Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
T4: Coarse-Graining Methods in Nonequilibrium
Systems
Markos Katsoulakis, University of Massachusetts, Amherst
RMMC, University of Wyoming, June 2014
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Collaborators and funding
Department of Energy (OASCR), National Science Foundation (CDI-Type II)
I M. Dobson, L. Rey-Bellet
(Math+Stat, University of Massachusetts, Amherst)
I P. Plechac (Math, University of Delaware)
I D.G. Vlachos (Chem. Engineering, University of Delaware)
References:
Markos Katsoulakis’ Homepage, ResearchGate Profile
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Path-Space Information Theory Methods - Outline
I
”Best-fit” Coarse-grained Dynamics to fine-scale data for
Nonequilibrium, Hi-Dimensional stochastic systems
I
Examples: Lattice Kinetic Monte Carlo, Langevin Dynamics
I
Developed Mathematical/Computational tools:
I
Rely on Non-equilibrium Statistical Mechanics and Info Theory
concepts:
I
I
I
I
I
I
I
Non-equilibrium Steady States (NESS): NESS 6= Ce −βH(σ)
Entropy Production: (lack of) Detailed Balance (path-space quantity)
Relative Entropy Rate: ”pseudo-metric” on path-space.
Path-Space Fisher Information Matrix
Model Selection, Information Criteria and Multibody interactions
Dynamic Inverse Monte Carlo and Force-matching
Relative Entropy, Observables and Transferability in Coarse-Graining.
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Outline
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Conclusions - Research Directions
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Relative Entropy
I
Pseudo-distance between probability measures P, Q:
Z
dP
R (P | Q) := log
dP
dQ
I
Properties: (i) R (P | Q) ≥ 0 and
(ii) R (P | Q) = 0 iff P = Q a.e.
Observables∗ : Csiszar-Kullback-Pinsker inequality:
p
|EP [φ] − EQ [φ]| ≤ ||φ||∞ 2R (P | Q)
I
∗ (sharper bounds also available–see later)
Coarse-Graining–Error Quantification and Parameterization using RE:
K., Vlachos (2003), K., Plechac, Rey-Bellet, Tsagkarogiannis (2007, 2008/9, 2013),
M.S. Shell (2008, 2012), Bilionis et al (2012), Zabaras et al (2013) ...
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Relative Entropy on Path Space
{σm }m∈Z+ : Time-series of Markov chain at steady state regime.
I
{σm }m∈Z+ , has path distribution in time instants 0, 1, ..., M given by
Q0,M σ0 , · · ·, σM = µ(σ0 )q(σ0 , σ1 ) · · · q(σM−1 , σM )
I
I
µ(σ) is the stationary PDF.
q(σ, σ 0 ) is the transition probability.
I
{σ̃m }m∈Z+ , corresponds to path distribution
P0,M σ0 , · · ·, σM = ν(σ0 )p(σ0 , σ1 ) · · · p(σM−1 , σM )
I
The path-wise relative entropy of Q0,M w.r.t. P0,M is
Z
dQ0,M
R (Q0,M | P0,M ) := log
dQ0,M
dP0,M
K., Pantazis, J. Chem. Phys. (2013); K., Plechac, J. Chem. Phys. (2013);
Kalligiannaki, K., Plechac, SIAM Sci. Comp., (2014)
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Relative Entropy Decomposition
Z
R (Q0,M | P0,M ) =
Z
···
E
× log
µ(σ0 )
M−1
Y
E
q(σi , σi+1 )
i=0
µ(σ0 )
QM−1
q(σi , σi+1 )
ν(σ0 )
QM−1
p(σi , σi+1 )
i=0
i=0
dσ0 · · · dσM
I
R (Q0,M | P0,M ) = MH (Q0,M | P0,M ) + R (µ | ν)
I
H (Q0,M | P0,M ) is the relative entropy rate (RER):
Z
q(σ, σ 0 )
0
0
H (Q0,M | P0,M ) = Eµ
q(σ, σ ) log
dσ
p(σ, σ 0 )
I
Space-time analogue of Gibbs states, relative entropy, etc.
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Relative Entropy Rate-RER
Properties:
I RER inherits properties of Relative Entropy:
I
I
I
I
I
H(P | Q) = R (µ ⊗ p | µ ⊗ q) ,
P
P
µ ⊗ p(A × B) = σ∈A µ(σ) σ0 ∈B p(σ, σ 0 ).
