Approximation of Nearly-Periodic Symplectic Maps via Structure-Preserving Neural Networks Valentin Duruisseaux1 Joshua W. Burby2 and Qi Tang2

2025-04-27 0 0 8.31MB 21 页 10玖币
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Approximation of Nearly-Periodic Symplectic Maps
via Structure-Preserving Neural Networks
Valentin Duruisseaux1,*, Joshua W. Burby2, and Qi Tang2
1Department of Mathematics, University of California San Diego, La Jolla, CA 92093
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545
*Corresponding author: vduruiss@ucsd.edu
ABSTRACT
A continuous-time dynamical system with parameter
ε
is nearly-periodic if all its trajectories are periodic with nowhere-vanishing
angular frequency as
ε
approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as
parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal
U(1)
symmetries to all
orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the
formal
U(1)
symmetry gives rise to a discrete-time adiabatic invariant. In this paper, we construct a novel structure-preserving
neural network to approximate nearly-periodic symplectic maps. This neural network architecture, which we call symplectic
gyroceptron, ensures that the resulting surrogate map is nearly-periodic and symplectic, and that it gives rise to a discrete-time
adiabatic invariant and a long-time stability. This new structure-preserving neural network provides a promising architecture for
surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing
spurious instabilities.
1 Introduction
Dynamical systems evolve according to the laws of physics, which can usually be described using differential equations. By
solving these differential equations, it is possible to predict the future states of the dynamical system. Identifying accurate and
efficient dynamic models based on observed trajectories is thus critical for the analysis, simulation and control of dynamical
systems. We consider here the problem of learning dynamics: given a dataset of trajectories followed by a dynamical system,
we wish to infer the dynamical law responsible for these trajectories and then possibly use that law to predict the evolution of
similar systems in different initial states. We are particularly interested in the surrogate modeling problem: the underlying
dynamical system is known, but traditional simulations are either too slow or expensive for some optimization task. This
problem can be addressed by learning a less expensive, but less accurate surrogate for the simulations.
Models obtained from first principles are extensively used across science and engineering. Unfortunately, due to incomplete
knowledge, these models based on physical laws tend to over-simplify or incorrectly describe the underlying structure of the
dynamical systems, and usually lead to high bias and modeling errors that cannot be corrected by optimizing over the few
parameters in the models.
Deep learning architectures can provide very expressive models for function approximation, and have proven very effective
in numerous contexts
1–3
. Unfortunately, standard non-structure-preserving neural networks struggle to learn the symmetries
and conservation laws underlying dynamical systems, and as a result do not generalize well. Indeed, they tend to prefer certain
representations of the dynamics where the symmetries and conservation laws of the system are not exactly enforced. As a result,
these models do not generalize well as they are often not capable of producing physically plausible results when applied to new
unseen states. Deep learning models capable of learning and generalizing dynamics effectively are typically over-parameterized,
and as a consequence tend to have high variance and can be very difficult to interpret
4
. Also, training these models usually
requires large datasets and a long computational time, which makes them prohibitively expensive for many applications.
A recent research direction is to consider a hybrid approach which combines knowledge of physics laws and deep learning
architectures
2,3,5,6
. The idea is to encode physics laws and the conservation of geometric properties of the underlying systems
in the design of the neural networks or in the learning process. Available physics prior knowledge can be used to construct
physics-constrained neural networks with improved design and efficiency and a better generalization capacity, which take
advantage of the function approximation power of neural networks to deal with incomplete knowledge.
In this paper, we will consider the problem of learning dynamics for highly-oscillatory Hamiltonian systems. Examples
include the Klein–Gordon equation in the weakly-relativistic regime, charged particles moving through a strong magnetic
field, and the rotating inviscid Euler equations in quasi-geostrophic scaling
7
. More generally, any Hamiltonian system may be
arXiv:2210.05087v2 [cs.LG] 10 May 2023
embedded as a normally-stable elliptic slow manifold in a nearly-periodic Hamiltonian system
8
. Highly-oscillatory Hamiltonian
systems exhibit two basic structural properties whose interactions play a crucial role in their long-term dynamics. First is
preservation of the symplectic form, as for all Hamiltonian systems. Second is timescale separation, corresponding to the
relatively short timescale of oscillations compared with slower secular drifts. Coexistence of these two structural properties
implies the existence of an adiabatic invariant
8–11
. Adiabatic invariants differ from true constants of motion, in particular
energy invariants, which do not change at all over arbitrary time intervals. Instead adiabatic invariants are conserved with
limited precision over very large time intervals. There are no learning frameworks available today that exactly preserve the
two structural properties whose interplay gives rise to adiabatic invariants. This work addresses this challenge by exploiting
a recently-developed theory of nearly-periodic symplectic maps
11
, which can be thought of as discrete-time analogues of
highly-oscillatory Hamiltonian systems9.
