Original: Hamiltonian Neural Koopman Operator
Revised: Hamiltonian Neural Koopman Operator

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Recently, physics-informed learning, a class of deep learning framework that incorporates the physics priors and the observational noise-perturbed data into theneural network models, has shown outstanding performances in learning physicalprinciples with higher accuracy, faster training speed, and better generalization ability.
Revised: Recently, physics-informed learning, a class of deep learning framework that incorporates the physics priors and the observational noise-perturbed data into the neural network models, has shown outstanding performances in learning physical principles with higher accuracy, faster training speed, and better generalization ability.

Original: Here, for the Hamiltonian mechanics and using the Koopman operator theory,we propose a typical physics-informed learning framework, named as H amiltonian
Revised: Here, for the Hamiltonian mechanics and using the Koopman operator theory, we propose a typical physics-informed learning framework, named as H amiltonian

Original: N eural K oopman O perator (HNKO) to learn the corresponding Koopman operatorautomatically satisfying the conservation laws.
Revised: N eural K oopman O perator (HNKO) to learn the corresponding Koopman operator automatically satisfying the conservation laws.

Original: We analytically investigate thedimension of the manifold induced by the orthogonal transformation, and use amodified auto-encoder to identify the nonlinear coordinate transformation that isrequired for approximating the Koopman operator.
Revised: We analytically investigate the dimension of the manifold induced by the orthogonal transformation, and use a modified auto-encoder to identify the nonlinear coordinate transformation that is required for approximating the Koopman operator.

Original: Taking the Kepler problemas an example, we demonstrate that the proposed HNKO in robustly learning the
Revised: Taking the Kepler problem as an example, we demonstrate that the proposed HNKO in robustly learning the

Original: Hamiltonian dynamics outperforms the representative methods developed in theliterature.
Revised: Hamiltonian dynamics outperforms the representative methods developed in the literature.

Original: Our results suggest that feeding the prior knowledge of the underlyingsystem and the mathematical theory appropriately to the learning framework canreinforce the capability of the deep learning.
Revised: Our results suggest that feeding the prior knowledge of the underlying system and the mathematical theory appropriately to the learning framework can reinforce the capability of the deep learning.

Original:  Introduction
Revised:  Introduction

Original: Reconstructing nonlinear dynamical systems solely using the observational noise-perturbed data isa focal challenge in various fields.
Revised: Reconstructing nonlinear dynamical systems solely using the observational noise-perturbed data is a focal challenge in various fields.

Original: The neural networks embedded with the induced biases haveremarkable abilities in learning and generalizing the intrinsic kinetics of the underlying systemsfrom the noisy data, such as the Hamiltonian neural networks [7, 8], the Lagrangian neural networks[5] and the physics-informed neural networks [13, 9, 6].
Revised: The neural networks embedded with the induced biases have remarkable abilities in learning and generalizing the intrinsic kinetics of the underlying systems from the noisy data, such as the Hamiltonian neural networks [7, 8], the Lagrangian neural networks [5] and the physics-informed neural networks [13, 9, 6].

Original: These methods are successfully appliedto a variety of tasks, such as the generative tasks [17] and the dynamics reconstruction [20, 15],sharing the similar design idea where utilisation of an appropriate loss function enforces the modelto nearly obey the physical principles.
Revised: These methods are successfully applied to a variety of tasks, such as the generative tasks [17] and the dynamics reconstruction [20, 15], sharing the similar design idea where utilisation of an appropriate loss function enforces the model to nearly obey the physical principles.

Original: Progress has been achieved; however, these methods havepoor generalization abilities, either enlarging the network complexity or over-fitting the noisy dataduring the training stage for decreasing the loss.
Revised: Progress has been achieved; however, these methods have poor generalization abilities, either enlarging the network complexity or over-fitting the noisy data during the training stage for decreasing the loss.

Original: To overcome these drawbacks, endowing the neuralnetworks with natural physical priors becomes one of the mainstream approaches to improving thesample efficiency, the robustness, and the generalization ability [11, 10].
Revised: To overcome these drawbacks, endowing the neural networks with natural physical priors becomes one of the mainstream approaches to improving the sample efficiency, the robustness, and the generalization ability [11, 10].

Original: Recent advances in the Koopman operator theory provide a new way to identify intrinsic kineticsusing linear representations for strongly nonlinear systems [3, 12, 14].
Revised: Recent advances in the Koopman operator theory provide a new way to identify intrinsic kinetics using linear representations for strongly nonlinear systems [3, 12, 14].

Original: There are several algorithmsfor approximating such an infinite-dimensional operator in a data-driven manner, including thedynamic mode decomposition (DMD) [16, 18, 14] and the extending dynamic mode decomposition(EDMD)
Revised: There are several algorithms for approximating such an infinite-dimensional operator in a data-driven manner, including the dynamic mode decomposition (DMD) [16, 18, 14] and the extending dynamic mode decomposition (EDMD) [19, 2].