Infers information regarding the path distribution: steady-state
distribution + dynamics.
RER is an observable + statistical estimators ⇒ computationally
tractable using (fast, scalable, etc) molecular solvers.
Not necessary to know the steady states µ: suitable for reaction
networks, reaction-diffusion and other non-equilibrium systems.
Same analysis can be carried out for SDE, spatial KMC, processes
with memory, etc. [K., Pantazis J. Chem. Phys. ’13]
Applicable to the transient regime.
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Outline
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Conclusions - Research Directions
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Off-lattice Models
Coarse-graining in molecular simulations
positions of atoms {X (k) } 7→ positions of metaparticles {Q (p) } = ΠX
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Coarse-graining of polymers; DPD methods
Effective Hamiltonian H̄(P, Q) using simplifying assumptions
Microscopic dynamics
I H̄(P, Q) used in dynamics relevant
only for long-time behaviour and
approach to equilibrium
I ad hoc CG: wrong predictions of
diffusion, crystallization, phase
transitions
Abrams, Kremer, J. Chem. Phys.
(2001), Pivkin, Karniadakis J. Chem.
Phys. (2006)
I without numerical analysis no
indication of wrong phenomenon
being deduced from simulation.
q̇ = ∇p H(p, q)
ṗ = −∇q H(p, q) − γp +
p
2/β Ẇ
CG map: (P, Q) = Π(p, q)
Effective equations of motion
Coarse-grained Hamiltonian H̄(P, Q)
I adaptive change of CG difficult
Praprotnik,Matysiak,Kremer,
Clementi, J. Phys. Cond. Matter
(2007).
Markos Katsoulakis, University of Massachusetts, Amherst
Q̇ = ∇P H̄(P, Q)
Ṗ = −∇Q H̄(P, Q) − γ̄P + σ¯˙W
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Coarse-Graining - Equilibrium
Spatial acceleration me
• Spatial adaptivity1
- Error estimates guide mesh refinement
1. Coarse-graining of
macromolecules
2. Stochastic lattice
dynamics/ KMC
• Multiscale MC methods for high accuracy2
– Higher order closures
– Multigrid
• Multicomponent interacting
systems
Time (s)
0.25
Markos Katsoulakis, University of Massachusetts, Amherst
Coarse
T4: lattice
Coarse-Graining Methods in Nonequilibrium Systems
Probability
0.20
Microscopic lattice
0.15
0.10
0.05
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Inverse Monte Carlo - Equilibrium
I
Build parametrized approx. of Gibbs states:
µ̄app ∼ e −β H̄
I
app
(η;θ)
Find optimal values of parameters θ∗ , such that
X
min
|Eµ [φi ] − Eµ̄app [φi ]|2
θ
i
for selected observables φi .
e.g. F. Muller-Plathe, Chem. Phys. Chem. (2002)
I
Parametrization depends on specific observable(s) φ.
I
Special case: Force-matching methods,
[G. Voth and collaborators]
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Coarse Graining-Parameterization-Transferability
I
Need to ensure accurate simulation of other observables, which are
not part of the parameterization, i.e.
I
Can we improve the ”transferability” of the parametrizations?
I
Recall the CKP inequality: for any observable φ,
p
|EP [φ] − EQ [φ]| ≤ ||φ||∞ 2R (P | Q)
Z
R (P | Q) :=
log
dP
dQ
dP
R (P | Q): Loss of Information during Coarse Graining
K., Vlachos, J. Chem. Phys., (2003)
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Coarse Graining-Parameterization-Equilibrium
Gibbs measure: µ ∼ e −βH(σ) , Coarse-grained Hamiltonian H̄(η)
Z
µ̄ ∼ e −β H̄(η) = E[e −βHN | η] ≡
e −βHN (σ) PN (dσ | η)
X
RG Map: Approximate H̄(η) ≈ H̄ app (η), i.e., µ̄(dη) ≈ µ̄app (dη)
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Coarse Graining-Parameterization-Equilibrium
Gibbs measure: µ ∼ e −βH(σ) , Coarse-grained Hamiltonian H̄(η)
Z
µ̄ ∼ e −β H̄(η) = E[e −βHN | η] ≡
e −βHN (σ) PN (dσ | η)
X
RG Map: Approximate H̄(η) ≈ H̄ app (η), i.e., µ̄(dη) ≈ µ̄app (dη)
Task: estimate & control the error in the relative entropy R (µ̄ | µ̄app );
however µ̄ is unknown...
