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# Hamiltonian flows on graphs

Hamiltonian flows play vital roles in dynamical systems. Famous examples include the Schrodinger equation, Schrodinger bridge problem and Mean field games. In this talk, we introduce these Hamiltonian flows on finite graphs. Our approach is based on the optimal transport metric in probability simplex over finite graphs, named probability manifold. We derive these Hamiltonian flows in probability manifold. The connection between Shannon-Boltzmann entropy and Fisher information will be established in these Hamiltonian flows. Several examples are provided.

January 11, 2019

## Transcript

1. ### Hamiltonian ﬂows on graphs Wuchen Li, UCLA 2019 Joint work

with Shui-Nee Chow (GT) and Haomin Zhou (GT).
2. ### Introduction: Hamiltonian ﬂow Consider a second order ODE ¨ x

= −∇V (x). Denote the momentum p = ˙ x, then (x, p) satisﬁes a Hamiltonian system ˙ x = p ˙ p = −∇V (x), which conserves the Hamiltonian H(x, p) = p2 2 + V (x). 2
3. ### Introduction: Law of Hamiltonian ﬂows The ﬁnite dimensional Hamiltonian ﬂow

connects to a pair of PDEs ∂t ρ(t, x) + ∇x · (ρ(t, x)∇x S(t, x)) = 0 ∂t S(t, x) + 1 2 |∇x S(t, x)|2 + V (x) = 0, which conserves the total Hamiltonian H(ρ, S) = Rd 1 2 |∇S(x)|2ρ(x) + V (x)ρ(x)dx. In this talk, we will build dynamical system viewpoint of Hamiltonian ﬂows via optimal transport: The law of Hamiltonian ﬂow is the Hamiltonian ﬂow in probability space. 3
4. ### Hamiltonian system+Optimal transport Related to Mean ﬁeld games (Larsy, Lions,

Gangbo); Related to weak KAM theory (Evans); Related to 2-Wasserstein metric (Brenier, Villani, Ambrosio); Related to Schr¨ odinger equations (Nelson, Carlen, Laﬀerty); Related to Schr¨ odinger Bridge problem (Carlen, Yause, Leonard). 4
5. ### History Remark Brownian motion (1905) Schrodinger equation (1926) Schrodinger bridge

(1931) Nelson process (1966) Optimal transport+ Hamiltonian system (Recently) 5
6. ### Optimal transport What is the optimal way to move or

transport the mountain with shape X, density ρ0(x) to another shape Y with density ρ1(y)? The problem was ﬁrst introduced by Monge in 1781 and relaxed by Kantorovich by 1940. It introduces a metric function on probability set, named optimal transport distance, Wasserstein metric or Earth Mover’s distance. 6
7. ### Overview The optimal transport has many diﬀerent formulations under various

angles: Mapping/Monge-Amp´ ere equation; Linear programming; Geometry/Fluid dynamics; which are considered by Otto, Kinderlehrer, Villani, McCann, Carlen, Lott, Strum, Gangbo, Jordan, Evans, Brenier, Benamou, Ambrosio, Gigli, Savare and many more. In this talk, we mainly follow its symplectic geometry formulation in a discrete setting, and build the Hamiltonian ﬂows for modeling and numerical computations. 7
8. ### Density manifold Rewrite x − y 2 = inf γ(t)

1 0 v(t)2dt : ˙ γ(t) = v(t), γ(0) = x, γ(1) = y . The distance has an optimal control formulation (Benamou-Brenier 2000). Let x = X0 (x), y = X1 (x), then inf v 1 0 Ex∼ρ0 v(t, Xt (x))2 dt, where E is the expectation operator and the inﬁmum runs over all vector ﬁelds vt , such that ˙ Xt (x) = v(t, Xt (x)), X0 ∼ ρ0, X1 ∼ ρ1. Under this metric, the probability set has a Riemannian geometry structure1. 1John D. Laﬀerty: the density manifold and conﬁguration space quantization, 1988. 8