As a result of being symplectic, a mapping assumes a number of special properties. In particular, symplectic mappings are
closely related to Hamiltonian systems: any solution to a Hamiltonian system is a symplectic flow
12
, and any symplectic flow
corresponds locally to an appropriate Hamiltonian system
13
. It is well-known that preserving the symplecticity of a Hamiltonian
system when constructing a discrete approximation of its flow map ensures the preservation of many aspects of the dynamical
system such as energy conservation, and leads to physically well-behaved discrete solutions over exponentially-long time
intervals13–17. It is thus important to have structure-preserving neural network architectures which can learn symplectic maps
and ensure that the learnt surrogate map preserves symplecticity. Many physics-informed and structure-preserving machine
learning approaches have recently been proposed to learn Hamiltonian dynamics and symplectic maps
2,3,18–35
. In particular,
Hénon Neural Networks (HénonNets)
2
can approximate arbitrary well any symplectic map via compositions of simple yet
expressive elementary symplectic mappings called Hénon-like mappings. In the numerical experiments conducted in this
paper, HénonNets
2
will be our preferred choice of symplectic map approximator to use as building block in our framework
for approximation of nearly-periodic symplectic maps, although some of the other approaches listed above for approximating
symplectic mappings can be used within our framework as well.
As shown by Kruskal
9
, every nearly-periodic system, Hamiltonian or not, admits an approximate
U(1)
-symmetry, deter-
mined to leading order by the unperturbed periodic dynamics. It is well-known that a Hamiltonian system which admits a
continuous family of symmetries also admits a corresponding conserved quantity. It is thus not surprising that a nearly-periodic
Hamiltonian system, which admits an approximate symmetry, must also have an approximate conservation law
11
, and the
approximately conserved quantity is referred to as an adiabatic invariant.
Nearly-periodic maps, first introduced by Burby et al.
11
, are natural discrete-time analogues of nearly-periodic systems,
and have important applications to numerical integration of nearly-periodic systems. Nearly-periodic maps may also be used
as tools for structure-preserving simulation of non-canonical Hamiltonian systems on exact symplectic manifolds
11
, which
have numerous applications across the physical sciences. Noncanonical Hamiltonian systems play an especially important
role in modeling weakly-dissipative plasma systems
36–42
. Similarly to the continuous-time case, nearly-periodic maps with
a Hamiltonian structure (that is symplecticity) admit an approximate symmetry and as a result also possess an adiabatic
invariant
11
. The adiabatic invariants that our networks target only arise in purely Hamiltonian systems. Just like dissipation
breaks the link between symmetries and conservation laws in Hamiltonian systems, dissipation also breaks the link between
approximate symmetries and approximate conservation laws in Hamiltonian systems. We are not considering systems with
symmetries that are broken by dissipation or some other mechanism, but rather considering systems which possess approximate
symmetries. This should be contrasted with other frameworks
43–45
which develop machine learning techniques for systems that
explicitly include dissipation.
We note that neural network architectures designed for multi-scale dynamics and long-time dependencies are available
46
,
and that many authors have introduced numerical algorithms specifically designed to efficiently step over high-frequency
oscillations
47–49
. However, the problem of developing surrogate models for dynamical systems that avoid resolving short
oscillations remains open. Such surrogates would accelerate optimization algorithms that require querying the dynamics of
an oscillatory system during the optimizer’s “inner loop". The network architecture presented in this article represents a first
important step toward a general solution of this problem. Some of its advantages are that it aims to learn a fast surrogate
model that can resolve long-time dynamics using very short time data, and that it is guaranteed to enjoy symplectic universal
approximation within the class of nearly periodic maps. As developed in this paper, our method applies to dynamical systems
that exhibit a single fast mode of oscillation. In particular, when initial conditions for the surrogate model are selected on the
zero level set of the learned adiabatic invariant, the network automatically integrates along the slow manifold50–54. While our
network architecture generalizes in a straightforward manner to handle multiple non-resonant modes, it cannot be applied to
dynamical systems that exhibit resonant surfaces.