Original: In these works, they all try to obtain parsimonious models, while maintaining anaccurate reconstruction of the unknown system in a data-driven manner.
Revised: In these works, they all try to obtain parsimonious models, while maintaining an accurate reconstruction of the unknown system in a data-driven manner.

Original: In this article, inspired by the advances of physics-informed learning and the Koopman operatortheory, we aim to develop a typical framework to efficiently and robustly learn the Hamiltonian dynamics solely using the observational noisy data.
Revised: In this article, inspired by the advances of physics-informed learning and the Koopman operator theory, we aim to develop a typical framework to efficiently and robustly learn the Hamiltonian dynamics solely using the observational noisy data.

Original: Specifically, based on the fact that the Koopmanoperator of the Hamiltonian dynamics is unitary, we use an orthogonal neural network to learnthe approximative Koopman opertaor that naturally keeps the conservation laws, and also use anauto-encoder to identify the nonlinear coordinate transformation, mapping the original data to alow-dimensional manifold on a high-dimensional sphere.
Revised: Specifically, based on the fact that the Koopman operator of the Hamiltonian dynamics is unitary, we use an orthogonal neural network to learn the approximated Koopman operator that naturally keeps the conservation laws, and also use an auto-encoder to identify the nonlinear coordinate transformation, mapping the original data to a low-dimensional manifold on a high-dimensional sphere.

Original: Taking the Kepler problem as an example,we demonstrate the efficacy of the proposed framework.
Revised: Taking the Kepler problem as an example, we demonstrate the efficacy of the proposed framework.

Original:  Methodology
Revised:  Methodology

Original: We consider a dynamical system whose state vector x
Revised: We consider a dynamical system whose state vector x

Original: =
Revised: =

Original: ( x 1 , · · · , x n ) ⊤ ∈ M ⊂ R n evolves alongsome smooth symplectic vector field f ( x ) .
Revised: ( x 1 , · · · , x n ) ⊤ ∈ M ⊂ R n evolves along some smooth symplectic vector field f ( x ) .

Original: Actually, the flow mappings of this autonomous dynamicalsystem form an operator group, denoted by { F t : F t ( x ) = x + (cid:82) t 0 f
Revised: Actually, the flow mappings of this autonomous dynamical system form an operator group, denoted by { F t : F t ( x ) = x + (cid:82) t 0 f

Original: ( x ( s ))
Revised: ( x ( s ))

Original: d s, x ∈ M , t > 0 } .
Revised: d s, x ∈ M , t > 0 } .

Original: The
Revised: The

Original: Koopman operator K t with regard to the flow F t is an infinite-dimensional linear operator, acting onthe function space F = { g : M → R } , and satisfying K t g = g ◦ F t for g ∈ F .
Revised: Koopman operator K t with regard to the flow F t is an infinite-dimensional linear operator, acting on the function space F = { g : M → R } , and satisfying K t g = g ◦ F t for g ∈ F .

Original: Specifically, if wefix the time interval ∆ t and given an initial state x 0 ∈ M , one can accordingly generate the statetrajectory by this flow, i.e., { x k : x k = F k ∆ t x 0 } mk =0 .
Revised: Specifically, if we fix the time interval ∆ t and given an initial state x 0 ∈ M , one can accordingly generate the state trajectory by this flow, i.e., { x k : x k = F k ∆ t x 0 } mk =0 .

Original: Thus, we have K ∆ t g ( x k ) =
Revised: Thus, we have K ∆ t g ( x k ) =

Original: g ( F ∆ t ( x k )) =
Revised: g ( F ∆ t ( x k )) =

Original: g ( x k +1 ) .
Revised: g ( x k +1 ) .

Original: In practice, the data { x i } mi =0 often contains noise perturbations due to the nature of datameasurement, which therefore motivates us to robustly approximate the Koopman operator, andpreserve the energy-like quantity in the original Hamiltonian dynamics.
Revised: In practice, the data { x i } mi =0 often contains noise perturbations due to the nature of data measurement, which therefore motivates us to robustly approximate the Koopman operator, and preserve the energy-like quantity in the original Hamiltonian dynamics.

Original: For a sake of simplicity, wedenote by K ∆ t for K when ∆ t is given.
Revised: For a sake of simplicity, we denote by K ∆ t for K when ∆ t is given.

Original: Connections with DMDand EDMD
Revised: Connections with DMD

Original:  The DMD [16, 18, 14] constructs two data matrices X and X ′ from the observational sequential data as X =
Revised: and EDMD The DMD [16, 18, 14] constructs two data matrices X and X ′ from the observational sequential data as X =

Original: ( x 0 , x 2 , · · · , x m − 1 ) , X ′ =(
Revised: ( x 0 , x 2 , · · · , x m − 1 ) , X ′ =

Original:  x 1 , x 2 , · · · , x m ) , X , X ′ ∈ R n × m .
Revised: ( x 1 , x 2 , · · · , x m ) , X , X ′ ∈ R n × m .

