equations. arXiv preprint arXiv:1910.05117 17, 2019.
Peter Battaglia, Razvan Pascanu, Matthew Lai, Danilo Jimenez Rezende, et al. Interaction networks for learning about objects, relations and physics. In Advances in Neural Information Processing Systems, pp. 4502-4510, 2016.
Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, et al. Relational inductive biases, deep learning, and graph networks. arXiv preprint 2018.
Luca Bertinetto, Joao F. Henriques, Philip Torr, and Andrea Vedaldi. Meta-learning with differen- tiable closed-form solvers. In International Conference Learning Reprexentations, 2019.
Josh Bongard and Hod Lipson. Automated reverse engineering of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 104(24):9943-99)) -948,, 2007.
Gert-Jan Both, Subham Choudhury, Pierre Sens, and Remy Kusters. Deepmod: Deep learning for /904.0990000 2019.
model discovery in noisy data. arXiv preprint
Steven I. Brunton, Joshua I. Proctor, and Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113((5):3932-3937, 2016.
Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary differential equations. In Advances in Neural Information Processing systems, pp. 6571-6583, 2018.
Zhengdao Chen, Jianyu Zhang, Martin Arjovsky, and Léon Bottou. Symplectic recurrent neural networks. In International Conference on Learning Representations, 2020.
Miles Cranmer, Alvaro Sanchez- Sanchez-Gonzaler, Peter Battaglia, Rui Xu, Kyle Cranmer, David Spergel, and Shirley Ho. Discovering symbolic models from deep learning with inductive biases. arXiv preprint arXiv:2006. 12287, 2020.
Miles D Cranmer, Rui Xu, Peter Battaglia, and Shirley Ho. Learning symbolic physies with graph networks. arXiv preprint arXiv:1909.0
2019.
Kang Feng and Mengzhao Qin. Symplectic geometric algorithms Hamiltonian systems. Springer, 2010.
Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In International Conference on Machine Learning, pp. 1126-1135, 2017.
Sebastian Flennerhag, Andrei A. Rusu, Razvan Pascanu, Francesco Visin, Hujun Yin, and Raia Hadsell. Meta-learning with warped gradient descent. International Conference on Learning Representations, 2020.
Herbert Goldstein, Charles Poole, and John Safko. Classical mechanics. American Association of Physics Teachers, 2002.
Samuel Greydanus, Misko Dzamba, and Jason Yosinski. Hamiltonian neural networks. In Advances in Neural Information Processing Systems, pp. 15379-15388, 2019.
10

---

Published as a conference paper at ICLR 2021
E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations 1 (2nd Revised. Ed.): Nonstiff Problems. Springer-Verlag, Berlin, Heidelberg, 1993. ISBN 0387566708.
Timothy Hospedales, Antreas Antoniou, Paul Micaelli, and Amos Storkey. Meta-learning in neural networks: A survey. arXiv preprini arXiv:2004.05439 2020.
Khurram Javed and Martha White. Meta-learning representations for continual learning. In Ad- vances in Neural Information Processing Systems, pp. 1820-1830, 2019.
Pengzhan Jin, Zhen Zhang, Aiqing Zhu, Yifa Tang, and George Em Karniadakis. Sympnets: Intrin- sic garucture-peeservie symplectic networks for identifying hamiltonian systems. Neural Net- works, 132:166 179, 2020.
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations, 2015.
Thomas N. Kipf, Ethan Fetaya, Kuan-Chieh Wang, Max Welling, and Richard S. Zemel. Neural relational inference for interacting systems. In International Conference on Machine Learning, pp. 2693-2702, 2018.
Yoonho Lee and Seungjin Choi. Gradient-based meta-learning with learned layerwise metric and subspace. In International Conference Machine Learning, pp. 2927-2936, 2018.
Zhenguo Li, Fengwei Zhou, Fei Chen, and Hang Li. Meta-sgd: Learning to learn quickly for few- shot learning. arXiv preprint /rXiv:/707.0900 2017.
Niall M Mangan, J Nathan Kutz, Steven I. Brunton, and Joshua Proctor. Model selection for dy- namical systems via sparse regression and information criteria. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2204):2010009)) 2017.
Alex Nichol, Joshua Achiam, and John Schulman. On first-order meta-learning algorithms. arXiv preprint arXiv:1803.02999 2018.