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Coarse Graining-Parameterization-Equilibrium
Gibbs measure: µ ∼ e −βH(σ) , Coarse-grained Hamiltonian H̄(η)
Z
µ̄ ∼ e −β H̄(η) = E[e −βHN | η] ≡
e −βHN (σ) PN (dσ | η)
X
RG Map: Approximate H̄(η) ≈ H̄ app (η), i.e., µ̄(dη) ≈ µ̄app (dη)
Task: estimate & control the error in the relative entropy R (µ̄ | µ̄app );
however µ̄ is unknown...
Reconstruction: νN (dσ|η) – conditional probability on CG variables η
app
µapp
N (dσ) := νN (dσ|η)µ̄M (dη) .
app
”Lifts” µ̄app
M (dη) to a new µN (dσ) on the microscopic space ΣN .
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Information-based CG parametrizations - Equilibrium
Coarse-Graining map Π : ΣN → ΣM ,
σ 7→ Πσ = η
Optimization Parametrized CG Hamiltonian: H̄ app (η; θ)
app
min R (µN | µapp
N (θ)) or min R (µN (θ) | µN )
θ
θ
Gibbs structure allows explicit calculations on R:
Z̄ app (θ)
R µ̄ | µ̄(0) ∼ Eµ [β(H̄ app (θ) − H)] + log
Z
Optimality condition: ∇θ R = 0
I
Solve using gradient methods, Newton-Raphson, etc: M.S. Shell
(2008) and (2012), Bilionis et al (2012), Zabaras et al (2013), ...
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Outline
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Conclusions - Research Directions
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Cluster Expansions–Parametrization of CG Hamiltonians
A key challenge for both equilibrium and non equilibrium CG:
I
Is the parametric CG model families (for example the CG
Hamiltonian H̄ app (θ) ) rich enough for reliable CG predictions?
I
Do we encounter ”information barriers” that need costly CG terms
to be overcome?
I
Can we find adaptive/multi-level CG strategies to overcome them?
Next we discuss two such examples:
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Model Selection–Parametrization of CG Hamiltonians
(0)
(1)
1. Cluster expansion CG Hamiltonian: H̄m (η) = H̄m (η) + H̄m (η)
Multi-body terms:
X X X
H̄ (1) (η) = β
[jk21 k2 k3 (−E1 (k1 )E2 (k2 )E1 (k3 ) + ...
k1 k2 >k1 k3 >k2
1
CGMC
Two−level ML−KMC
Thermodynamic limit sol.
0.9
Are, K., Plechac , Rey-Bellet SIAM Sci. Comp. (2008), Math. Comp. (2013)
Crucial for adaptive CG:
K. et al., J. Non-Newt. Fluid. Mech.
0.8
0.7
Average coverage <c>
Typically omitted, but essential to capture
phase transitions, switching times:
0.6
0.5
0.4
0.3
(2008)
0.2
0.1
0
−1
0
1
2
3
4
5
External field h
2. DFT-parametrized KMC: Many-body terms are necessary for
correct parametrization Schmidt et al., J. Chem. Theory Comput. (2012)
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
6
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Outline
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Conclusions - Research Directions
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Coarse-graining of Non-equilibrium systems
I
Find the ”best fit” coarse-grained Markovian (and possibly
approximation.
Non-equilibrium Steady State (NESS) 6= Ce −βH(σ) . Typical:
non-Markovian?)
I
I
I
I
Multiple mechanisms: reaction-diffusion-adsoption-desorption, etc.
Reaction networks
Driven or multi-physics systems:
Schematic of driven process (top) and particles (bottom) for reactive adsorption.
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Dynamics and Non-equilibrium systems
Example: Continuous-time jump dynamics (KMC):
X
∂t P(σ, t; ζ) =
[c(σ 0 , σ)P(σ 0 , t; ζ) − c(σ, σ 0 )P(σ, t; ζ)] ,
σ0
I
I
I
P
Stationary states:
∂t P = 0 =⇒ σ0 js (σ 0 , σ) = 0
js (σ 0 , σ) = c(σ 0 , σ)µ(σ 0 ) − c(σ, σ 0 )µ(σ)
Current σ 0 → σ:
Reversible dynamics =⇒ Detailed Balance condition w.r.t to µ(σ)
(e.g., µ ∼ e −βH(σ) )
c(σ 0 , σ)µ(σ 0 ) = c(σ, σ 0 )µ(σ)
I
Irreversible dynamics =⇒ Non-equlibrium steady states-NESS, µ:
X
X
js (σ 0 , σ) =
(c(σ 0 , σ)µ(σ 0 ) − c(σ, σ 0 )µ(σ)) = 0
σ0
I Other examples:
σ0
SDEs, discrete-time Markov Chains, semi-Markov, etc.