10. ### Brownian motion and Entropy Production The gradient ﬂow of entropy

H(ρ) = Rd ρ(x)log ρ(x)dx, w.r.t. optimal transport metric distance is: ∂ρ ∂t = ∇ · (ρ∇log ρ) = ∆ρ. Entropy dissipation: − d dt H(ρ) = Rd (∇ log ρ)2ρdx = I(ρ). 10
11. ### Goal: Hamiltonian ﬂows on graphs Question: Can we consider the

Hamiltonian ﬂows, e.g. Schr¨ odinger equation or bridge problem, on ﬁnite graphs? Answer: Yes, we need to build a discrete optimal transport metric on probability simplex over ﬁnite states. Using this metric, we build the associated Hamiltonian ﬂows in probability simplex. Recent Developments on Hamiltonian ﬂows and Optimal transport: Chow, Li, Zhou, Gangbo, Leger, Mou... 11
12. ### Basic setting Graph with ﬁnite vertices G = (V, E,

ω), V = {1, · · · , n}, E is the edge set, ω is the weight set; Probability set P(G) = (ρi )n i=1 | n i=1 ρi = 1, ρi ≥ 0 . Example: Consider a discrete space {1, 2, 3} with the graph structure: The probability simplex set forms 12
13. ### Deﬁnition I We plan to ﬁnd the discrete analog of

density manifold (Maas, Mielke, Chow). First, it is natural to deﬁne a vector ﬁeld on a graph v = (vij )(i,j)∈E , satisfying vij = −vji . Given a potential S = (Si )n i=1 , a gradient vector ﬁeld refers to ∇G Sij = √ ωij (Si − Sj ), where ωij = ωji is the weight function on an edge. 13
14. ### Deﬁnition II We next deﬁne an inner product of two

vector ﬁelds v1, v2: (v1, v2)ρ := 1 2 (i,j)∈E v1 ij v2 ij θij (ρ); and a divergence of a vector ﬁeld v at ρ ∈ P(G): divG (ρv) := − j∈N(i) √ ωij vij θij (ρ) n i=1 . Here θ represents the probability weight on the edge θij (ρ) = 1 2 ( ωij i ∈N(i) ωii ρi + ωij j ∈N(j) ωjj ρj ). θ has some other choices. 14
15. ### Optimal transport distance on a graph The metric for any

ρ0, ρ1 ∈ Po (G) is W(ρ0, ρ1)2 := inf v { 1 0 (v, v)ρ dt : dρ dt + div(ρv) = 0, ρ(0) = ρ0, ρ(1) = ρ1}. Here W is the proposed Wasserstein metric on graph. Later on, we will show it introduces a metric tensor structure in density manifold. 15
16. ### Hodge decomposition Continuous state v(x) = ∇S(x) + u(x), where

v(x) is a given vector ﬁeld, ∇S is the gradient vector ﬁeld and u is the divergence free with respect to a density ρ, i.e. div(ρu) = 0. Graph v = ∇G S + u where v = (vij ) is a discrete vector ﬁeld, ∇G S = (Si − Sj )ωij is the discrete gradient vector ﬁeld, and the divergence free on a graph means divG (ρu) = 0. 16
17. ### Variational formulation Lemma The discrete Wasserstein metric is equivalent to

W(ρ0, ρ1)2 = inf ∇GS 1 0 (∇G S, ∇G S)ρ dt, where the inﬁmum is taken among all discrete potential vector ﬁelds ∇G S, such that dρ dt + divG (ρ∇G S) = 0, ρ(0) = ρ0, ρ(1) = ρ1. This metric gives Po (G) Riemannian geometry structure. 17
18. ### Discrete probability manifold Denote −divG (ρ∇G S) = L(ρ)S. Then

the metric in Po (G) is equivalent to W(ρ0, ρ1)2 = inf{ 1 0 ˙ ρT L(ρ)−1 ˙ ρ dt : ρ(0) = ρ0, ρ(1) = ρ1}. 18
19. ### Wasserstein metric tensor Here L(ρ) ∈ Rn×n is the linear

