Data-driven discovery of non-Newtonian astronomy via learning non-Euclidean Hamiltonian Oswin So

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Data-driven discovery of non-Newtonian

astronomy via learning non-Euclidean Hamiltonian

Oswin So∗

Massachusetts Institute of Technology, USA

oswinso@mit.edu

Gongjie Li

Georgia Institute of Technology, USA

gongjie.li@physics.gatech.edu

Evangelos A. Theodorou

Georgia Institute of Technology, USA

evangelos.theodorou@gatech.edu

Molei Tao

Georgia Institute of Technology, USA

mtao@gatech.edu

Abstract

Incorporating the Hamiltonian structure of physical dynamics into deep learning

models provides a powerful way to improve the interpretability and prediction accu-

racy. While previous works are mostly limited to the Euclidean spaces, their exten-

sion to the Lie group manifold is needed when rotations form a key component of the

dynamics, such as the higher-order physics beyond simple point-mass dynamics for

-body celestial interactions. Moreover, the multiscale nature of these processes

presents a challenge to existing methods as a long time horizon is required. By lever-

aging a symplectic Lie-group manifold preserving integrator, we present a method

for data-driven discovery of non-Newtonian astronomy. Preliminary results show

the importance of both these properties in training stability and prediction accuracy.

1 Introduction

Point

Rigid

Lie T2

Figure 1: One planet’s orbit

around a star: rigid body correc-

tion results in a precession,i.e.

slow rotation of the orbital axis.

Our method, ‘Lie T2’, learns

from data and predicts a trajec-

tory that matches the ground truth

with the rigid body potential.

Deep Neural Networks (DNN) have been demonstrated to be

effective tools for learning dynamical systems from data. One

important class of systems to be learned have dynamics described

by physical laws, whose structure can be exploited by learning the

Hamiltonian of the system instead of the vector ﬁeld [

]. An

appropriately learned Hamiltonian can endow the learned system

with properties such as superior long prediction accuracy [

] and

applicability to chaotic systems [3–5].

To learn continuous dynamics from discrete data, one important

step is to bridge the continuous and discrete times. Seminal work

initially approximated the time derivative via ﬁnite differences

and then matched it with a learned (Hamiltonian) vector ﬁeld[

Recent efforts avoid the inaccuracy of ﬁnite difference by numer-

ically integrating the learned vector ﬁeld. Especially relevant

here is SRNN [

], which uses a symplectic integrator to ensure

the learned dynamics is symplectic (a necessity for Hamiltonian

systems). Although SRNN only demonstrated learning separable Hamiltonians, breakthrough in

symplectic integration of arbitrary Hamiltonians [

] was used to extend SRNN [

]. Further efforts on

improving the time integration error have also been made [

–

]. Meanwhile, alternative approaches

based on learning a symplectic map instead of the Hamiltonian also demonstrated efﬁcacy [

although these approaches have not been extended to non-Euclidean problems.

∗Work was done while Oswin was at Georgia Tech.

Preprint. Under review.

arXiv:2210.00090v1 [cs.LG] 30 Sep 2022

In fact, one relatively under-explored area is learning Hamiltonian dynamics on manifolds like the Lie

group manifold family

. One important member of this family is

SO(n)

, which describes isometries

and is important for, e.g., dynamical astronomy. The evolution of celestial bodies correspond to

a mechanical system, and the 2- and 3-body problems have been a staple problem in works on learning

Hamiltonian (e.g., [

]); however, the Newtonian (point-mass) gravity considered is already

well understood. Practical problems in planetary dynamics are complicated by higher-order physics

such as planet spin-orbit interaction, tidal dissipation, and general relativistic correction. While it is

unclear what would be a perfect scientiﬁc model for these effects, planetary rotation is a necessary

component to account for spin-orbit interaction and tidal forcing, creating an

SO(3)⊗N

component of

the conﬁguration space. To learn these physics from data, we need to learn on the Lie group.

Rigid body dynamics also play important roles in other applications such as robotics. In a seminal

work [

], Hamiltonian dynamics on

SO(3)

are used to learn rigid body dynamics for a quadrotor.

In that work, Runge-Kutta 4 integrator is used. Consequently, the method is applicable to short

time-horizon (see Sec.3and last paragraph of Sec.2).