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Relative Entropy Rate and Dynamics Parametrization
Example: Fine-scale Markov Chain with transition probability p(σ, σ 0 )
I Define parametrized family CG transition probabilities q θ (σ, σ 0 ):
I
I
I
CG map: Πσ = η; Parametrized CG transition probabilities: p̄ θ (η, η 0 )
1
Reconstruction scheme: ν(σ 0 |Πσ 0 ), e.g. uniform:
|{σ:Πσ=η 0 }|
q θ (σ, σ 0 ) = ν(σ 0 |Πσ 0 )p̄ θ (Πσ, Tσ 0 ) ,
I R P | Q θ = Loss of Information (in time-series) due to CG
I For long times M >> 1, RER is dominant:
R P | Q θ = MH(P | Q θ ) + R µ | µθ
H(P | Q θ ) =
X
µ(σ)
σ∈X
X
σ 0 ∈X
p(σ, σ 0 ) log
p(σ, σ 0 )
].
q θ (σ, σ 0 )
I No need for explicit knowledge of NESS: suitable for reaction networks, SDEs
(e.g. forced Langevin), driven systems, reaction-diffusion, etc.
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Inverse Dynamic Monte Carlo
I
”Best-fit” obtained by minimizing RER:
θ∗ = arg min H(P | Q θ ) ,
θ
I
I
Optimality condition ∇θ H(P | Q θ ) = 0; minimization scheme:
α
θ(n+1) = θ(n) − G (n+1) ,
n
α > 0 and G (n+1) any ”alternative” to the gradient ∇θ H(P | Q θ ):
Newton-Raphson and Path-space Fisher Information Matrix, FH
n
n
G n = Hess(H(P | Q θ ))−1 ∇θ H(P | Q θ ) .
"
FH Q
θ
θ
= Hess(H(P | Q )) = −Eµ
#
X
p(σ, σ)∇2θ
θ
0
log q (σ, σ ) .
σ0
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Data-based parametrization of CG dynamics
I
Unbiased estimator for RER,
ĤN (P| Q θ ) :=
N
1 X
p(σi , σi+1 )
log θ
,
N
q (σi , σi+1 )
i=1
I
Minimization of RER:
min ĤN (P| Q θ ) = max
θ
I
θ
N
N
1 X
1 X
log q θ (σi , σi+1 )−
log p(σi , σi+1 ) ,
N
N
i=1
i=1
Coarse-grained path space Log-Likelihood maximization
max L(θ; {σi }N
i=0 ) := max
θ
θ
I No need for microscopic reconstruction:
Markos Katsoulakis, University of Massachusetts, Amherst
N
1 X
log p̄ θ (Πσi , Πσi+1 ) .
N
i=1
q θ (σ, σ 0 )
= ν(σ 0 |Πσ 0 )p̄ θ (Πσ, Πσ 0 )
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Fisher Information Matrix and Parameter Identifiability
Since RER is a relative entropy,
I
H(P | Q) = R (µ ⊗ p | µ ⊗ q) :
Asymptotic Gaussianity of the Maximum Likelihood Estimator:
∗
I
I
θ̂N → θ∗ a.s. and N −1/2 (θ̂N − θ∗ )*N (0, FH −1 (Q θ )),
∗
Variance determined by the path-space FIM FH Q θ , or
asymptotically by FH Q θ̂N .
Estimating the FIM FH Q θ̂N provides rigorous error bars on
computed optimal parameter values θ∗ .
K., Plechac, J.Chem. Phys. (2013)
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Outline
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Conclusions - Research Directions
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Multi-scale Diffusions and Stochastic Averaging
I
Coarse-graining for diffusion processes Xt ∈ Rn × Rm :
p
dXt = a(Xt ) dt + 2β −1 dWt , X0 = x .
