Koopman Operators and Dynamic Mode Decomposition Shubhendu Trivedi - PowerPoint PPT Presentation

Extended Koopman Operator Can extend the Koopman operator to this larger space Denote F = ( f 1 , . . . , f K ) T ∈ F K Then U K : F K → F K   [ Uf 1 ]( p ) . . [ U K F ]( p ) :=   .   [ Uf K ]( p ) K � Then U K = U 1 F K is the space of C K -valued observables on the state space M More generally: F : M → V where V is a vector space Shubhendu Trivedi (TTI-C) Koopman Operators 8 / 50

Koopman Operators in Continuous Time D.S. Consider the continuous time dynamical system p = T ( p ) ˙ Shubhendu Trivedi (TTI-C) Koopman Operators 9 / 50

Example: Cyclic Group Shubhendu Trivedi (TTI-C) Koopman Operators 10 / 50

Setup Reminder: Group that can be obtained by a single generator Shubhendu Trivedi (TTI-C) Koopman Operators 11 / 50

Setup Reminder: Group that can be obtained by a single generator Let M = { e, a, a 2 } be a cyclic group of order 3 Shubhendu Trivedi (TTI-C) Koopman Operators 11 / 50

Setup Reminder: Group that can be obtained by a single generator Let M = { e, a, a 2 } be a cyclic group of order 3 Define T : M → M as T(p) = a · p Shubhendu Trivedi (TTI-C) Koopman Operators 11 / 50

Setup Reminder: Group that can be obtained by a single generator Let M = { e, a, a 2 } be a cyclic group of order 3 Define T : M → M as T(p) = a · p Entire state space is a periodic orbit with period 3 Shubhendu Trivedi (TTI-C) Koopman Operators 11 / 50

Setup Reminder: Group that can be obtained by a single generator Let M = { e, a, a 2 } be a cyclic group of order 3 Define T : M → M as T(p) = a · p Entire state space is a periodic orbit with period 3 Let F be C -valued functions on M Shubhendu Trivedi (TTI-C) Koopman Operators 11 / 50

Setup Reminder: Group that can be obtained by a single generator Let M = { e, a, a 2 } be a cyclic group of order 3 Define T : M → M as T(p) = a · p Entire state space is a periodic orbit with period 3 Let F be C -valued functions on M Space of observables is C 3 Shubhendu Trivedi (TTI-C) Koopman Operators 11 / 50

Setup Let f 1 , f 2 , f 3 be indicator functions on e, a, a 2 : � 1 if p = e f 1 ( p ) = 0 if p � = e � 1 if p = a f 2 ( p ) = 0 if p � = a � if p = a 2 1 f 3 ( p ) = if p � = a 2 0 Shubhendu Trivedi (TTI-C) Koopman Operators 12 / 50

Setup Let f 1 , f 2 , f 3 be indicator functions on e, a, a 2 : � 1 if p = e f 1 ( p ) = 0 if p � = e � 1 if p = a f 2 ( p ) = 0 if p � = a � if p = a 2 1 f 3 ( p ) = if p � = a 2 0 Form a basis for F Shubhendu Trivedi (TTI-C) Koopman Operators 12 / 50

Example: Cyclic Group Action of the Koopman operator on this basis: [ Uf 1 ]( p ) = f 1 ( a · p ) = f 3 ( p ) [ Uf 2 ]( p ) = f 2 ( a · p ) = f 1 ( p ) [ Uf 3 ]( p ) = f 3 ( a · p ) = f 2 ( p ) Shubhendu Trivedi (TTI-C) Koopman Operators 13 / 50

Example: Cyclic Group Action of the Koopman operator on this basis: [ Uf 1 ]( p ) = f 1 ( a · p ) = f 3 ( p ) [ Uf 2 ]( p ) = f 2 ( a · p ) = f 1 ( p ) [ Uf 3 ]( p ) = f 3 ( a · p ) = f 2 ( p ) Consider arbitrary observable f ∈ F i.e. f = c 1 f 1 + c 2 f 2 + c 3 f 3 Shubhendu Trivedi (TTI-C) Koopman Operators 13 / 50

Example: Cyclic Group Action of the Koopman operator on this basis: [ Uf 1 ]( p ) = f 1 ( a · p ) = f 3 ( p ) [ Uf 2 ]( p ) = f 2 ( a · p ) = f 1 ( p ) [ Uf 3 ]( p ) = f 3 ( a · p ) = f 2 ( p ) Consider arbitrary observable f ∈ F i.e. f = c 1 f 1 + c 2 f 2 + c 3 f 3 Consider the action of the Koopman operator on f : Uf = U ( c 1 f 1 + c 2 f 2 + c 3 f 3 ) = c 1 f 3 + c 2 f 1 + c 3 f 2 Shubhendu Trivedi (TTI-C) Koopman Operators 13 / 50