For our problem of learning non-Newtonian astronomy, the time-horizon has to be long. Hence,

we use a different approach by leveraging a Lie-group preserving symplectic integrator. Structure-

preserving integration of dynamical systems on manifolds has been extensively studied in literature,

for example for Lie groups [17–21] and more broadly, geometric integration [22–25].

In summary, we propose a deep learning methodology for performing data-driven discovery of

non-Newtonian astronomy. By leveraging the use of a symplectic Lie-group manifold preserving

integrator, we show how a non-Euclidean Hamiltonian can be learned for accurate prediction of

non-Newtonian effects. Moreover, we provide insights that show the importance of both symplecticity

and exact preservation of the Lie-group manifold in training stability.

2 Method

Given observations of a dynamically evolving system, our goal is to learn the physics that governs

its evolution from the data. Denote by

(qk,l,Rk,l,pk,l,Πk,l)K

k=1,l ∈[L]

a dataset of snapshots of

continuous-time trajectories of a system with Ninteracting rigid bodies. That is,

(qk,l,Rk,l,pk,l,Πk,l)=ql(k∆t),Rl(k∆t),pl(k∆t),Πl(k∆t),

where

ql(t),Rl(t),pl(t),Πl(t)

is a solution of some latent Hamiltonian ODE to be learned corre-

sponding to mechanical dynamics on

T∗SE(3)⊗N

∆t

is a (possibly large) observation timestep,

R∈SO(3)⊗N

is the rotational conﬁguration of the

rigid bodies, and

Π∈so(3)⊗N

denotes each’s

angular momentum in their respective body frames.

Importantly, since the conﬁguration space

Q=SE(3)⊗N

is not ﬂat, the mechanical dynamics are not

given by

q=∂H

∂p,˙

p=−∂H

∂q

for some Hamiltonian

that depends on the generalized coordinates

q∈Q

and generalized momentum

p∈T∗

. Instead, the equations of motion can be derived via

either Lagrange multipliers [

] or a Lie group variational principle [

], which will be

qi=pi/mi,(1a)

pi=−∂V

∂pi

+Fpi(1b)

Ri=Ri

J−1

iΠi(1c)

Πi=Πi×J−1

iΠi−R|

∂V

∂Ri−∂V

∂Ri|

Ri∨+FΠi(1d)

assuming a physical Hamiltonian

H(q,R,p,Π) = PN

i=1 1

2pT

ipi/mi+Pi1

2ΠT

iJ−1

iΠi+V(q,R)

that sums total (translation and rotational) kinetic energy and interaction potential

, where

mi,Ji

denote the mass and inertial tensor of the

th body, and

Fp,FΠ

are forcing terms to model

nonconervative forces.

Πi∈R3

is a vector,

∧

is the map from

Skew3

and

∨

is its inverse ([

]

for more details). By learning the potential

, external forcing

and torque

FΠ

, we can learn the

physics of the system.

2.1 Machine Learning Challenges Posed by Dynamical Astronomy

We study this setup because it helps answer scientiﬁc questions like: what physics governs the motions

of celestial bodies, such as planets in a planetary system? The leading order physics is of course

already well known, namely these bodies can be approximated by point masses that are interacting

2We note extensions to include holonomic constraints in [13] and to handle contact in [14]).

(q1, R1, p1,Π1)

(q0, R0, p0,Π0)

ϕθ,Lie T2

. . . ϕθ,Lie T2

ϕθ,Lie T2

. . . ϕθ,Lie T2

Hlayer SRNN

(ˆq1,ˆ

R1,ˆp1,ˆ

Π1)

(ˆq2,ˆ

R2,ˆp2,ˆ

Π2)L(θ)

Figure 2: Inputs are fed through a recurrent layer with Lie

. Prediction error is used as a loss on

through a

1/r

gravitational potential. However, planets are not point masses, and their rotations

matter because they shape planetary climates [

] and even feedback to their orbits [

]. This

already starts to alter

even if one only considers classical gravity. For example, the gravitational

potential Vfor interacting bodies of ﬁnite sizes should be V(q,R)=Pi<j Vi,j , where

Vi,j (q,R)=ZBiZBj−Gρ(xi)ρ(xj)

kqi+Rixi−qj−Rjxjkdxidxj=−Gmimj

kqi−qjk

| {z }

Vi,j,point

+O 1

kqi−qjk2!

| {z }

Vi,j,resid

.(2)

Working with the full potential is complicated since

is not known and the integral is not analytically

known. Can we directly learn Vresid from time-series data?

