Predictive Coarse-Graining M. Schberl (TUM), N.Zabaras - PowerPoint PPT Presentation

Predictive Coarse-Graining M. Schöberl (TUM), N.Zabaras (Warwick/TUM-IAS), P .S. Koutsourelakis (TUM) Predictive Multiscale Materials Modelling Turing Gateway to Mathematics, Isaac Newton Institute for Mathematical Sciences Decemeber 1 2015 p.s.koutsourelakis@tum.de Predictive Coarse-Graining 1 / 32

Problem Definition - Equilibrium Statistical Mechanics Fine scale p f ( x ) ∝ e − β V f ( x ) • x : fine -scale dofs • V f ( x ) : atomistic potential Observables: E p f [ a ] = � a ( x ) p f ( x ) d x p.s.koutsourelakis@tum.de Predictive Coarse-Graining 2 / 32

Problem Definition - Equilibrium Statistical Mechanics Fine scale p f ( x ) ∝ e − β V f ( x ) • x : fine -scale dofs • V f ( x ) : atomistic potential � Observables: E p f [ a ] = a ( x ) p f ( x ) d x Coarse scale X = R ( x ) , dim ( X ) << dim ( x ) • X : coarse -scale dofs • R : restriction operator ( fine → coarse ) p.s.koutsourelakis@tum.de Predictive Coarse-Graining 2 / 32

Problem Definition - Equilibrium Statistical Mechanics Fine scale p f ( x ) ∝ e − β V f ( x ) • x : fine -scale dofs • V f ( x ) : atomistic potential � Observables: E p f [ a ] = a ( x ) p f ( x ) d x Coarse scale X = R ( x ) , dim ( X ) << dim ( x ) • X : coarse -scale dofs • R : restriction operator ( fine → coarse ) Goal: How can one simulate X and still predict E p f [ a ] ? p.s.koutsourelakis@tum.de Predictive Coarse-Graining 2 / 32

Problem Definition - Equilibrium Statistical Mechanics • Suppose the observable of interest a ( x ) depends on X i.e.: a ( x ) = A ( X ) = A ( R ( x )) • Then: � E p f [ a ] = a ( x ) p f ( x ) d x � = A ( R ( x ) p f ( x ) d x � �� � = A ( X ) δ ( X − R ( x )) d X p f ( x ) d x � �� � = A ( X ) δ ( X − R ( x )) p f ( x ) d x d X � = A ( X ) p c ( X ) d X where p c is the (marginal) PDF of the CG variables X : � δ ( X − R ( x )) p f ( x ) d x ∝ e − β V c ( X ) p c ( X ) = and the CG potential V c ( X ) is the potential of mean force (PMF) w.r.t. X : � V c ( X ) = − β − 1 log δ ( X − R ( x )) p f ( x ) d x p.s.koutsourelakis@tum.de Predictive Coarse-Graining 3 / 32

Existing Methods • Free-energy methods (for low-dimensional X ) [ Lelièvre et al 2010 ] • Lattice systems [ Katsoulakis 2003 ], Soft matter ] Peter & Kremer 2010 ] • Inversion-based methods: Iterative Boltzmann Inversion [ Reith et al. (2003) ], Inverse Monte Carlo [ Lyubartsev & Laaksonen (1995), Soper (1996) ], Molecular RG-CG [ Savelyev & Papoian 2009 ] • Variational methods: Multiscale CG [ Izvekov &et al. (2005), Noid et al. (2007) ] , Relative Entropy [ Shell (2008) ], Ultra-Coarse-Graining [ Dama et al. 2013 ] p.s.koutsourelakis@tum.de Predictive Coarse-Graining 4 / 32

Motivation • What are good coarse-grained variables X (how many, what is fine-to-coarse mapping R ) • What is the right CG potential (or CG model)? • How much information is lost during coarse-graining and how does this affect predictive uncertainty? • Given finite simulation data at the fine-scale, how (un)certain can we be in our predictions? • Can one use the same CG variables X to make predictions about observables a ( x ) � = A ( X ) ? p.s.koutsourelakis@tum.de Predictive Coarse-Graining 5 / 32