Aniruddh Raghu, Maithra Raghu, Samy Bengio, and Oriol Vinyals. Rapid learning or feature reuse? towards understanding the effectiveness of maml. In International Conference Learning Rep- resentations, 2020.
Linda E Reichl. A modern course in statistical physics. American Association of Physics Teachers, 1999.
Samuel H Rudy, Steven L Brunton, Joshua L Proctor, and Nathan Kutz. Data-driven discovery of partial differential equations. Science Advances, (4)):1160614 2017.
Andrei A. Rusu, Dushyant Rao, Jakub Sygnowski, Oriol Vinyals, Razvan Pascanu, Simon Osindero, and Raia Hadsell. Meta-learning with latent embedding optimization. In International Conference on Learning Representations, 2019.
Subham Sahoo, Christoph Lampert, and Georg Martius. Learning equations for extrapolation and control. In International Conference on Machine Learning, pp. 4442 4450, 2018.
Jun John Sakurai and Eugene D Commins. Modern quantum mechanics, revised edition. American Association of Physies Teachers, 1995.
Rick Salmon. Hamiltonian fluid mechanics. Annual review of fluid mechanics, 20(1):225-256. 1988.
Alvaro Sanchez-Gonzalez, Nicolas Heess, Jost Tobias Springenberg, Josh Merel, Martin A. Ried- miller, Raia Hadsell, and Peter Battaglia. Graph networks as learnable physics engines for infer- ence and control. In International Conference Machine Learning, pp. 4467-4476. 2018.
Alvaro Sanchez- Fanchez-Gonzalez. Victor Bapst, Kyle Cranmer, and Peter Battaglia. Hamiltonian graph networks with ode integrators. arXiv preprint 1rXXv:/909.127000 2019.

---

Published as a conference paper at ICLR 2021
Alvaro
Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovec, and Pe- ter W Battaglia. Learning to simulate complex physics with graph networks. arXiv preprint arXiv:2002.09405. 2020.
Adam Santoro, Sergey Bartunov, Matthew Botvinick, Daan Wierstra, and Timothy Lillicrap. Meta- learning with memory-augmentedd neural networks. In International Conference on Machine Learning. pp. 1842-1850, 2016.
Adam Santoro, David Raposo, David G Barrett, Mateusz Malinowski, Razvan Pascanu, Peter Battaglia, and Timothy Lillicrap. A simple neural network module for relational reasoning. In Advances in Neural Information Processing systems. pp. 4967-4976, 2017.
Hayden Schaeffer.
Learning partial differential equations via data discovery and sparse optimiza- tion. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473 (2197):20160446 2017.
Michael Schmidt and Hod Lipson. Distilling free-form natural laws from experimental data. science, 324(5923):81 2009.
Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Advances in Neural Information Processing Systems, pp. 4077-4087,2000
Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip HS Torr, and Timothy M Hospedales. Learning to compare: Relation network for few-shot learning. In Proceedings of the IEEE Con- ference Computer Vision and Putlern Recognition, pp. 1199-1208, 2018
Peter Toth, Danilo Rezende, Andrew Jaegle, Sébastien Racanière, Aleksandar Botev, and Irina Higgins. Hamiltonian generative networks. In International Conference on Learning Represen- tations, 2020.
Sarviu-Marian Udrescu and Max Tegmark. feynman: A physics-inspired method for symbolic regression. Science Advances, 6(16))eaay2631, 2020.
Zhongwen Xu, Hado P van Hasselt, and David Silver. Meta-gradient reinforcement learning. In
Advances in Neural Information Processing Systems, pp. 2396-2407,2015 2018.
Yaofeng Desmond Zhong, Biswadip Dey, and Amit Chakraborty. Symplectic ode-net: Learning hamiltonian dynamics with control. International Conference on Learning Representations, 2020a.
Yaofeng Desmond Zhong, Biswadip Dey, and Amit Chakraborty. Dissipative symoden: En- coding hamiltonian dynamics with dissipation and control into deep learning. arXiv preprint arXiv:2002.08866) 2020b.
Allan Zhou, Tom Knowles, and Chelsea Finn. Meta-learning symmetries by reparameterization. arXiv preprint arXiv:2007.02933, 2020.