2/21
Note that many of the approaches listed earlier for physics-based or structure-preserving learning of Hamiltonian dynamics
focus on learning the vector field associated to the continuous-time Hamiltonian system, while others learn a discrete-time sym-
plectic approximation to the flow map of the Hamiltonian system. In many contexts, we do not need to infer the continuous-time
dynamics, and only need a surrogate model which can rapidly generate accurate predictions which remain physically consistent
for a long time. Learning a discrete-time approximation to the evolution or flow map, instead of learning the continuous-time
vector field, allows for fast prediction and simulation without the need to integrate differential equations or use neural ODEs and
adjoint techniques (which can be very expensive and can introduce additional errors due to discretization). In this paper, we will
learn nearly-periodic symplectic approximations to the flow maps of nearly-periodic Hamiltonian systems, with the intention of
obtaining algorithms which can generate accurate and physically-consistent simulations much faster than traditional integrators.
Outline.
We first review briefly some background notions from differential geometry in Section 2.1. Then, we discuss how
symplectic maps can be approximated using HénonNets in Section 2.2, before defining nearly-periodic systems and maps and
reviewing their important properties in Section 2.3. In Section 3, we introduce novel neural network architectures, gyroceptrons
and symplectic gyroceptrons, to approximate symplectic and non-symplectic nearly-periodic maps. We then show in Section 4
that symplectic gyroceptrons admit adiabatic invariants regardless of the values of their weights. Finally, in Section 5, we
demonstrate how the proposed architecture can be used to learn surrogate maps for the nearly-periodic symplectic flow maps
associated to two different systems: a nearly-periodic Hamiltonian system composed of two nonlinearly coupled oscillators (in
Section 5.1), and the nearly-periodic Hamiltonian system describing the evolution of a charged particle interacting with its
self-generated electromagnetic field (in Section 5.2).
2 Preliminaries
2.1 Differential Geometry Background
In this paper, we reserve the symbol
M
for a smooth manifold equipped with a smooth auxiliary Riemannian metric
g
, and
E
will always denote a vector space for the parameter
ε
. We will now briefly introduce some standard concepts from differential
geometry that will be used throughout this paper (more details can be found in introductory differential geometry books
55–57
).
A smooth map
h∶M1→M2
between smooth manifolds
M1,M2
is a
diffeomorphism
if it is bijective with a smooth inverse.
We say that
fε∶M1→M2
,
ε∈E
, is a smooth
ε
-dependent mapping when the mapping
M1×R→M2∶(m,ε)↦fε(m)
is smooth.
A
vector field
on a manifold
M
is a map
X∶M→T M
such that
X(m)∈TmM
for all
m∈M
, where
TmM
denotes the
tangent
space
to
M
at
m
and
T M ={(m,v)m∈M,v∈TmM}
is the
tangent bundle T M
of
M
. The vector space dual to
TmM
is the
cotangent space T∗
mM
, and the
cotangent bundle
of
M
is
T∗M={(m,p)m∈M,p∈T∗
mM}
. The integral curve at
m
of a
vector field
X
is the smooth curve
c
on
M
such that
c(0)=m
and
c′(t)=X(c(t))
. The
flow
of a vector field
X
is the collection
of maps ϕt∶M→Msuch that ϕt(m)is the integral curve of Xwith initial condition m∈M.
A
k
k
k-form
on a manifold
M
is a map which assigns to every point
m∈M
a skew-symmetric
k
-multilinear map on
TmM
. Let
αbe a k-form and βbe a s-form βon a manifold M. Their tensor product α⊗βat m∈Mis defined via
(α⊗β)m(v1,...,vk+s)=αm(v1,...,vk)βm(vk+1,...,vk+s).
The alternating operator Alt acts on a k-form αvia
Alt(α)(v1,...,vk)=1
k!∑
π∈Sk
sgn(π)α(vπ(1),...,vπ(k)),
where
Sk
is the group of all the permutations of
{1,...,k}
and
sgn(π)
is the sign of the permutation. The
wedge product α∧β
is then defined via
α∧β=(k+s)!
k!s!Alt(α⊗β).
The
exterior derivative
of a smooth function
f∶M→R
is its differential
df
, and the
exterior derivative dα
of a
k
-form
α
with k>0 is the (k+1)-form defined by
d
∑
i1,...,ik
αi1...ikdxi1∧...∧dxik
=∑
j∑
i1,...,ik
∂jαi1...ikdxj∧dxi1∧...∧dxik.