I
Approximating Markov Chain:
X k+1 = X k + a(X k )h +
p
√
2β −1 Z h ,
I
CG (reduced) dynamics: x̄ ≡ Πx ∈ Rn , x̃ ≡ Π⊥ x ∈ Rm
p
√
ΠX k+1 = ΠX k + Πa(X k )h + 2β −1 ΠZ h ,
I
Markovian approximation:
X̄ k+1 = X̄ k + ā(X̄ k ; θ)h +
Markos Katsoulakis, University of Massachusetts, Amherst
p
√
2β −1 Z̄ h ,
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Rigorous ”Dynamic” Force Matching
Relative Entropy Rate: Approximating Markov chain
β
ph (x, x 0 )dx 0 ∼ e − h |x
0
−x−ha(x)|2
0
0
β
dx 0 , p̄h (x̄, x̄ 0 ; θ) ∼ e − h |x̄
0
0
−x̄−ā(x̄;θ)|2
0
qh (x, x ; θ) = p̄h (Πx, Πx ; θ)ν(x |Πx )
Z Z
θ
H(P | P ) =
µ(x)ph (x, x 0 ) log
Z
θ
min H(P | P ) ⇐⇒ min
θ
θ
ph (x, x 0 )
dx 0 dx .
qh (x, x 0 ; θ)
|ā(Πx; θ) − Πa(x)|2 µ(x) dx
i.e., for Gaussian fluctuations: “Force-matching”
M. Saunders, G. Voth, Ann. Rev. Biophysics, (2013)
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
d x̄ 0
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Example 1b: Two-scale systems - Stochastic Averaging
dXt = a(X , Y )dt + dWt1
dYt = −1 b(X , Y )dt + −1/2 dWt2 ,
Theory: asymptotics → 0 – averaging principle (Khasminskii, etc)
Z
d X̄t = ā(X̄t ) + dWt , ā(x) = lim a(x, y ) µx (dy )
→0
Minimization of RER: µ (dx dy ) = µ̄ (dx)µ(dy |x) and for 1
µ (dx dy ) ≈ µ̄(dx)µx (dy )
Z
min |a(x, y ) − ā(x)|2 µx (dy )µ̄(x) dx ,
ā
Unique minimizer as → 0 (Γ-convergence argument!)
Z
ā(x) = a(x, y )µx (dy ) .
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Example: Force-matching (derived
via RER minimization!) for various
values of :
Fitted drift coefficient āǫ (x)
3
a(x, y ) = −y − y
CG fitted āǫ , ǫ = 1.0e − 02
CG fitted āǫ , ǫ = 1.0e + 00
CG fitted āǫ , ǫ = 1.0e − 01
3
2
b(x, y ) = x − y − y 3
2
− 14 y 4
0
aǫ (x)
1
µx (dy ) ∼ e − 2 (y −x)
1
−1
−2
Note that as → 0
−3
−4
−3
ā(x) = −x
Markos Katsoulakis, University of Massachusetts, Amherst
−2
−1
0
x
1
T4: Coarse-Graining Methods in Nonequilibrium Systems
2
3
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Example: a(x, y ) = −y − y 3 , b(x, y ) = x − y − y 3
1
µx (dy ) ∼ e − 2 (y −x)
2
− 14 y 4
→ 0 ā(x) = −x
Autocorrelation functions ǫ = 0.010
1
Autocorrelation functions ǫ = 1.000
1.2
Marginals of invariant distributions (Xtǫ , Ytǫ ) and CG X̄tǫ
0.5
0.5
0.8
0.4
1
pdf µ ǫy (dx)
0.8
pdf µ ǫx (dy)
pdf CG µ̄ ǫ(dx)
autocorrelation
autocorrelation
pdf
0.1
0.5
0
0.4
5
0.2
0.1
0.2
0.6
0
−5
10
0.6
0
−10
−5
0
x
5
10
0.4
x
0.2
0.3
0.2
0
acf Xtǫ
0.1
acf Xtǫ
acf CG X̄tǫ
acf CG X̄tǫ
−0.2
0
pdf µ ǫx (dy)
pdf CG µ̄ ǫ(dx)
0.3
0.3
0.7
pdf µ ǫy (dx)
0.4
pdf
0.9
0
0.5
1
1.5
lag
2
2.5
3
Figure : Autocorrelation function of
the CG stationary process X̄t
Markos Katsoulakis, University of Massachusetts, Amherst
0
0.5
1
1.5
lag
2
2.5
Figure : Autocorrelation function of
the CG stationary process X̄t