weighted Laplacian matrix function L(ρ) = −divG (ρ∇G ) = −DT Θ(ρ)D, where D ∈ R|E|×|V | is a discrete gradient matrix with Dij,k =      1 i = k, j ∈ N(i) −1 j = k, j ∈ N(i) 0 otherwise, DT ∈ R|V |×|E| is a discrete divergence matrix, and Θ ∈ R|E|×|E| is a diagonal weight matrix Θ(i,j)∈E,(k,l)∈E = ρi+ρj 2 if (i, j) = (k, l) ∈ E; 0 otherwise. 19
20. ### Probability manifold: Geodesics Consider the Lagrangian L(ρ, ˙ ρ) =

1 2 ˙ ρT L(ρ)−1 ˙ ρ. The geodesic satisﬁes the Euler-Lagrangian equation d dt ∇ ˙ ρ L(ρ, ˙ ρ) = ∇ρ L(ρ, ˙ ρ), which can be written into the following second order ODE2 ¨ ρ + ΓW ρ ( ˙ ρ, ˙ ρ) = 0, Here ΓW ρ is the Christoﬀel symbol in probability manifold. In other words, we write the Euler-Lagrangian equation into a second order ODE. 2Wuchen Li, Geometry of probability simplex via optimal transport, 2018. 20
21. ### Christoﬀel symbol The Christoﬀel symbol is the coeﬃcient of the

quadratic form ΓW ρ ( ˙ ρ, ˙ ρ) = −L( ˙ ρ)L(ρ)−1 ˙ ρ + 1 2 L(ρ) ∇G L(ρ)−1 ˙ ρ, ∇G L(ρ)−1 ˙ ρ . It is also new in the continuous domain: ΓW ρ ( ˙ ρ, ˙ ρ) = −∆ ˙ ρ ∆−1 ρ ˙ ρ − 1 2 ∆ρ ∇∆−1 ρ ˙ ρ, ∇∆−1 ρ ˙ ρ . Thus we propose to study the following second order ODE: ¨ ρ − ∆ ˙ ρ ∆−1 ρ ˙ ρ − 1 2 ∆ρ ∇∆−1 ρ ˙ ρ, ∇∆−1 ρ ˙ ρ = 0, where ∆ρ = ∇ · (ρ∇). Interestingly, it can be written into the ﬁrst order ODE system later on. 21
22. ### Hamiltonian formulation By the Legendre transform, i.e. H(ρ, S) =

sup ˙ ρ ˙ ρT S − L(ρ, ˙ ρ), then the geodesics satisﬁes the Hamiltonian system d dt ρ S = 0 I −I 0 ∂ ∂ρ H ∂ ∂S H , where H(ρ, S) = 1 2 ST L(ρ)S. 22
23. ### Hamiltonian system on probability manifold We write the above Hamiltonian

system explicitly. Denote H(ρ, S) = 1 2 (i,j)∈E ωij (Si − Sj )2θij (ρ), then the geodesics satisﬁes            ˙ ρ + j∈N(i) (Si − Sj )ωij θij (ρ) = 0 ˙ S + 1 2 j∈N(i) (Si − Sj )2ωij ∂θij (ρ) ∂ρi = 0. They are discrete analog of H(ρ, S) = 1 2 (∇S(x), ∇S(x))ρ(x)dx, with continuity equation and Hamilton-Jacobi equation    ∂t ρ(t, x) + ∇ · (ρ(t, x)∇S(t, x)) = 0 ∂t S(t, x) + 1 2 (∇S(t, x))2 = 0. 23
24. ### General Hamiltonian ﬂow We mainly consider the second order ODE