Classical gravity (i.e. Newtonian physics) is not the only driver of planetary motion — tidal forces

and general relativity (GR) matter too. The former provides a dissipation mechanism and plays

critical roles in altering planetary orbits [

]; the latter doesn’t need much explanation and has

been demonstrated by, e.g., Mercury’s precessing orbit [

]. Tidal forces depend on celestial bodies’

rotations [

] and thus is a function of both

q,R

. GR’s effects cannot be fully characterized with

classical coordinates

q,R,p,Π

, but post-Newtonian approximations based purely on these coordinates

are popular [35]. Can we learn both purely from data if we did not have theories for either?

In addition to the scientiﬁc questions, there are also signiﬁcant machine learning challenges:

Multiscale dynamics.

Rigid-body correction (

Vresid

), tidal force, and GR correction are all much

smaller forces compared to point-mass gravity. Consequently, their effects do not manifest until long

time. Thus, one challenge for learning them is that the dynamical system exhibits different behaviors

over

multiple timescales

. It is reasonable to require long time series data for the small effects to be

learned; meanwhile, when observations are expensive to make, the observation time step

∆t

can be

much longer than the smallest timescales. Can we still learn the physics in this case? We will leverage

symplectic integrator and its mild growth of error over long time [

] to provide a positive answer.

Respecting the Lie group manifold.

However, even having a symplectic integrator is not enough

because the position variable of the latent dynamics (i.e. truth) stays on

SE(3)⊗N

. If the integrated

solution falls off this manifold such that

R|R=I

no longer holds, it is not only incorrect but likely

misleading for the learning of

V(q,R)

. Popular integrators such as forward Euler, Runge-Kutta 4

(RK4) and Leapfrog [1,6,36] unfortunately do no maintain the manifold structure.

2.2 Learning with Lie Symplectic RNNs

Our method can be viewed as a Lie-group generalization of the seminal work of SRNN [

], where a

good integrator that is both symplectic and Lie-group preserving is employed as a recurrent block.

Lie T2: A Symplectic Lie-Group Preserving Integrator.

To construct an integrator that achieves

both properties, we borrow from [

] the idea of Lie-group and symplecticity preserving splitting, and

split our Hamiltonian as

H=HKE +HPE +Hasym

, which contains the axial-symmetric kinetic energy,

potential energy and asymmetric kinetic energy correction terms. This enables computing the exact

integrators

φ[KE]

t,φ[PE]

and

φ[asym]

(see App Bfor details). We then construct a 2nd-order symplectic

integrator Lie

by applying the Strang composition scheme. To account for non-conservative forces,

the corresponding non-conservative momentum update

φ[force]:(p,Π)←F(q,R,p,Π)

is inserted in

the middle of the composition [20]. This gives φ[Lie T2]

hfor stepsize has

φLie T2

h:=φ[KE]

h/2◦φ[PE]

h/2◦φ[asym]

h/2◦φ[force]

h◦φ[asym]

h/2◦φ[PE]

h/2◦φ[KE]

h/2(3)

A Recurrent Architecture for Nonlinear Regression.

Given the simplicity of

Vpoint

, we assume this

is known and learn

Vresid

and

Fθ

with multi-layer perceptron (MLP)

Vθ

resid

and

Fθ

without assuming

any pairwise structure (see App Cfor discussion). We then use

φθ,Lie T2

to integrate dynamics

forward, where

denotes the dependence on the networks. However, when the temporal spacing

between observations

∆t

is large, using a single

φθ,Lie T2

∆t

will result in large errors for the fast

timescale dynamics. Instead, we compose

φθ,Lie T2

times as

(ˆ

k+1,l,ˆ

k+1,l) = φθ,Lie T2

h◦···◦

φθ,Lie T2

h(q

k,l,p

k,l)

, where

H=h/∆t∈Z

determines the integration stepsize

. We perform training

by minimizing the following empirical loss over random minibatches of size Nb

L(θ):=1

NbK

l=1

k=1n

q

k,l −ˆ

qθ

k,l

2

2+

p

k,l −ˆ

pθ

k,l

2

2o(4)

Note that we do not assume access to the true derivatives

k,j

and

k,j

used in the loss function of

some works [1,37,38]. Our training process in summarized in Fig. 2(see App Cfor details).

Beneﬁt.