Example: Cyclic Group Matrix representation of the Koopman operator U in the { f 1 , f 2 , f 3 } basis:  c 1   0 1 0   c 1   = U c 2 0 0 1 c 2      c 3 1 0 0 c 3 Shubhendu Trivedi (TTI-C) Koopman Operators 14 / 50

Example: Linear Diagonalizable Systems Shubhendu Trivedi (TTI-C) Koopman Operators 15 / 50

Setup Let M = R d , and define T : M → M as : ( T ( x )) i = µ i x i Shubhendu Trivedi (TTI-C) Koopman Operators 16 / 50

Setup Let M = R d , and define T : M → M as : ( T ( x )) i = µ i x i x = ( x 1 , . . . , x d ) T ∈ M and µ i ∈ R Shubhendu Trivedi (TTI-C) Koopman Operators 16 / 50

Setup Let M = R d , and define T : M → M as : ( T ( x )) i = µ i x i x = ( x 1 , . . . , x d ) T ∈ M and µ i ∈ R Let F denote space of functions R d → C Shubhendu Trivedi (TTI-C) Koopman Operators 16 / 50

Setup Let M = R d , and define T : M → M as : ( T ( x )) i = µ i x i x = ( x 1 , . . . , x d ) T ∈ M and µ i ∈ R Let F denote space of functions R d → C Let { b 1 . . . , b d } ⊂ M be a basis for M ; define f i ( x ) = � b i , x � Shubhendu Trivedi (TTI-C) Koopman Operators 16 / 50

Example: Linear Diagonalizable Systems The action of the Koopman operator U : F → F on f i is   µ 1 x 1 . � � . [ Uf i ]( x ) = � b i , T ( x ) � = b i, 1 . . . b i,d   .   µ d x d Shubhendu Trivedi (TTI-C) Koopman Operators 17 / 50

Example: Linear Diagonalizable Systems The action of the Koopman operator U : F → F on f i is   µ 1 x 1 . � � . [ Uf i ]( x ) = � b i , T ( x ) � = b i, 1 . . . b i,d   .   µ d x d   µ 1 0 . . . 0   x 1 0 µ 2 . . . 0   . � � . [ Uf i ]( x ) = b i, 1 . . . b i,d  . . .    ... . . . .     . . .   x d 0 0 . . . µ d Shubhendu Trivedi (TTI-C) Koopman Operators 17 / 50

Example: Linear Diagonalizable Systems d Recall F d = � F , define U d as earlier, then for F = ( f 1 , . . . , f d ) T 1 Shubhendu Trivedi (TTI-C) Koopman Operators 18 / 50

Example: Linear Diagonalizable Systems d Recall F d = � F , define U d as earlier, then for F = ( f 1 , . . . , f d ) T 1 Then the action of the extended Koopman operator Shubhendu Trivedi (TTI-C) Koopman Operators 18 / 50

Example: Linear Diagonalizable Systems d Recall F d = � F , define U d as earlier, then for F = ( f 1 , . . . , f d ) T 1 Then the action of the extended Koopman operator   µ 1 0 . . . 0     b 1 , 1 . . . b 1 ,d x 1 0 µ 2 . . . 0 . . . ...   . . . [ U d F ]( x ) =    . . .    . . ... . . . .       . . .   b d, 1 . . . b d,d x d 0 0 . . . µ d Shubhendu Trivedi (TTI-C) Koopman Operators 18 / 50

Example: Linear Diagonalizable Systems d Recall F d = � F , define U d as earlier, then for F = ( f 1 , . . . , f d ) T 1 Then the action of the extended Koopman operator   µ 1 0 . . . 0     b 1 , 1 . . . b 1 ,d x 1 0 µ 2 . . . 0 . . . ...   . . . [ U d F ]( x ) =    . . .    . . ... . . . .       . . .   b d, 1 . . . b d,d x d 0 0 . . . µ d This is the action of the Koopman operator on the particular observable F , not the entire observable space F Shubhendu Trivedi (TTI-C) Koopman Operators 18 / 50

Example: Heat equation with periodic boundary conditions Shubhendu Trivedi (TTI-C) Koopman Operators 19 / 50

Mode Analysis Shubhendu Trivedi (TTI-C) Koopman Operators 20 / 50

Eigenfunctions and Koopman Modes We have put no structure on F so far Shubhendu Trivedi (TTI-C) Koopman Operators 21 / 50