T4: Coarse-Graining Methods in Nonequilibrium Systems
3
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
CG Langevin Systems, RER and Dynamic Force Matching
I
Langevin process (qt , pt ) ∈ Rd×d (MD sampling of the canonical
ensemble)
(
dqt = M1 pt dt ,
γ
dpt = −F (qt )dt − M
pt dt + σdBt
I
discretization: explicit EM-Verlet implicit EM scheme (the BBK
scheme)

γ
h
h

pi+1/2 = pi − F (qi ) 2 − M pi 2 + σ∆Wi ,
qi+1 = qi + M −1 pi+1/2 h ,


γ
pi+1 = pi+1/2 − F (qi+1 ) h2 − M
pi+1 h2 + σ∆Wi+1/2
I
Markov chain: (q, p) → (q 0 , p 0 ):
P h (q, p, q 0 , p 0 ) = P h (q 0 |q, p)P h (p 0 |q 0 , q, p)
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
I
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Discretized dynamics
h 0 0
P h (q, p, q 0 , p0 ) = P h (q 0 |q, p)P
(p |q , q, p)

h 0
P (q |q, p) =
1
Zh
0
exp


M2
0
h
2
|q − ∆0 (q, p)|
−
and
 σ 2 h3

)
γh
0
h
0 2
,
|p (1 +
) − ∆1 (q, q )|
2M
#
h
hγ
h
M 0
h
h
0
0
∆0 (q, p) = q − h p − F (q)
+p
and ∆1 (q, q ) =
(q − q) − F (q ) .
2M
2M
h
2
h 0 0
P (p |q , q, p) =
I
(
1
Zh
1
"
exp
−
1
σ2 h
CG mapping Π : Rd → R m , (Πqi , Πpi ) Discretized CG dynamics
(q̄i , p̄i )

γ
h
h

p̄i+1/2 = p̄i − F̄ (q̄i ; θ) 2 − M p̄i 2 + σ∆W̄i ,
−1
q̄i+1 = q̄i + M p̄i+1/2 h ,


γ
p̄i+1 = p̄i+1/2 − F̄ (q̄i+1 ; θ) h2 − M
p̄i+1 h2 + σ∆W̄i+1/2
P̄θh (q̄, p̄, q̄ 0 , p̄ 0 ) = P̄θh (q̄ 0 |q̄, p̄)P̄θh (p̄ 0 |q̄ 0 , q̄, p̄) ,
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Best-fit CG Langevin dynamics
Find the parametrized force field F̄ (q̄; θ)
argminθ H(P h |Pθh ) = argminθ [C (θ) + Dh (θ)]
2 i
1 h −1
Eµ σ (ΠF (q) − F̄ (q̄; θ)) and
4 Z
2 0
1
− 21 3 |q 0 −∆h0 (q,p)|2 −1
0
0
dq
σ h
Dh (θ) = Eµ
σ
(ΠF
(q
)
−
F̄
(q̄
;
θ))
e
Z0h
C (θ) =
As h → 0 we obtain the dynamic force-matching:
lim ∇θ H(P h |Pθh ) =
h→0
h
2 i
5
∇θ Eµ σ −1 (ΠF (q) − F̄ (q̄; θ))
4
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Driven Arrhenius diffusion
I
I
I
Diffusion (hopping)Rates: c(x, y , σ) ∼ d e −β(U(x,σ))
P
Energy barrier: U(x, σ) = z6=x J(x − z)σ(z) − h
J(z) = J0 , for |z| ≤ L and J = 0 otherwise.
Parametrized CG potential:
X
Ū(k, η) =
J̄(k, l)η(k) + J̄(0, 0)(η(k) − 1) − h̄
l
I
Parametrized CG rates: assume local equilibrium, σ(x) ≈ q −1 η(k)
c̄(k, l, η) =
I
1
η(k)(q − η(l))d e −β Ū(k,η)
q
θ = (βJ0 , d, J̄(k, l), J̄(k, l, m), ...).
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Coverage profile at initial and optimal
1
J0 = 4.00
0.9
q=32
q=16
q= 8
q= 4
0.8
J = 5.70
0
J0 = 4.53
J0 = 4.16
J = 4.04
0
0.7
q=8, L=32
coverage
0.6
q=32, L=32
0.5
q=16, L=32
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
lattice site
0.6
0.7
0.8
0.9
1
Non-equilibrium Statistical Mechanics theory for this system:
Bonetto, Lebowitz, Rey-Bellet, (2000), Presutti et al (2012), Maes et al (2013),...