Learning an accurate

Vθ,F θ

requires accurate numerical simulation which also leads to

a trainable model. Without preservation of the manifold structure, training can lead to ‘shortcuts’

outside the manifold that seemingly match the data but completely mislead the learning. Symplecticity

also plays a vital role in controlling the long time integration error — under reasonable conditions, a

th-order symplectic integrator has linear

O(∆thp)

error bound, whereas a

th-order nonsymplectic

one has an exponential

O(eC∆thp)

error bound [

]. While these bounds do not matter for small

∆t

, they are signiﬁcant for multiscale problems where

∆t

is macroscopic but

is microscopic.

Consequently, improving error estimates for a nonsymplectic integrator by reducing

makes the

RNN exponentially deep — this often renders training difﬁcult [39] and is not desirable.

3 Results

0 100 200 300 400 500

Integrator steps

10−10

10−6

10−2

kRTR−Ik2

0 20 40 60 80 100

Integrator steps

10−7

10−4

10−1

Herror

Euler

RK4

Verlet

Lie RK2

Lie RK4

Lie T2

Figure 3: Results on TRAPPIST-1 with short

data separation.

Top

SO(3)

manifold er-

ror.

Bottom

: Hamiltonian error over the inte-

grated trajectory. Only Lie T2 achieves low

errors in both metrics.

We aim to answer two questions.

Can we learn

multiscale physics? Q2 How important are symplec-

ticity (

) and Lie-group preservation (

) for learn-

ing? The closest baseline for our problem is work of

[16], which learns short timescale rigid-body Hamil-

tonian dynamics for robotics. Placed in our frame-

work, their work corresponds to using RK4 for the re-

current block, which is neither

nor

. Therefore, to

investigate

, we vary the choice of integrator in our

framework as follows: Normal: Explicit Euler, RK4.

: Verlet.

: Lie RK2(CF2) and Lie RK4(CF4) [

We leave the precise details to Apps Cand D.

Toy Two-Body Problem.

We consider an illustrative

two-body problem to demonstrate the effects of

Vrigid

In Fig. 1‘Point’ & ‘Rigid’ denote exact solutions for

a point-mass and rigid-body potential, and ‘Lie T2’

the prediction of our method based on a

learned

from data. Compared to ‘Point’, ‘Rigid’ induces an

apsidal precession (rotation of the orbital axis) due to spin-orbit couplings. Our method successfully

predict this interaction and matches the trajectory of ‘Rigid’.

We next test our method by learning the dynamics of the

TRAPPIST-1

system [

] which consists of

seven earth-sized planets and is notable for potential habitability for terrestrial forms of lives.

TRAPPIST-1, Large ∆t.

To answer

, we choose a large data timestep

∆t= 2.4e−3yr

. The

closest planet has an orbital period of

∼2∆t

(

4.1e−3yr

), while the rigid body correction, tidal force

and GR correction act on much longer scales. Only Lie

successfully trains. All other methods

diverge during training (denoted by

∞

) despite attempts at stabilization with techniques such as

normalization (LayerNorm [

], GroupNorm [

]). Reducing

improves integration accuracy, but

increases the RNN depth and makes training more unstable. We compare with the solution for point-

mass potential only (No Correction). Our method reduces the error up to two orders of magnitude in

measures of trajectory error and potential gradients (Table 1). See App Efor column deﬁnitions.

TRAPPIST-1, Small ∆t.

To gain more insight on

, we shrink

∆t

until almost all methods can

converge and only consider conservative forces (i.e. no tidal force or GR). The mean errors in the

predicted trajectory and derivatives of the learned potential

after

500

integrator steps are shown in

Table 2. Both

methods achieve small errors in position related terms. Verlet has a large rotational

error since it does not integrate on the rotation manifold.

methods achieve lower rotational errors

but are worse elsewhere. Lie T2being both SLachieves the lowest error on both fronts.

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Data-drivendiscoveryofnon-Newtonianastronomyvialearningnon-EuclideanHamiltonianOswinSoMassachusettsInstituteofTechnology,USAoswinso@mit.eduGongjieLiGeorgiaInstituteofTechnology,USAgongjie.li@physics.gatech.eduEvangelosA.TheodorouGeorgiaInstituteofTechnology,USAevangelos.theodorou@gatech.eduMoleiTao...

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Data-driven discovery of non-Newtonian astronomy via learning non-Euclidean Hamiltonian Oswin So

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