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The interior product ιXαwhere Xis a vector field on Mand αis a k-form is the (k−1)-form defined via
(ιXα)m(v2,...,vk)=αm(X(m),v2,...,vk).
The pull-back ψ∗αof αby a smooth map ψ∶M→Nis the k-form defined by
(ψ∗α)m(v1,...,vk)=αψ(m)(dψ⋅v1,...,dψ⋅vk).
The
Lie derivative LXα
of the
k
-form
α
along a vector field
X
with flow
ϕt
is
LXα=d
dt t=0ϕ∗
tα
, and for a smooth function
f∶M→R,LXfis the directional derivative LXf=df⋅X.
The
circle group U(1)
, also known as first unitary group, is the one-dimensional Lie group of complex numbers of unit
modulus with the standard multiplication operation. It can be parametrized via
eiθ
for
θ∈[0,2π)
, and is isomorphic to the
special orthogonal group
SO(2)
of rotations in the plane. A
circle action
on a manifold
M
is a one-parameter family of smooth
diffeomorphisms Φθ∶M→Mthat satisfies the following three properties for any θ,θ1,θ2∈U(1)≅Rmod 2π:
Φθ+2π=Φθ(periodicity), Φ0=IdM(identity), Φθ1+θ2=Φθ1○Φθ2(additivity).
The infinitesimal generator of a circle action Φθon Mis the vector field on Mdefined by m↦d
dθθ=0
Φθ(m).
2.2 Approximation of Symplectic Maps via Hénon Neural Networks
Let
U⊂Rn×Rn=R2n
be an open set in an even-dimensional Euclidean space. Denote points in
Rn×Rn
using the notation
(x,y), with x,y∈Rn. A smooth mapping Φ∶U→R2nwith components Φ(x,y)=(¯x(x,y),¯y(x,y))is symplectic if
n
∑
i=1
dxi∧dyi=n
∑
i=1
d¯xi∧d¯yi.(2.1)
The symplectic condition
(2.1)
implies that the mapping
Φ
has a number of special properties. In particular, there is a
close relation between Hamiltonian systems and symplecticity of flows: Poincaré’s Theorem
12
states that any solution to a
Hamiltonian system is a symplectic flow, and it can also be shown that any symplectic flow corresponds locally to an appropriate
Hamiltonian system. Preserving the symplecticity of a Hamiltonian system when constructing a discrete approximation of its
flow map ensures the preservation of many aspects of the dynamical system such as energy conservation, and leads to physically
well-behaved discrete solutions
13–17
. It is thus important to have structure-preserving network architectures which can learn
symplectic maps.
The space of all symplectic maps is infinite dimensional
58
, so the problem of approximating an arbitrary symplectic map
using compositions of simpler symplectic mappings is inherently interesting. Turaev
59
showed that every symplectic map may
be approximated arbitrarily well by compositions of Hénon-like maps, which are special elementary symplectic maps.
Definition 2.1
Let
V∶Rn→R
be a smooth function on
Rn
and let
η∈Rn
be a constant. We define the
Hénon-like map
H[V,η]∶Rn×Rn→Rn×Rnwith potential V and shift ηvia
H[V,η]x
y=y+η
−x+∇V(y).(2.2)
Theorem 2.1 (Turaev59)
Let
Φ∶U→Rn×Rn
be a
Cr+1
symplectic mapping. For each compact set
C⊂U
and
δ>0
there is a
smooth function
V∶Rn→R
, a constant
η
, and a positive integer
N
such that
H[V,η]4N
approximates the mapping
Φ
within
δ
in the Crtopology.
Remark 2.1
The significance of the number
4
in this theorem follows from the fact that the fourth iterate of the Hénon-like
map with trivial potential V =0is the identity map: H[0,η]4=IdRn×Rn.
Turaev’s result suggests the specific neural network architecture to approximate symplectic mappings using Hénon-like
maps2. We review the construction of HénonNets2, starting with the notion of a Hénon layer.
Definition 2.2
Let
η∈Rn
be a constant vector, and let
V
be a scalar feed-forward neural network on
Rn
, that is., a smooth
mapping
V∶W×Rn→R
, where
W
is a space of neural network weights. The
Hénon layer
with potential
V
, shift
η
, and weight
W is the iterated Hénon-like map
L[V[W],η]=H[V[W],η]4,(2.3)
where we use the notation V[W]to denote the mapping V [W](y)=V(W,y),for any y ∈Rn,W∈W.