Original: Then, obtain the optimal linear operator K
Revised: Then, obtain the optimal linear operator K

Original: = X ′ X
Revised: = X ′ X

Original: + ∈
Revised: + ∈

Original: R n × n 1 for KX ≈ X ′ as the approximation of K [16].
Revised: R n × n 2 for KX ≈ X ′ as the approximation of K [16].

Original: Particularly, the Koopman operator for theconservative dynamical systems governed by the Hamiltonian dynamics is unitary.
Revised: Particularly, the Koopman operator for the conservative dynamical systems governed by the Hamiltonian dynamics is unitary.

Original: This property implies that we should restrict the candidate K in the orthogonal group SO ( n ) =
Revised: This property implies that we should restrict the candidate K in the orthogonal group SO ( n ) =

Original: { B ∈ R n × n | BB ⊤ =
Revised: { B ∈ R n × n | BB ⊤ =

Original: I, det ( B ) = 1 } , then we formally have the following vanilla optimal problem
Revised: I, det ( B ) = 1 } , then we formally have the following vanilla optimal problem

Original: [Equation]
Revised: [Equation]

Original: where ∥·∥ F denotes the Frobenius norm.
Revised: where ∥·∥ F denotes the Frobenius norm.

Original: The vanilla surrogate of the DMD has two major weaknesses:
Revised: The vanilla surrogate of the DMD has two major weaknesses:

Original: The size of K is limited by the dimension n of the original system, which is not sufficiently large enough to approximate the intrinsic infinite-dimenisonal operator K ;2.
Revised: The size of K is limited by the dimension n of the original system, which is not sufficiently large enough to approximate the intrinsic infinite-dimensional operator K ; 2.

Original: The orthogonal transformation preserves the norm of the state and hence induces its dynamics{ K i x 0 } mi =0 embedded on a sphere, while the conserved orbit of the Hamiltonian dynamicsmay not be on some n -dimensional sphere.
Revised: The orthogonal transformation preserves the norm of the state and hence induces its dynamics { K i x 0 } mi =0 embedded on a sphere, while the conserved orbit of the Hamiltonian dynamics may not be on some n -dimensional sphere.

Original: For the first weakness, the EDMD [19, 2] lifts the dimension of K by introducing adictionary of nonlinear observation functions { g i } pi =1 to obtain the augmented state y =(
Revised: For the first weakness, the EDMD [19, 2] lifts the dimension of K by introducing a dictionary of nonlinear observation functions { g i } pi =1 to obtain the augmented state y =

Original:  g 1 ( x ) , g 2 ( x ) , · · · , g p ( x ))
Revised: ( g 1 ( x ) , g 2 ( x ) , · · · , g p ( x ))

Original: ⊤ ∈ R p , p > n .
Revised: ⊤ ∈ R p , p > n .

Original: Analogous to the DMD, the two data matrices areconstructed as Y =
Revised: Analogous to the DMD, the two data matrices are constructed as Y =

Original: ( y 0 , · · · , y m − 1 ) and Y ′
Revised: ( y 0 , · · · , y m − 1 ) and Y ′

Original: =
Revised: =

Original: ( y 1 , · · · , y m ) , yielding an approximated Koopmanoperator as K = Y ′ Y + ∈ R p × p .
Revised: ( y 1 , · · · , y m ) , yielding an approximated Koopman operator as K = Y ′ Y + ∈ R p × p .

Original: For the second weakness, in this work, we embed the augmentedstate y to some higher-dimensional sphere, denoted by S p ( r ) =
Revised: For the second weakness, in this work, we embed the augmented state y to some higher-dimensional sphere, denoted by S p ( r ) =

Original: { y ∈ R p : ∥
Revised: { y ∈ R p : ∥

Original: y ∥ 2 = r 2 } .
Revised: y ∥ 2 = r 2 } .

Original: Then, weobtain the following constrained optimization problem:
Revised: Then, we obtain the following constrained optimization problem:

Original: [Equation]
Revised: [Equation]

Original: In general, it is difficult to solve the optimization problem (1) with the classical optimization methodsdue to the highly nonconvex property.
Revised: In general, it is difficult to solve the optimization problem (1) with the classical optimization methods due to the highly nonconvex property.

Original: Hence, we transform the above problem into a neural networkframework to efficiently address it.
Revised: Hence, we transform the above problem into a neural network framework to efficiently address it.

Original: X + = ( X ⊤ X ) − 1 X ⊤ is the Auto-encoder structure of dictionary functions In the EDMD, the dictionary { g i } pi =1 is regardedas a basis of F and the coordinate function is reconstructed as the optimal linear combinationof these functions.
Revised: Auto-encoder structure of dictionary functions In the EDMD, the dictionary { g i } pi =1 is regarded as a basis of F and the coordinate function is reconstructed as the optimal linear combination of these functions.