Motivation Existing methods Fine-Scale x R ( x )= X → ¯ p f ( x ) − − − − − p c ( X ) R ( x ) Fine to Coarse � �� � � �� � coarse fine Coarse-Scale X Proposed (Generative model) Coarse-Scale X � p cf ( x | X ) → ¯ p c ( X ) − − − − − p f ( x ) = p cf ( x | X ) p c ( X ) d X Coarse to Fine � �� � � �� � coarse fine Fine-Scale x Notes • No restriction operator (fine-to-coarse R ( x ) = X ). • A probabilistic coarse-to-fine map p cf ( x | X ) is prescribed • The coarse model p c ( X ) is not the marginal of X (given R ( x ) = X ) p.s.koutsourelakis@tum.de Predictive Coarse-Graining 6 / 32

Motivation Relative Entropy CG [Shell, 2008] • A fine → coarse map R and an (approximate) CG density ¯ p c ( X ) imply: p f ( x ) = δ ( R ( x ) − X ) ¯ ¯ p c ( R ( x )) Ω( R ( x )) � where: Ω( R ( x )) = δ ( R ( x ) − X ) d x • Find ¯ p c ( X ) that minimizes KL-divergence between p f ( x ) (exact) and ¯ p f ( x ) (approximate): KL ( p f ( x ) || ¯ p f ( x )) = KL ( p c ( X ) || ¯ p c ( X )) + S map ( R ) � �� � � �� � inf. loss due to error in PMF inf. loss due to map R � where S map = p f ( x ) log Ω( R ( x )) d x p.s.koutsourelakis@tum.de Predictive Coarse-Graining 7 / 32

Motivation Proposed Probabilistic Generative model • A probabilistic coarse → fine map p cf ( x | X ) and a CG model p c ( X ) , imply: � ¯ p cf ( x | X ) p c ( X ) d X p f ( x ) = • Find p cf ( x | X ) and p c ( X ) that minimize: � � p cf ( x | X ) p c ( X ) d X KL ( p f ( x ) || ¯ p f ( x )) = − p f ( x ) log d x p f ( x ) p.s.koutsourelakis@tum.de Predictive Coarse-Graining 8 / 32

Learning Proposed Probabilistic Generative model p c ( X | θ c ) , p cf ( x | X , θ cf ) • Parametrize: � �� � � �� � coarse model coarse → fine map • Optimize: θ c , θ cf KL ( p f ( x ) || ¯ p f ( x | θ c , θ cf )) min � � p cf ( x | X , θ cf ) p c ( X | θ c ) d X ↔ min θ c , θ cf − p f ( x ) log d x p f ( x ) � � � � ↔ max p f ( x ) p cf ( x | X , θ cf ) p c ( X | θ c ) d X log d x θ c , θ cf � N � p cf ( x ( i ) | X , θ cf ) p c ( X | θ c ) d X ↔ max i = 1 log θ c , θ cf ↔ max θ c , θ cf L ( θ c , θ cf ) , (MLE) • MAP estimate: max θ c , θ cf L ( θ c , θ cf ) + log p ( θ c , θ cf ) � �� � log − prior • Fully Bayesian i.e. posterior: p ( θ c , θ cf | x ( 1 : N ) ) ∝ exp {L ( θ c , θ cf ) p ( θ c , θ cf ) } p.s.koutsourelakis@tum.de Predictive Coarse-Graining 9 / 32