Eigenfunctions and Koopman Modes We have put no structure on F so far When F is a vector space, the Koopman operator is linear Shubhendu Trivedi (TTI-C) Koopman Operators 21 / 50

Eigenfunctions and Koopman Modes We have put no structure on F so far When F is a vector space, the Koopman operator is linear Interest: Study spectral properties of the Koopman Operator to probe into the dynamics of the system Shubhendu Trivedi (TTI-C) Koopman Operators 21 / 50

Eigenfunctions and Koopman Modes We have put no structure on F so far When F is a vector space, the Koopman operator is linear Interest: Study spectral properties of the Koopman Operator to probe into the dynamics of the system Assume: F is a Banach space Shubhendu Trivedi (TTI-C) Koopman Operators 21 / 50

Eigenfunctions and Koopman Modes We have put no structure on F so far When F is a vector space, the Koopman operator is linear Interest: Study spectral properties of the Koopman Operator to probe into the dynamics of the system Assume: F is a Banach space Assume: U is a bounded, continuous operator on F Shubhendu Trivedi (TTI-C) Koopman Operators 21 / 50

Eigenfunctions and Koopman Modes Let { φ 1 , . . . , φ n } be a set of eigenfunctions of U , where n = 1 , 2 , .., ∞ Shubhendu Trivedi (TTI-C) Koopman Operators 22 / 50

Eigenfunctions and Koopman Modes Let { φ 1 , . . . , φ n } be a set of eigenfunctions of U , where n = 1 , 2 , .., ∞ For the discrete case: Shubhendu Trivedi (TTI-C) Koopman Operators 22 / 50

Eigenfunctions and Koopman Modes Let { φ 1 , . . . , φ n } be a set of eigenfunctions of U , where n = 1 , 2 , .., ∞ For the discrete case: [ Uφ i ]( p ) = λ i φ i ( p ) Shubhendu Trivedi (TTI-C) Koopman Operators 22 / 50

Eigenfunctions and Koopman Modes Let { φ 1 , . . . , φ n } be a set of eigenfunctions of U , where n = 1 , 2 , .., ∞ For the discrete case: [ Uφ i ]( p ) = λ i φ i ( p ) For the continuous case: Shubhendu Trivedi (TTI-C) Koopman Operators 22 / 50

Eigenfunctions and Koopman Modes Let { φ 1 , . . . , φ n } be a set of eigenfunctions of U , where n = 1 , 2 , .., ∞ For the discrete case: [ Uφ i ]( p ) = λ i φ i ( p ) For the continuous case: [ U t φ i ]( p ) = e λ i t φ i ( p ) Shubhendu Trivedi (TTI-C) Koopman Operators 22 / 50

Eigenfunctions and Koopman Modes Let { φ 1 , . . . , φ n } be a set of eigenfunctions of U , where n = 1 , 2 , .., ∞ For the discrete case: [ Uφ i ]( p ) = λ i φ i ( p ) For the continuous case: [ U t φ i ]( p ) = e λ i t φ i ( p ) λ ’s are the eigenvalues of the generator U , and { e λ i } of the Koopman semi-group Shubhendu Trivedi (TTI-C) Koopman Operators 22 / 50

Algebraic Structure of Eigenfunctions Assume that F is a subset of all C valued functions on M Shubhendu Trivedi (TTI-C) Koopman Operators 23 / 50

Algebraic Structure of Eigenfunctions Assume that F is a subset of all C valued functions on M Also assume that it forms a vector space that is closed under pointwise products of functions Shubhendu Trivedi (TTI-C) Koopman Operators 23 / 50

Algebraic Structure of Eigenfunctions Assume that F is a subset of all C valued functions on M Also assume that it forms a vector space that is closed under pointwise products of functions = ⇒ set of eigenfunctions forms an abelian semigroup under pointwise products of functions Shubhendu Trivedi (TTI-C) Koopman Operators 23 / 50

Algebraic Structure of Eigenfunctions Assume that F is a subset of all C valued functions on M Also assume that it forms a vector space that is closed under pointwise products of functions = ⇒ set of eigenfunctions forms an abelian semigroup under pointwise products of functions Concretely: If φ 1 , φ 2 ∈ F are eigenfunctions of U with eigenvalues λ 1 and λ 2 , then φ 1 φ 2 is an eigenfunction of U with eignevalue λ 1 λ 2 Shubhendu Trivedi (TTI-C) Koopman Operators 23 / 50