Original: In practice, it always require a large dictionary to approximate a set of basisfunctions.
Revised: In practice, it always requires a large dictionary to approximate a set of basis functions.

Original: To reduce the computational cost and sustain the representation ability, we adopt an autoencoder structure to encode y
Revised: To reduce the computational cost and sustain the representation ability, we adopt an autoencoder structure to encode y

Original: =
Revised: =

Original: ( g 1 ( x ) , · · · , g p ( x ))
Revised: ( g 1 ( x ) , · · · , g p ( x ))

Original: ⊤ as y =
Revised: ⊤ as y =

Original: ϕ θ 1 ( x ) , and decode the coordinatesas x = ϕ − 1 θ 2 ( y ) .
Revised: ϕ θ 1 ( x ) , and decode the coordinates as x = ϕ − 1 θ 2 ( y ) .

Original: We train the parameters θ = ( θ 1 , θ 2 ) of this auto-encoder using the loss function as
Revised: We train the parameters θ = ( θ 1 , θ 2 ) of this auto-encoder using the loss function as

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Introduction of the neural orthogonal method The Lie exponent map A → exp( A ) :
Revised: Introduction of the neural orthogonal method The Lie exponent map A → exp( A ) :

Original: = I +
Revised: = I +

Original: A + 12 A 2 + · · · from the skew-symmetric group so ( n ) =
Revised: A + 12 A 2 + · · · from the skew-symmetric group so ( n ) =

Original: { A ∈ R n × n : A + A ⊤ = 0 } to SO ( n )is surjective [11], and there exists an isomorphism α from R n ( n − 1)to so ( n ) as α ( A ) =
Revised: { A ∈ R n × n : A + A ⊤ = 0 } to SO ( n ) is surjective [11], and there exists an isomorphism α from R n ( n − 1)to so ( n ) as α ( A ) =

Original: A − A ⊤ ,where A ∈ R n ( n − 1)is identified as an upper triangular matrix with the zero diagonal elements.
Revised: A − A ⊤ , where A ∈ R n ( n − 1)is identified as an upper triangular matrix with the zero diagonal elements.

Original: Thus, we can represent the approximative orthogonal Koopman operator as a parameterized form
Revised: Thus, we can represent the approximative orthogonal Koopman operator as a parameterized form

Original: K = exp
Revised: K = exp

Original: ( α ( A )) , where A owns n ( n − 1)learnable parameters.
Revised: ( α ( A )) , where A owns n ( n − 1)learnable parameters.

Original: Then we train the matrix K usingthe loss function as
Revised: Then we train the matrix K using the loss function as

Original: [Equation]
Revised: [Equation]

Original: Notice that, in our framework, the orthogonality of the operator K is automatically guaranteed by itsconstruction procedure.
Revised: Notice that, in our framework, the orthogonality of the operator K is automatically guaranteed by its construction procedure.

Original: Indeed, this framework does not require an injection of the regularizationterms into the loss function as those usually required in the literature [7, 5].
Revised: Indeed, this framework does not require an injection of the regularization terms into the loss function as those usually required in the literature [7, 5].

Original: Constrain the dimension of the embedded manifold To ensure the augmented states { y i } mi =0 onsome p -dimensional sphere, we set the radius r of the embedded sphere as a learnable parameter andsimply set the distance to the origin as the loss function
Revised: Constrain the dimension of the embedded manifold To ensure the augmented states { y i } mi =0 on some p -dimensional sphere, we set the radius r of the embedded sphere as a learnable parameter and simply set the distance to the origin as the loss function

Original: [Equation]
Revised: [Equation]

Original: From Theorem 3.1, it follows that the trajectory generated by K ∈ SO ( p ) belongs to an at most⌊ p/ 2 ⌋ -dimensional manifold.
Revised: From Theorem 3.1, it follows that the trajectory generated by K ∈ SO ( p ) belongs to an at most ⌊ p/ 2 ⌋ -dimensional manifold.

Original: Hence, the freedom degree of the augmented state trajectory { y i } mi =0should be lower than ⌊ p/ 2 ⌋ .
Revised: Hence, the freedom degree of the augmented state trajectory { y i } mi = should be lower than ⌊ p/ 2 ⌋ .

Original: For any nonzero vector v , the hyperplane equation ⟨ v , y ⟩ = [-
 minusone degree of-] the freedom  for y .
Revised: Notice that, once the hyperplane equation ⟨ v , y ⟩ = [-
 minusone degree of-]0 is satisfied for any nonzero vector v , the freedom degree for y decreases by one order.