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Published as a conference paper at ICLR 2021
A APPENDIX
A.1 SYSTEM DETAILS
Spring-Mass. The physical parameters are randomly sampled from (m, go) ([0.5, 5]. (0.5, (-,,5))))) The initial states are randomly sampled from (g.p) E ([-10, 10 Fine- spaced test sets of new systems consist of 50 equally spaced grids for each coordinate in the region of the phase space where we sampled the point states. Therefore, there are 2,500 grids points in the test sets.
Pendulum.
The physical parameters are randomly sampled from (.)
([0.5, 5). 00.5, 5), [-v, π]). We fix the gravitational acceleration as g = The initial states randomly sampled from (g.p) E ([-2x, 2 [-20, 20)). Fine-spaced test sets of new systems consist of 50 equally spaced grids for each coordinate in the region of the phase space where we sampled the point states. Therefore, there 2,500 grids points the test sets.
Kepler Problem. The physical parameters are randomly sampled from (M, m, qu qy) E ([0.5, 2.5) [-2.5, 2.5]²). We fix the gravitational constant as domly sampled from (q.p) ([-5, 5)*). Fine-spaced test sets of new systems consist of 10 equally spaced grids for each coordinate in the region of the phase space where we sampled the point states. Therefore, there are 10,000 grids points in the test sets.
The initial states are ran-
A.2 IMPLEMENTATION DETAILS
For all tasks, we took the baseline model as fully connected neural networks with the size of state dimensions 64 Softplus 64 Softplus 64 Softplus state dimensions and the HNN model as fully connected neural networks with the size of state dimensions 64 Softplus 64 Softplus 64 Softplus dimension. We searched the hyperparameters, exploring the size of hidden dimensions from {32, 64, 128, 256]. the number of layers from {1, 2, 3, 4), and the activation function from (Sigmoid, Tanh, Relu, Softplus). During meta-training or pretraining, we use Adam optimizer (Kingma & Ba 2015) outer-loop with learning rate of 0.001 and use gradient descent inner- loop with learning rate of 0.002. For all systems, we set the number of task batches of 10, inner gradient updates of 5, and episodes of outer loop of 100 for meta-optimizatton. During the meta- testing, we also use the Adam optimizer with a learning rate 0.002. There is no weight decay for all.
A.3 ADDITIONAL RESULTS
More results and video could be available at httpsss//github.comm//75 7ogpo//mmmmmmnmmtt / Ident:-fyinng-Pyss.cca-
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Published as a conference paper at ICLR 2021
Pretrained Nalve MAML
HAMAMI
Ground Truth
(a) Before adaptation
25 Points
50 Points
Trajectories
Trajecuriis
(b) After gradient step
Points
50 S
Trajectoii
Trajectortos
(c) After 10 gradient steps
Figure 5: Predicted vector fields (gray streamlines) by adapting the learners observations of spring-mass systems given as point dynamics (red arrows) or trajectories (red dots) after the corre- sponding gradient steps. The r-axis and y-axis denote q and p, respectively.
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Reviews:
[{"review_id": "lJnICu9wC_t", "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 combine Hamiltonian neural networks (HNN) with the MAML and ANIL neural learning approaches for Meta-Learning. The results are mostly consistent with what would be expected when combining them.\nThe paper is easy to follow and nice to read, and seems to have sufficient implementation details for reproducibility.\n\nMy main criticism is that there is not really novel technical content: the model is a trivial combination of MAML/NIL with HNNs. This could be ok in the following scenarios:\n* If results were given on well-known important domains achieving state of the art. Currently all domains are very toy.\n* If the paper gave a hint of what real-world problem settings could benefit from this specific combination, which does not seem to be part of the discussion.\n* If the paper did a very thorough empirical investigation. The results are nice (I really like Figure 2), but otherwise none of the insights are really new, but just the union of the conclusions of the MAML/NIL papers and the HNN paper.\n\nbut those are not really well covered. \n\nProbably the biggest value of the paper is that it brings together the HNN community (often closer to natural sciences communities) to the meta-learning community (often closer to data-science communities), and this paper may be a gateway from people from the former one to learn more about meta-learning, but I am not if this is sufficient to grant acceptance. I would not oppose an acceptance decision if the AC, or other reviewers decide to accept it on these grounds in its current form.\n\nAdditional comments:\n\nI am a bit surprised HANIL is so much better than HAMAML, considering Naive ANIL is not so much better than Naive MAML comparatively (specially in SpringMass and and Pendulum point dynamics).