Algebraic Structure of Eigenfunctions If p > 0 and φ is an eigenfunction with eigenvalue λ , then φ p is an eigenfunction with eigenvalue λ p Shubhendu Trivedi (TTI-C) Koopman Operators 24 / 50

Algebraic Structure of Eigenfunctions If p > 0 and φ is an eigenfunction with eigenvalue λ , then φ p is an eigenfunction with eigenvalue λ p If φ is an eigenfunction that vanishes nowhere and r ∈ R , then φ r is an eigenfunction with eigenvalue λ r Shubhendu Trivedi (TTI-C) Koopman Operators 24 / 50

Algebraic Structure of Eigenfunctions If p > 0 and φ is an eigenfunction with eigenvalue λ , then φ p is an eigenfunction with eigenvalue λ p If φ is an eigenfunction that vanishes nowhere and r ∈ R , then φ r is an eigenfunction with eigenvalue λ r Eigenfunctions that vanish nowhere form an Abelian group Shubhendu Trivedi (TTI-C) Koopman Operators 24 / 50

Spectral Equivalence of Topologically Conjugate Systems Proposition Let S : M → M and T : N → N be topologically conjugate; i.e. ∃ a homomorphism h : N → M such that S ◦ h = h ◦ T . If φ is an eigenfunction of U S with eigenvalue λ , then φ ◦ h is an eigenfunction of U T at eigenvalue λ Shubhendu Trivedi (TTI-C) Koopman Operators 25 / 50

Example: Linear Diagonalizable Systems Let y ( k ) = ( y ( k ) 2 ) T ( ( k ) indexes time) 1 , y ( k ) Shubhendu Trivedi (TTI-C) Koopman Operators 26 / 50

Example: Linear Diagonalizable Systems Let y ( k ) = ( y ( k ) 2 ) T ( ( k ) indexes time) 1 , y ( k ) Let y ( k +1) = T y ( k ) Shubhendu Trivedi (TTI-C) Koopman Operators 26 / 50

Example: Linear Diagonalizable Systems Let y ( k ) = ( y ( k ) 2 ) T ( ( k ) indexes time) 1 , y ( k ) Let y ( k +1) = T y ( k ) T is a matrix with eigenvectors v 1 , v 2 at eigenvalues λ 1 , λ 2 with v i � = e j Shubhendu Trivedi (TTI-C) Koopman Operators 26 / 50

Example: Linear Diagonalizable Systems Let y ( k ) = ( y ( k ) 2 ) T ( ( k ) indexes time) 1 , y ( k ) Let y ( k +1) = T y ( k ) T is a matrix with eigenvectors v 1 , v 2 at eigenvalues λ 1 , λ 2 with v i � = e j If V = [ v 1 v 2 ] , then with new coordinates x ( k ) = ( x k 2 ) T = V − 1 y ( k ) 1 , x ( k ) Shubhendu Trivedi (TTI-C) Koopman Operators 26 / 50

Example: Linear Diagonalizable Systems Let y ( k ) = ( y ( k ) 2 ) T ( ( k ) indexes time) 1 , y ( k ) Let y ( k +1) = T y ( k ) T is a matrix with eigenvectors v 1 , v 2 at eigenvalues λ 1 , λ 2 with v i � = e j If V = [ v 1 v 2 ] , then with new coordinates x ( k ) = ( x k 2 ) T = V − 1 y ( k ) 1 , x ( k ) � x ( k +1) � � � x ( k ) � � x ( k ) � � λ 1 0 1 1 1 = := Λ x ( k +1) x ( k ) x ( k ) 0 λ 2 2 2 2 Shubhendu Trivedi (TTI-C) Koopman Operators 26 / 50

Example: Linear Diagonalizable Systems Let y ( k ) = ( y ( k ) 2 ) T ( ( k ) indexes time) 1 , y ( k ) Let y ( k +1) = T y ( k ) T is a matrix with eigenvectors v 1 , v 2 at eigenvalues λ 1 , λ 2 with v i � = e j If V = [ v 1 v 2 ] , then with new coordinates x ( k ) = ( x k 2 ) T = V − 1 y ( k ) 1 , x ( k ) � x ( k +1) � � � x ( k ) � � x ( k ) � � λ 1 0 1 1 1 = := Λ x ( k +1) x ( k ) x ( k ) 0 λ 2 2 2 2 Maps Λ and T are topologically conjugate by Λ V − 1 = V − 1 T Shubhendu Trivedi (TTI-C) Koopman Operators 26 / 50