Original: Thus, to restrict the freedom degree of the augmented states, weintroduce the loss as follows:
Revised: Thus, to restrict the freedom degree of the augmented states, we introduce the loss as follows:

Original: [Equation]
Revised: [Equation]

Revised: = q (cid:88) k

Revised: = m (cid:88)

Revised: i

Revised: [Equation]

Revised: (cid:28)

Original:  where { v k ∈ R p , k = 1 , · · · , q } are learnable parameters and q ∈ Z + holds q ≤ p − 2 .
Revised: [Equation] (cid:29) [Equation] , where { v k ∈ R p , k = 1 , · · · , q } are learnable parameters and q ∈ Z + holds q ≤ p − 2 .

Original: Toguarantee that { v k } qk =1 are linearly independent, we design an orthogonal regularization term as
Revised: To guarantee that { v k } qk =1 are linearly independent, we design an orthogonal regularization term as

Original: L ind = (cid:80) k ̸
Revised: L ind = (cid:80) k ̸

Original: = j ⟨ v k , v j ⟩ 2 .
Revised: = j ⟨ v k , v j ⟩ 2 .

Original: Finally, we train the parameters { θ , K, r, V = ( v 1 , ..., v q ) } using thefollowing integrated loss function
Revised: Finally, we train the parameters { θ , K, r, V = ( v 1 , ..., v q ) } using the following integrated loss function

Original: [Equation]
Revised: [Equation]

Original: where q is an adjustable hyperparameter.
Revised: where q is an adjustable hyperparameter.

Original: We use L sphere and L deg to embed the original n -dimensionaldata to the ( p − q − 1)
Revised: We use L sphere and L deg to embed the original n -dimensional data to the ( p − q − 1)

Original: -dimensional manifold on S p ( r ) .
Revised: -dimensional manifold on S p ( r ) .

Original: Here we select q such that p − ⌊ p/ 2 ⌋ − 1 ≤q ≤ p − 2 with p − q − 1 ≤ ⌊ p/ 2 ⌋ .
Revised: Here we select q such that p − ⌊ p/ 2 ⌋ − 1 ≤ q ≤ p − 2 with p − q − 1 ≤ ⌊ p/ 2 ⌋ .

Original: Robustness learning from noisy data The proposed framework can be directly applied to thenoise perturbed case, i.e., { ˜ x i = x
Revised: Robustness learning from noisy data The proposed framework can be directly applied to the noise perturbed case, i.e., { ˜ x i = x

Original: i + ξ i } mi =0 , where { ξ i } are the samples of independent randomvariables.
Revised: i + ξ i } mi =0 , where { ξ i } are the samples of independent random variables.

Original: Generally, the existing models, such as the DMD, the EDMD and the other learning-basedmethods, likely to overfit the noise perturbed data due to a lack of prior knowledge.
Revised: Generally, the existing models, such as the DMD, the EDMD and the other learning-based methods, likely to overfit the noise perturbed data due to a lack of prior knowledge.

Original: For example,it may break the conservative law in the Hamiltonian dynamics as the noise perturbed data act asan effect of a diffusion dynamics.
Revised: For example, it may break the conservative law in the Hamiltonian dynamics as the noise perturbed data act as an effect of a diffusion dynamics.

Original: On the contrary, our framework can identify the noisy sampleseffectively, because the conservation is naturally rooted in the orthogonal transform.
Revised: On the contrary, our framework can identify the noisy samples effectively, because the conservation is naturally rooted in the orthogonal transform.

Original: We numericallyvalidate the natural conservation in Appendix A.2.2.
Revised: We numerically validate the natural conservation in Appendix A.2.2.

Original:  Theoretical Results
Revised:  Theoretical Results

Original: In this section, we focus on the lowest dimension of manifold M ⊂ R n that covers the trajectory Γ ={ x , Ax , A 2 x , · · · } generated by the orthogonal transform A ∈ SO ( n ) .
Revised: In this section, we focus on the lowest dimension of manifold M ⊂ R n that covers the trajectory Γ = { x , Ax , A 2 x , · · · } generated by the orthogonal transform A ∈ SO ( n ) .

Original: Although the orthogonaltransform restricts the trajectory Γ to some n -dimensional sphere, this constraint is too rough todescribe the manifold M .
Revised: Although the orthogonal transform restricts the trajectory Γ to some n -dimensional sphere, this constraint is too rough to describe the manifold M .

Original: In particular, Γ may have degrees of freedom lower than n − 1 .
Revised: In particular, Γ may have degrees of freedom lower than n − 1 .

Original: Thefollowing theorem precisely provides the admitted dimension for M .
Revised: The following theorem precisely provides the admitted dimension for M .

Original: The proof is presented in
Revised: The proof is presented in

Original: Appendix A.1.1.
Revised: Appendix A.1.1.