in probability manifold: ¨ ρ + ΓW ρ ( ˙ ρ, ˙ ρ) = −gradW F(ρ). In Hamiltonian formalism, it represents          ˙ ρ + j∈N(i) (Si − Sj )ωij θij (ρ) = 0 ˙ S + 1 2 j∈N(i) (Si − Sj )2ωij ∂θij ∂ρi = ∇ρ F(ρ). Two important examples of Hamiltonian ﬂows include: Schr¨ odinger equation; Schr¨ odinger bridge problem. 24
25. ### Entropy production on graphs An important energy in Hamiltonian ﬂow

also evolves in the gradient ﬂow, known as the entropy production. The gradient ﬂow of the Shannon entropy S(ρ) = n i=1 ρi log ρi in (P(G), W) is the diﬀusion process on a graph: dρ dt = −gradW S(ρ) = divG (ρ∇G log ρ). The dissipation of entropy deﬁnes the Fisher information on a graph: I(ρ) = (gradW S(ρ), gradW S(ρ))ρ = 1 2 (i,j)∈E (log ρi − log ρj )2θij (ρ). Many interesting topics have been extracted from this observation. E.g. entropy dissipation, Log-Sobolev inequalities, Ricci curvature, Yano formula (Annals of Mathematics, 1952, 7 pages). 25
26. ### Example 1: Schr¨ odinger equations Consider ¨ ρ + ΓW

ρ ( ˙ ρ, ˙ ρ) = −gradW F(ρ) with F(ρ) = −I(ρ). Then its Hamiltonian formulation (ρ, S) recovers the following quantum hydrodynamics on graphs            ˙ ρ + j∈N(i) (Si − Sj )ωij θij (ρ) = 0 ˙ S + 1 2 j∈N(i) (Si − Sj )2ωij ∂θij (ρ) ∂ρi = −∂ρi I(ρ) 26
27. ### Hamiltonian structure Laplacian operator Interestingly, the Madelung transform3 on graphs

Ψ = √ ρe √ −1S, introduces the following Hamilton-structure-keeping Laplacian operator: √ −1 dΨi dt = 1 2 Ψi { j∈N(i) (log Ψi − log Ψj ) θij |Ψi |2 + j∈N(i) | log Ψi − log Ψj |2 ∂θij ∂|Ψi |2 }. It is a consistent scheme for Schr¨ odinger equation: √ −1∂t Ψt = 1 2 ∆Ψt . 3Nelson, Quantum diﬀusion, 1985. 27

29. ### Example: Ground state Compute the ground state via min ρ∈P(G)

h2 8 I(ρ) + V(ρ) + W(ρ). −1 −0.5 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x ρ h=1 h=0.1 h=0.01 Figure: The plot of ground state’s density function. The blue, black, red curves represents h = 1, 0.1, 0.01, respectively. 29
30. ### Example 2: Schr¨ odinger bridge problem Consider ¨ ρ +

ΓW ρ ( ˙ ρ, ˙ ρ) = −gradW F(ρ) with F(ρ) = I(ρ). Then its Hamiltonian formulation (ρ, S) recovers the Schr¨ odinger bridge on graphs. It has many other formulations connecting with Mean ﬁeld games and diﬀusion process. Its Lagrangian formulation can be viewed as the regularized version of optimal transport. 30
31. ### Discussion We establish the Hamiltonian ﬂows on probability simplex over

ﬁnite graphs. The Fisher information is added as the diﬀusion perturbation into the proposed Hamiltonian ﬂows. 31
32. ### Main references Edward Nelson Derivation of the Schr¨ odinger Equation

from Newtonian Mechanics, 1966. Shui-Nee Chow, Wuchen Li and Haomin Zhou Entropy dissipation of Fokker-Planck equations on ﬁnite graphs, 2017. Shui-Nee Chow, Wuchen Li and Haomin Zhou A Schr¨ odinger equation on ﬁnite graphs via optimal transport, 2017. Shui-Nee Chow, Wuchen Li, Chenchen Mou and Haomin Zhou A Schr¨ odinger bridge problem on ﬁnite graphs via optimal transport, 2018. Wuchen Li, Geometry of probability simplex via optimal transport, 2018. 32