Example: Linear Diagonalizable Systems Let y ( k ) = ( y ( k ) 2 ) T ( ( k ) indexes time) 1 , y ( k ) Let y ( k +1) = T y ( k ) T is a matrix with eigenvectors v 1 , v 2 at eigenvalues λ 1 , λ 2 with v i � = e j If V = [ v 1 v 2 ] , then with new coordinates x ( k ) = ( x k 2 ) T = V − 1 y ( k ) 1 , x ( k ) � x ( k +1) � � � x ( k ) � � x ( k ) � � λ 1 0 1 1 1 = := Λ x ( k +1) x ( k ) x ( k ) 0 λ 2 2 2 2 Maps Λ and T are topologically conjugate by Λ V − 1 = V − 1 T V − 1 is now the h from the proposition Shubhendu Trivedi (TTI-C) Koopman Operators 26 / 50

Koopman Modes Assume f ∈ F is an observable in the linear span of a set of eigenfunctions { φ i } n 1 , then for c i ( f ) ∈ C : Shubhendu Trivedi (TTI-C) Koopman Operators 27 / 50

Koopman Modes Assume f ∈ F is an observable in the linear span of a set of eigenfunctions { φ i } n 1 , then for c i ( f ) ∈ C : n � f ( p ) = c i ( f ) φ i ( p ) i =1 Shubhendu Trivedi (TTI-C) Koopman Operators 27 / 50

Koopman Modes Assume f ∈ F is an observable in the linear span of a set of eigenfunctions { φ i } n 1 , then for c i ( f ) ∈ C : n � f ( p ) = c i ( f ) φ i ( p ) i =1 Dynamics of f have a simple form: n n � � [ Uf ]( p ) = f ( T ( p )) = c i ( f ) φ i ( T ( p )) = c i ( f )[ Uφ i ]( p ) i =1 i =1 Shubhendu Trivedi (TTI-C) Koopman Operators 27 / 50

Koopman Modes Assume f ∈ F is an observable in the linear span of a set of eigenfunctions { φ i } n 1 , then for c i ( f ) ∈ C : n � f ( p ) = c i ( f ) φ i ( p ) i =1 Dynamics of f have a simple form: n n � � [ Uf ]( p ) = f ( T ( p )) = c i ( f ) φ i ( T ( p )) = c i ( f )[ Uφ i ]( p ) i =1 i =1 n � [ Uf ]( p ) = f ( T ( p )) = λ i c i ( f ) φ i ( p ) i =1 Shubhendu Trivedi (TTI-C) Koopman Operators 27 / 50

Koopman Modes Dynamics of f have a simple form: n n � � [ Uf ]( p ) = f ( T ( p )) = c i ( f ) φ i ( T ( p )) = c i ( f )[ Uφ i ]( p ) i =1 i =1 Shubhendu Trivedi (TTI-C) Koopman Operators 28 / 50

Koopman Modes Dynamics of f have a simple form: n n � � [ Uf ]( p ) = f ( T ( p )) = c i ( f ) φ i ( T ( p )) = c i ( f )[ Uφ i ]( p ) i =1 i =1 n � [ Uf ]( p ) = f ( T ( p )) = λ i c i ( f ) φ i ( p ) i =1 Shubhendu Trivedi (TTI-C) Koopman Operators 28 / 50

Koopman Modes Dynamics of f have a simple form: n n � � [ Uf ]( p ) = f ( T ( p )) = c i ( f ) φ i ( T ( p )) = c i ( f )[ Uφ i ]( p ) i =1 i =1 n � [ Uf ]( p ) = f ( T ( p )) = λ i c i ( f ) φ i ( p ) i =1 Likewise n [ U m f ]( p ) = � λ m i c i ( f ) φ i ( p ) i =1 Shubhendu Trivedi (TTI-C) Koopman Operators 28 / 50

Koopman Modes Extension to vector valued observables F = ( f 1 , . . . , f K ) T , with each f i in the closed linear span of eigenfunctions: Shubhendu Trivedi (TTI-C) Koopman Operators 29 / 50

Koopman Modes Extension to vector valued observables F = ( f 1 , . . . , f K ) T , with each f i in the closed linear span of eigenfunctions:   c i ( f 1 ) n . � [ U k F ]( p ) = λ m . i φ i ( p )   .   i =1 c i ( f K ) Shubhendu Trivedi (TTI-C) Koopman Operators 29 / 50

Koopman Modes Extension to vector valued observables F = ( f 1 , . . . , f K ) T , with each f i in the closed linear span of eigenfunctions:   c i ( f 1 ) n . � [ U k F ]( p ) = λ m . i φ i ( p )   .   i =1 c i ( f K ) Written compactly: Shubhendu Trivedi (TTI-C) Koopman Operators 29 / 50