Original: Theorem 3.1 For each A ∈ SO ( n ) , if the orthogonal normal form of A has s diagonal blocks,denoted by A i , i = 1 , · · · , s with A i = ± 1 , (( a i , − b i ) ⊤ , ( b i , a i ) ⊤ ) , and b i ̸
Revised: Theorem 3.1 For each A ∈ SO ( n ) , if the orthogonal normal form of A has s diagonal blocks, denoted by A i , i = 1 , · · · , s with A i = ± 1 , (( a i , − b i ) ⊤ , ( b i , a i ) ⊤ ) , and b i ̸

Original: = 0 .
Revised: = 0 .

Original: Then, for any initialstate x , the trajectory Γ = { A j x } ∞ j =0 belongs to an ( n − s ) -dimensional manifold.
Revised: Then, for any initial state x , the trajectory Γ = { A j x } ∞ j =0 belongs to an ( n − s ) -dimensional manifold.

Original: Morever, Γbelongs to an at most ⌊ n 2 ⌋ -dimensional manifold, where ⌊·⌋ represents the ceil function.
Revised: Moreover, Γ belongs to an at most ⌊ n 2 ⌋ -dimensional manifold, where ⌊·⌋ represents the floor function.

Original: Using the result in Theorem 3.1, we can determine all the admitted dimensions for M as1 , 2 , · · · , ⌊ n/ 2 ⌋ according to the number of diagonal blocks in A ’s Jordan normal form.
Revised: Using the result in Theorem 3.1, we can determine all the admitted dimensions for M as 1 , 2 , · · · , ⌊ n/ 2 ⌋ according to the number of diagonal blocks in A ’s Jordan normal form.

Original: Thistheorem suggests that the degrees of freedom in the embedding procedure should be restricted suchthat the augmented states are distributed on some at most ⌊ p/ 2 ⌋ -dimensional manifold.
Revised: This theorem suggests that the degrees of freedom in the embedding procedure should be restricted such that the augmented states are distributed on some at most ⌊ p/ 2 ⌋ -dimensional manifold [Figure]

Original:  Experiments
Revised:  Experiments

Original: We consider the planar Kepler problem with the Hamiltonian quantity H =
Revised: We consider the planar Kepler problem with the Hamiltonian quantity H =

Original: m 2 ( ˙ x 2 + ˙ y 2 ) − [Table] ,which is a special case for the classical n -body Hamiltonian system with n = 1 [1].
Revised: m 2 ( ˙ x 2 + ˙ y 2 ) − gm 2 √ x 2 + y 2 , which is a special case for the classical n -body Hamiltonian system with n = 1 [1].

Original: Here, ( x, y ) and( ˙ x, ˙ y ) represent the coordinate and the velocity, respectively.
Revised: Here, ( x, y ) and ( ˙ x, ˙ y ) represent the coordinate and the velocity, respectively.

Original: Here, we set m =
Revised: Here [Figure] we set m =

Original: g = 1 and sample thetrajectory of this 4 -D equation with the odeint package [4] and add the Gaussian noise N (0 , 0 . 03)to each data point.
Revised: g = 1 and sample the trajectory of this 4 -D equation with the odeint package [4] and add the Gaussian noise N (0 , 0 . 03) to each data point.

Original: We compare our HNKO with the currently mainstream methods, the HNN [7]and the EDMD [19], in terms of the trajectory prediction and the energy conservation.
Revised: We compare our HNKO with the currently mainstream methods, the HNN [7] and the EDMD [19], in terms of the trajectory prediction and the energy conservation.

Original: As shown in
Revised: As shown in

Original: Figure 1, our HNKO outperforms the other methods in terms of robustness and accuarcy.
Revised: Figure 1, our HNKO outperforms the other methods in terms of robustness and accuracy

Original: [Figure]
Revised: [Figure]

Original:  Conclusion
Revised:  Conclusion

Original: We propose a learning-based framework to approximate the Koopman operator for the Hamiltoniandynamics.
Revised: We propose a learning-based framework to approximate the Koopman operator for the Hamiltonian dynamics.

Original: We provide theoretical analysis for the dimension of manifold induced by the orthogonaltransform and using an auto-encoder to encode the original data to some low-dimensional manifoldrestricted in some high-dimensional sphere, then decode it back to the original data space.
Revised: We provide theoretical analysis for the dimension of manifold induced by the orthogonal transform and using an auto-encoder to encode the original data to some low-dimensional manifold restricted in some high-dimensional sphere, then decode it back to the original data space.

Original: We showthe proposed framework outperforms the existing methods through the numerical comparison on the
Revised: We show the proposed framework outperforms the existing methods through the numerical comparison on the

Original: Kepler problem.
Revised: Kepler problem.

Original:  A Appendix
Revised:  Acknowledgments

Revised: We thank the anonymous reviewers for their valuable and constructive comments that helped us to improve the work. Q.Z. is supported by the Shanghai Postdoctoral Excellence Program (No. 2021091) and by the STCSM (Nos. 21511100200 and 22ZR1407300). W.L. is supported by the National

Revised: Natural Science Foundation of China (No. 11925103) and by the STCSM (Nos. 19511101404, 22JC1402500, 22JC1401402, and 2021SHZDZX0103)

Original:  A.1 ProofsA.1.1 Proof of Theorem 3.1 
Revised: A Appendix A.1 Proofs A.1.1 Proof of Theorem 3.