Model system for diffusion through membranes: K., Vlachos, Phys.Rev.Lett. (2000),
...
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Path-FIM, Curvature and Identifiability
Gradient H
Error indicator for different
Estimation error bar: gradient H
2
0.6
J
0
0.4
10
Conf. interval
Rel. error
0.5
0.4
0.35
0
10
0.3
0.3
0.2
−2
10
0.1
0
0.25
−4
8
8.2
8.4
8.6
8.8
10
9
8
8.2
8.4
8.6
8.8
9
H(.|.)
−0.1
Error indicator for different
0.2
Estimation error bar: error indicator for different
−1
10
0.25
0.15
0.2
0.1
H(.|.)
H(.|.)
−2
10
0.15
0.1
−3
10
0.05
0.05
0
−4
8
8.2
8.4
8.6
8.8
9
10
8
8.2
8.4
8.6
8.8
Markos Katsoulakis, University of Massachusetts, Amherst
9
0
7.5
8
8.5
J0
9
9.5
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Outline
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Multi-scale Diffusions and Stochastic Averaging
CG Langevin Dynamics and Dynamic Force Matching
Driven KMC diffusion of interacting particles
Conclusions - Research Directions
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
Conclusions
I
Path-Space Information Theory Methods for Hi-Dim. stochastic
systems
I
I
Mathematical/Computational tools
I
I
I
I
I
”Best-fit” Coarse-grained Dynamics to fine-scale data
Computational Non-equilibrium Stat. Mech. methods
Relative Entropy Rate, Path-Space Fisher Information Matrix
Efficient statistical estimators for RER
Enabling Tools: SPPARKS, LAMMPS, etc
Current and Future Research
I
I
I
I
I
Observables and risk-sensitive bounds
Adaptive Coarse-Graining at non-equilibrium
Model selection, information criteria and multi-body interactions
Synergies with other CG methods:
Mori-Zwanzig and other parametrizations with memory
Parametrized CG models and DFT data.
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems
Background: Information Theory in Path Space
Background: Some Existing Methods in Coarse-Graining
Model Selection Challenges
Coarse-Graining: Non-equilibrium Systems
Examples
Conclusions - Research Directions
References
I
Sensitivity Analysis in Path Space (information-theoretic, goal-oriented)
A Relative Entropy Rate Method for Path Space Sensitivity Analysis of Stationary Complex Stochastic Dynamics, Y. Pantazis,
M.K., J. Chem. Phys. (2013).
Parametric Sensitivity Analysis for Biochemical Reaction Networks based on Pathwise Information Theory, M. K., D. Vlachos, Y.
Pantazis, BMC Bioinformatics, (2013).
Measuring the irreversibility of numerical schemes for reversible stochastic differential equations, M. K. Y. Pantazis, L.
Rey-Bellet., ESAIM: Math. Model. Num. Analysis, (2014).
Goal-oriented sensitivity analysis for lattice kinetic Monte Carlo simulations, M. K. and G. Arampatzis, J. Chem. Phys., to
appear, (2014).
I
Coarse-graining (path-space, multi-body effects)
Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems, M. K., P. Plechac,
J. Chem. Phys. (2013).
Coarse-graining schemes for stochastic lattice systems with short and long-range interactions, M. K., P. Plechac, L. Rey-Bellet
and D. Tsagkarogiannis, Math. Comp., to appear (2014).
Spatial two-level interacting particle simulations and information theory-based error quantification, E. Kalligiannaki, M. K. , P.
Plechac SIAM Sci. Comp., 36, A634A667 (2014).
I
Parallel KMC and numerical analysis for SPPARKS-type algorithms
Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms, G. Arampatzis, M. K., P. Plecháč,
Michela Taufer and Lifan Xu Journal of Computational Physics, (2012).
Parallelization, processor communication and error analysis in lattice kinetic Monte Carlo, G. Arampatzis, M. K. and P. Plecháč,
to appear, SIAM Numerical Analysis, to appear (2014).
See also: Markos Katsoulakis’ Homepage, ResearchGate Profile
Markos Katsoulakis, University of Massachusetts, Amherst
T4: Coarse-Graining Methods in Nonequilibrium Systems