\n\nWas the number of gradient steps during evaluation fine-tuned for each baseline separately? Could this be the reason why MAML is comparatively worse? This is hinted in “while HAMAML is slower than HANIL to adapt the true vector fields because of the larger number of parameters to update in adaptation process”, but it would be nicer to see more supporting evidence or additional investigations.\n\nAll of the results in Table 1 corresponds to the evaluation of the loss of the derivatives, and not the error over an integrated trajectory right? I find it a bit unintuitive that the Naive network is on pair with the model on Kepler, but not on the other two domains. I would expect most of the advantage of the Hamiltonian model would come from the stability when integrating over longer trajectories, but not so noticeable on the predictions of gradients at single points. In fact in Table 1 from the original HNNs paper, they precisely show that the differences between the Baseline and the HNN are mostly w.r.t. preservation of energy, but that the losses on the derivatives are nearly identical.\n\nIt would be nice to see some experiments where the functional form of the Hamiltonian itself (rather than just hamiltonian parameters change across time). The title, introduction and related work, somehow makes it look like different physical laws will be inferred during the meta learning stage, however in practice the meta-learning task covers the same physical laws than those seen during training with different parameters. Otherwise the title is highly misleading, since it does not identify physical laws, but to identify specific instances of the same physical law.\n\nHow were the trajectories from Figure 3 chosen, fully at random?\n\nMinor comments/typos (have not affected my decision):\n\nTable 1: Possibly bold font for best model, or even better, some bar plots would make it easier to visualize the data.\n\nweird grammar: “with sufficient data to discover Hamiltonian.”\nmissing space: “partial differentiations(Rudy”\n\nAre the numbers in the last row of the results table truly exactly the same for Point dynamics and Trajectories (0.33±0.19 0.33±0.18 0.33±0.19 0.33±0.18), or is this an editing error?\n\n> When observing 25-shot point dynamics and 5-shot trajectories, the number of given samples in the phase space is the same, and the same is true for observing 50-shot point dynamics and 10-shot trajectories. \nWould be good to refer to L=5 again at this point.\n\nFig 2 (b-c) I would recommend to add some sort of labels indicating what each of the 4 rows on the left of the plot.\n\nOne model seems to be missing from Figure 3(a).\n\nFigure 3(a) caption mentions “after 50 gradient steps” but other places say “10 gradient steps”, not sure if this was changed for Fig 3, or this is a typo.", "extra_scores": {}, "extra_text": {}}, {"review_id": "fsj7SSqYEio", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 7, "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 paper introduces a meta-learning approach in Hamiltonian Neural Networks to find the structure of the Hamiltonian that can be adapted quickly to a new instance of a physical system.\n\nThe contribution is novel and the paper is well written. The presentation is mostly clear, however, some improvements are needed.\n\nStrength:\n- Clean methodology\n- good performance on the task of few-shot learning on a new system\n- nice visualization of the vector fields\n\nWeaknesses:\n- No real-world problem\n- The meta-learner, in particular HANIL, is clearly not adjusting the parameters of the Hamiltonian as we would expect it, like the physical parameters, which would be more at the beginning of the network.  It only adjusts the read-out layer. In this way, it is hard to understand what is really happening.   \n- You need a dense sampling of the meta-world. Really you have to look at 10000 tasks? A latent variable estimation model would have been a good baseline, or alternative to the Meta-learning framework, for instance, a VAE.\n  \nDetails:\n- below Eq 3: consider calling beta not a learning rate, because it is more a step-size since nothing is really learned in the inner loop\n- related work: machine learning methods to perform symbolic regression directly, such as \"Learning Equations for Extrapolation and Control\"\nSahoo et al, ICML 2018, might be good to add\n- Fig 3: font size is far too small, lines are too thin\n- Fig 3: consider using a log-scale for the MSE plots\n\n----\nPost rebuttal update:\nI  read the response and commented on it. The authors clarified my questions and updated the paper accordingly. So I think my score of 7 is supported.", "extra_scores": {}, "extra_text": {}}, {"review_id": "8Fpj2bTLzFd", "reviewer": "AnonReviewer4", "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": "The paper presents a meta-learning method for learning Hamiltonian dynamic systems from data. More specifically, the novelty is incorporating Hamiltonian Neural Networks (HNNs) within known meta-learning methods (MAML and ANIL) in order to model new dynamical systems (with previously known structures but unknown parameters) from partially observed data. The results from the experimental evaluation (on three well-known systems) show that such an approach, and in particular HNNs w/ ANIL (HANIL), leads to more accurate models of unseen dynamics compared to other benchmarks methods such as \"vanilla\" HNNs and HNNs w/ MAML (HAMAML).