Original: First, denote the orthogonal normal form of A as J diag ( A 1 , · · · , A s ) , and there exists transformmatrix P ∈
Revised: First, denote the orthogonal normal form of A as J diag ( A 1 , · · · , A s ) , and there exists transform matrix P ∈

Original: SO
Revised: SO

Original: ( n ) such that A = P ⊤ JP .
Revised: ( n ) such that A = P ⊤ JP .

Original: Then the transform P induce an isomorphism betweenΓ = { A j x } ∞ j =0 and ˜Γ =
Revised: Then the transform P induce an isomorphism between Γ = { A j x } ∞ j =0 and ˜Γ =

Original: { J j y } ∞ j =0 , where y = P x .
Revised: { J j y } ∞ j =0 , where y = P x .

Original: Hence, we only need to identify the degree offreedom of ˜Γ .
Revised: Hence, we only need to identify the degree of freedom of ˜Γ .

Original: Next, split the state y as y =
Revised: Next, split the state y as y =

Original: ( y ⊤ 1 , · · · , y ⊤ s )
Revised: ( y ⊤ 1 , · · · , y ⊤ s )

Original: ⊤ such that the number of variables of y s is the same as the order of A s .
Revised: ⊤ such that the number of variables of y s is the same as the order of A s .

Original: Then we have Jy =
Revised: Then we have Jy =

Original: (( A 1 y 1 ) ⊤ , · · · , ( A s y s ) ⊤ ) .
Revised: (( A 1 y 1 ) ⊤ , · · · , ( A s y s ) ⊤ ) .

Original: Since A i , i = 1 , · · · , s are also orthogonal matrix, then preserve the norm of the corresponding vector, i.e. 
Revised: Since A i , i = 1 , · · · , s are also orthogonal matrix, then preserve the norm of the corresponding vector, i.e. ∥ A i y i ∥ =

Original: [Equation]
Revised: ∥ y i ∥ , A i ∈ SO (2) , or | A i y

Revised: i

Revised: | = | y

Revised: i | , A i ∈ SO (1) .

Original: Each of the above equality constraint will minus one degree from ˜Γ .
Revised: Each of the above equality constraint will remove one degree from ˜Γ .

Original: Then these s equations willrestrict ˜Γ on some ( n − s ) -dimensional manifold, which completes the proof.
Revised: Then these s equations will restrict ˜Γ on some ( n − s ) -dimensional manifold, which completes the proof.

Original: A.2 Experimental Configurations
Revised: A.2 Experimental Configurations

Original: Numerical Implementation For parameters learning of the constructed orthogonal Koopmanoperator K , we use the GeoTorch package for constraints on the manifold [10].
Revised: Numerical Implementation For parameters learning of the constructed orthogonal Koopman operator K , we use the GeoTorch package for constraints on the manifold [10].

Original: All the necessarymodels can be trained within several minutes on a computing device with a single i7-10870 CPUwith 16GB memory.
Revised: All the necessary models can be trained within several minutes on a computing device with a single i7-10870 CPU with 16GB memory.

Original: A.2.1 Kepler Problem
Revised: A.2.1 Kepler Problem

Original: Mathematically, the kepler problem is written as
Revised: Mathematically, the kepler problem is written as

Original: [Equation]
Revised: [Equation]

Original: The corresponding physical quantities are
Revised: The corresponding physical quantities are

Original: Kinetic energy:

Original: [Equation]
Revised: [Equation]

Original: Potential energy: E p

Original: [Equation]

Original: Total energy: E = E

Original: [Equation]

Original: We uniformly sample 50 observables { x i } 49 i =

Revised: We uniformly sample 50 observables { x i } 49 i =

Original: from the initial point (1 . 0 , 0 . 0 , 0 . 0 , 0 . 9) ⊤ on timeinterval [0 , 5] , then we add Gaussian noise N (0 , σ 2 ) with σ 2 = 0 .
Revised: from the initial point (1 . 0 , 0 . 0 , 0 . 0 , 0 . 9) ⊤ on time interval [0 , 5] , then we add Gaussian noise N (0 , σ 2 ) with σ 2 = 0 .

Original: 03 to the samples.
Revised: 03 to the samples.

Original: In the learning procedure, we use a 4 × 40 × 40 × 40 × 13 fully connected neural network (FNN)with Tanh activation to encode the original 4 -D data to 13 -D sphere and use a 13 × 40 × 40 ×[-
-]
Revised: In the learning procedure, we use a 4 × 40 × 40 × 40 × 13 fully connected neural network (FNN) with Tanh activation to encode the original 4 -D data to 13 -D sphere and use a 13 × 40 × 40 ×[-
-]

Original: FNN as decoder.
Revised: FNN as decoder.