4/21
There are various network architectures for the potential
V[W]
that are capable of approximating any smooth function
V∶Rn→R
with any desired level of accuracy. For example, a fully-connected neural network with a single hidden layer of sufficient width
can approximate any smooth function. Therefore a corollary of Theorem 2.1 is that any symplectic map may be approximated
arbitrarily well by the composition of sufficiently many Hénon layers with various potentials and shifts. This leads to the notion
of a Hénon Neural Network.
Definition 2.3 Let N be a positive integer and
•V
V
V={Vk}k∈{1,...,N}be a family of scalar feed-forward neural networks on Rn
•W
W
W={Wk}k∈{1,...,N}be a family of network weights for V
V
V
•η
η
η={ηk}k∈{1,...,N}be a family of constants in Rn
The Hénon neural network (HénonNet) with layer potentials V
V
V , layer weights W
W
W , and layer shifts η
η
ηis the mapping
H[V
V
V[W
W
W],η
η
η]=L[VN[WN],ηN]○... ○L[V2[W2],η2]○L[V1[W1],η1](2.4)
=H[VN[WN],ηN]4○... ○H[V2[W2],η2]4○H[V1[W1],η1]4.(2.5)
A composition of symplectic mappings is also symplectic, so every HénonNet is a symplectic mapping, regardless of
the architectures for the networks
Vk
and of the weights
Wk
. Furthermore, Turaev’s Theorem 2.1 implies that the family of
HénonNets is sufficiently expressive to approximate any symplectic mapping:
Lemma 2.1
Let
Φ∶U→Rn×Rn
be a
Cr+1
symplectic mapping. For each compact set
C⊂U
and
δ>0
there is a HénonNet
H
that approximates Φwithin δin the Crtopology.
Remark 2.2 Note that Hénon-like maps are easily invertible,
H[V,η]x
y=y+η
−x+∇V(y)⇒H−1[V,η]x
y=∇V(x−η)−y
x−η,(2.6)
so we can also easily invert Hénon networks by composing inverses of Hénon-like maps.
We also introduce here modified versions of Hénon-like maps and HénonNets to approximate symplectic maps possessing a
near-identity property:
Definition 2.4
Let
V∶Rn→R
be a smooth function and let
η∈Rn
be a constant. We define the
near-identity Hénon-like map
Hε[V,η]∶Rn×Rn→Rn×Rnwith potential V and shift ηvia
Hε[V,η]x
y=y+η
−x+ε∇V(y).(2.7)
Near-identity Hénon-like maps satisfy the near-identity property H0[V,η]4=IdRn×Rn.
Definition 2.5 Let N be a positive integer and
•V
V
V={Vk}k∈{1,...,N}be a family of scalar feed-forward neural networks on Rn
•W
W
W={Wk}k∈{1,...,N}be a family of network weights for V
V
V
•η
η
η={ηk}k∈{1,...,N}be a family of constants in Rn
The near-identity Hénon network with layer potentials V
V
V , layer weights W
W
W , and layer shifts η
η
ηis the mapping defined via
Hε[V
V
V[W
W
W],η
η
η]=Hε[VN[WN],ηN]4○... ○Hε[V2[W2],η2]4○Hε[V1[W1],η1]4,(2.8)
and it satisfies the near-identity property H0[V
V
V[W
W
W],η
η
η]=IdRn×Rn.
2.3 Nearly-Periodic Systems and Nearly-Periodic Maps
2.3.1 Nearly-Periodic Systems
Intuitively, a continuous-time dynamical system with parameter
ε
is nearly-periodic if all of its trajectories are periodic with
nowhere-vanishing angular frequency in the limit
ε→0
. Such a system characteristically displays limiting short-timescale
dynamics that ergodically cover circles in phase space. More precisely, a nearly-periodic systems can be defined as follows:
5/21
摘要：

ApproximationofNearly-PeriodicSymplecticMapsviaStructure-PreservingNeuralNetworksValentinDuruisseaux1,*,JoshuaW.Burby2,andQiTang21DepartmentofMathematics,UniversityofCaliforniaSanDiego,LaJolla,CA920932TheoreticalDivision,LosAlamosNationalLaboratory,LosAlamos,NM87545*Correspondingauthor:vduruiss@ucsd...

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