\n\nOverall, the paper is well written and the contributions are solid: leading to reasonable improvements over recent work on modeling Hamiltonian systems while offering better understanding of the underlying modeling problem. However, I have some concerns regarding the experimental set-up as well as the discussion of related work (and the motivation, thereof), which are reflected in my score. \n\nFirst, it is unclear to me how the meta-training is performed: The authors state that 10K tasks are being sampled into the meta-training sets -  are these 10K sampled for each of the 3 systems (leading to 30K in total) or 10K in total (what is the task/system distribution in this case)? Is the meta-leaning performed on tasks sampled from all 3 dynamic systems (i.e. one meta-optimization for all 3 systems) or this is done per system (as Fig1 suggests)? The choice of 10K seems a bit arbitrary, how does this choice affects the overall performance? An ablation study on the size of the meta-sets will further highlight the strengths of the proposed method.  \n\nSecond, the authors chose two benchmarks which address modeling Hamiltonian systems (HNNs) \"by design\" i.e. vanilla HNNs trained from scratch (for a given task) and Pretrained HNNs (using the meta-sets). Recent work (for instance Symplectic Recurrent Neural Networks (Chen et. al 2020); Hamiltonian Graph NNs (Sanchez-Gonzalez et al. 2019); Symplectic ODE-net (Zhong et al 2020); SympNets (Jin et al 2020)) has shown improvements over the vanilla HNNs both in terms of accuracy and stability. Although these methods are very briefly mentioned in the paper, they are never in-depth discussed nor considered. It would be helpful if (at least) such a discussion is given, as well as maybe discussing possible extensions(eg. SRNN with ANIL).\n\nNext, the authors should consider extending the related work section in few directions. First, since one of the major contributions is related to meta-learning, the authors should provide a related-work segment discussing related meta-learning approaches which would further justify their design choices (eg. why wasn't Reptile considered). Next, in Sec 4.1., the authors state that the \"most related\" are HNN but omit any further discussion regarding eg. SRNNs (or HGNNs) - why SRNNs are not up-there with HNNs? IMO such a discussion will further motivate the contributions of the paper.\n\nI respectfully disagree with the last statement in Sec 4.2,  - that \"In contrast to our work, the existing methods of identifying the governing representations used the symbolic representation underlying the assumptions that the unknown physical laws are expressed by combinations of mathematically known expressions.\" Namely, the methods in this study also 'implicitly' build upon the very same physical laws (which are combinations of known expressions). The meta-tasks sampling is performed over systems with known structures therefore the very same assumptions still hold.\n\nOther comments:\n- Figure 2, while taking a whole page, is a bit unreadable and confusing to follow. More elaborative discussion will help with conveying the message better.\n- Can you clarify the difference in the results presented in Table 1 and Figure 3? More specifically, in Figure 3a, the coordinates MSE shows that HAMAML is diverging (while Pre-trained HNNs seem more stable). In Table 1, on the other hand, PreHNN are worse than HAMAML. Are these the same 10 \"new\" system but at different gradient steps, or? \n- In Sec 6, can you clarify this statement \"That is because the standard supervised learning scheme is not efficient for the model to learn appropriate shared representation across the systems. \"? What is a \"standard\" supervised scheme and why is not efficient?\n- The very next sentence (in Sec 6) states that \"Naive NNs ... also fail ... because are hard to grasp the continuous and conservative structure of the vector fields\". Can you also clarify this? \n\n----\nPost Rebuttal Update:\n\nThe authors addressed and clarified many of my concerns, therefore I updated my score. ", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes a meta-learning approach for inferring the Hamiltonian governing the dynamics of physical systems from observational data, and using it to adapt to new systems from the same class of dynamics quickly. The paper does this by effectively combining the previously published Hamiltonian Neural Networks and MAML/ANIL. The reviewers agree that the paper is well written, and the experiments are comprehensive, however, they also have reservations about the technical novelty of the proposed solution, given that it appears to be combination of pre-existing models. Saying this, the authors were able to  address a lot of the reviewers' concerns during the discussion period, hence I recommend this paper for acceptance.
Decision: Accept (Poster)