Original: The Koopman operator K is constructed as a linear layer in FNN without biasunder constraint geotorch.orthogonal .
Revised: The Koopman operator K is constructed as a linear layer in FNN without bias under constraint geotorch.orthogonal .

Original: We train the parameters simultaneously with the lossfunction (2).
Revised: We train the parameters simultaneously with the loss function (2).

Original: In the predicting procedure, we select any point x on the original continuous trajectory as initialpoint and encode it to y =
Revised: In the predicting procedure, we select any point x on the original continuous trajectory as initial point and encode it to y =

Original: ϕ θ 1 ( x ) , then we use K to generate the high-dimensional predictivetrajectory { K j y } lj =0 .
Revised: ϕ θ 1 ( x ) , then we use K to generate the high-dimensional predictive trajectory { K j y } l j =0 .

Original: Finally, we use decoder to obtain the predictive trajectory in original space{ ˆ x i = ϕ − 1 θ 2 ( K j y ) } lj =0 .
Revised: Finally, we use decoder to obtain the predictive trajectory in original space { ˆ x i = ϕ − 1 θ 2 ( K j y ) } lj =0 .

Original: Specifically, we choose x = x 0 and l = 50 to test the performance of longterm prediction of learned K .
Revised: Specifically, we choose x = x 0 and l = 50 to test the performance of long term prediction of learned K .

Original: For HNN, we use 4 × 100 × 100 × 1 FNN with Tanh activation and without the last layer biasas proposed by [7].
Revised: For HNN, we use 4 × 100 × 100 × 1 FNN with Tanh activation and without the last layer bias as proposed by [7].

Original: For EDMD, we use Hermite polynomial functions up to order 3 as dictionaryfunctions, and this dictionary corresponds to a 256 order Koopman matrix, which is far greater thanour HNKO.
Revised: For EDMD, we use Hermite polynomial functions up to order 3 as dictionary functions, and this dictionary corresponds to a 256 order Koopman matrix, which is far greater than our HNKO.

Original: The predictive procedure is the same as above.
Revised: The predictive procedure is the same as above.

Original: A.2.2 Planar Rotation
Revised: A.2.2 Planar Rotation

Original: We compare the vanilla version of our HNKO and the classic DMD method in an planar rotationmodel to demonstrate the robustness of our HNKO.
Revised: We compare the vanilla version of our HNKO and the classic DMD method in an planar rotation model to demonstrate the robustness of our HNKO.

Original: First, we define a rotation matrix as A =
Revised: First, we define a rotation matrix as A =

Original: ((cos( a ) , sin( a )) ⊤ , ( − sin( a ) , cos( a )) ⊤ ) .
Revised: ((cos( a ) , sin( a )) ⊤ , ( − sin( a ) , cos( a )) ⊤ ) .

Original: Then we obtainm observables as { A j x } mj =0 .
Revised: Then we obtain m observables as { A j x } mj =0 .

Original: We add Gaussian noise N (0 , σ 2 ) to this dataset.
Revised: We add Gaussian noise N (0 , σ 2 ) to this dataset.

Original: Now we test theperformance of HNKO and DMD.
Revised: Now we test the performance of HNKO and DMD.

Original: We select a = 0 .
Revised: We select a = 0 .

Original: 2 , m = 99 , σ 2 = 0 . 0 , 0 . 1 .
Revised: 2 , m = 99 , σ 2 = 0 . 0 , 0 . 1 .

Original: Notice that A itselfis the unique solution of the following optimal problem
Revised: Notice that A itself is the unique solution of the following optimal problem

Original: [Equation]
Revised: [Equation]

Original: Hence DMD can exactly learn the dynamic A in noise-free case.
Revised: Hence DMD can exactly learn the dynamic A in noise-free case.

Original: However, DMD solution K ∗ =
Revised: However, DMD solution K ∗ =

Original: X ′ X
Revised: X ′ X

Original: + generally cannot equal to A in the noisy case because A cannot minimize the objectivefunction of the following oprimal problem
Revised: + generally cannot equal to A in the noisy case because A cannot minimize the objective function of the following oprimal problem

Original: [Equation]
Revised: [Equation]

Original: We demonstrate the above analysis in Figure 2.
Revised: We demonstrate the above analysis in Figure 2.

Original: To guarantee the predictive stability of HNKO, we modify L koop as
Revised: To guarantee the predictive stability of HNKO, we modify L koop as

Original: [Equation]
Revised: [Equation]

Original: We can see that HNKO learn the exact transform A in both noisy and noise-free case, while DMDfail to learn the A in the noisy case due to overfit on the noisy data.
Revised: We can see that HNKO learn the exact transform A in both noisy and noise-free case, while DMD fail to learn the A in the noisy case due to overfit on the noisy data.

Original: [Figure]
Revised: [Figure]

Revised: [Figure]