Learning Proposed Probabilistic Generative model p c ( X | θ c ) , p cf ( x | X , θ cf ) • Parametrize: � �� � � �� � coarse model coarse → fine map • Optimize: θ c , θ cf KL ( p f ( x ) || ¯ p f ( x | θ c , θ cf )) min � � p cf ( x | X , θ cf ) p c ( X | θ c ) d X ↔ min θ c , θ cf − p f ( x ) log d x p f ( x ) � � � � ↔ max p f ( x ) p cf ( x | X , θ cf ) p c ( X | θ c ) d X log d x θ c , θ cf � N � p cf ( x ( i ) | X , θ cf ) p c ( X | θ c ) d X ↔ max i = 1 log θ c , θ cf ↔ max θ c , θ cf L ( θ c , θ cf ) , (MLE) • MAP estimate: max θ c , θ cf L ( θ c , θ cf ) + log p ( θ c , θ cf ) � �� � log − prior • Fully Bayesian i.e. posterior: p ( θ c , θ cf | x ( 1 : N ) ) ∝ exp {L ( θ c , θ cf ) p ( θ c , θ cf ) } p.s.koutsourelakis@tum.de Predictive Coarse-Graining 9 / 32

Learning Proposed Probabilistic Generative model p c ( X | θ c ) , p cf ( x | X , θ cf ) • Parametrize: � �� � � �� � coarse model coarse → fine map • Optimize: θ c , θ cf KL ( p f ( x ) || ¯ p f ( x | θ c , θ cf )) min � � p cf ( x | X , θ cf ) p c ( X | θ c ) d X ↔ min θ c , θ cf − p f ( x ) log d x p f ( x ) � � � � ↔ max p f ( x ) p cf ( x | X , θ cf ) p c ( X | θ c ) d X log d x θ c , θ cf � N � p cf ( x ( i ) | X , θ cf ) p c ( X | θ c ) d X ↔ max i = 1 log θ c , θ cf ↔ max θ c , θ cf L ( θ c , θ cf ) , (MLE) • MAP estimate: max θ c , θ cf L ( θ c , θ cf ) + log p ( θ c , θ cf ) � �� � log − prior • Fully Bayesian i.e. posterior: p ( θ c , θ cf | x ( 1 : N ) ) ∝ exp {L ( θ c , θ cf ) p ( θ c , θ cf ) } p.s.koutsourelakis@tum.de Predictive Coarse-Graining 9 / 32

Prediction Probabilistic Prediction • For an observable a ( x ) (reconstruction, [ Katsoulakis et al. 2006, Trashorras et al. 2010 ]): data � �� � x ( 1 : N ) ] ≈ E ¯ p f [ a | E p f [ a ] � a ( x ) ¯ p f ( x | x ( 1 : N ) ) d x = • For each θ c , θ cf from the posterior, one gets an estimate of the observable • Not just point-estimates anymore, but whole distributions! p.s.koutsourelakis@tum.de Predictive Coarse-Graining 10 / 32

Prediction Probabilistic Prediction • For an observable a ( x ) (reconstruction, [ Katsoulakis et al. 2006, Trashorras et al. 2010 ]): data � �� � x ( 1 : N ) ] E p f [ a ] ≈ E ¯ p f [ a | � p f ( x | x ( 1 : N ) ) d x a ( x ) ¯ = � a ( x ) ¯ p f ( x , θ c , θ cf | x ( 1 : N ) ) d θ c d θ cf d x = � p f ( x | θ c , θ cf ) p ( θ c , θ cf | x ( 1 : N ) ) d θ c d θ cf d x a ( x ) ¯ = � �� � p ( θ c , θ cf | x ( 1 : N ) ) d θ c d θ cf d x = a ( x ) p cf ( x | X , θ cf ) p c ( X | θ c ) d X � �� � p ( θ c , θ cf | x ( 1 : N ) ) d θ c d θ cf = a ( x ) p cf ( x | X , θ cf ) p c ( X | θ c ) d X d x � ˆ a ( θ c , θ cf ) p ( θ c , θ cf | x ( 1 : N ) ) = d θ c d θ cf � �� � posterior • For each θ c , θ cf from the posterior, one gets an estimate of the observable • Not just point-estimates anymore, but whole distributions! p.s.koutsourelakis@tum.de Predictive Coarse-Graining 10 / 32