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Towards a Searchable Space of Dynamical System Models

Towards a Searchable Space of Dynamical System Models. Eric Mjolsness Scientific Inference Systems Laboratory (SISL) University of California, Irvine www.ics.uci.edu/~emj In collaboration with: Guy Yosiphon NKS June 2006. Motivations shared with NKS.

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Towards a Searchable Space of Dynamical System Models

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  1. Towards a Searchable Space of Dynamical System Models Eric Mjolsness Scientific Inference Systems Laboratory (SISL) University of California, Irvine www.ics.uci.edu/~emj In collaboration with: Guy Yosiphon NKS June 2006 NKS Washington DC 06/15/06

  2. Motivations shared with NKS • Objective exploration of properties of “simple” computational systems • Relation of such to the sciences • Example: bit string lexical ordering of cellular automata rules; reducibility relationships; applications to fluid flow NKS Washington DC 06/15/06

  3. Criteria for a space of simple formal systems • C1: Demonstrated expressive power in scientific modeling • C2: Representation as discrete labeled graph structure • that can be searched and explored computationally • E.g. Bayes nets, Markov Random Fields • roughly in order of increasing size - with index nodes (DD’s) • C3: Self-applicability • useful transformations and searches of such dynamical systems should be expressible • … as discrete-time dynamical systems that compute • So major changes of representation during learning are not excluded. NKS Washington DC 06/15/06

  4. C1: Demonstration of expressive power in scientific modeling NKS Washington DC 06/15/06

  5. Elementary Processes • A(x)  B(y) + C(z)withrf (x, y, z) • B(y) + C(z)  A(x) withrr (y, z, x) • Examples • Chemical reaction networks w/o params • . • XXX from paper • Effective conservation laws • E.g. ∫ NA(x) dx + ∫ NB(y) dy , ∫ NA(x) dx + ∫ NC(z) dz NKS Washington DC 06/15/06

  6. t R N A - L e u L e u t R N A - A l a A l a T C A c y c l e G l u c o s e G l y c o l y s i s P y r A s p V a l t R N A - V a l I l e t R N A - I l e a K B + T h r L y s M e t t R N A - T h r Amino Acid Syntheses Kmech: Yang, et al. Bioinformatics 21: 774-780, 2005 Amino acid synthesis: Yang et al., J. Biological Chemistry, 280(12):11224-32, , Mar 25 2005. GMWC modeling: Najdi et al., J. Bioinformatics and Comp. Biol., to appear 2006. NKS Washington DC 06/15/06

  7. Example: AnabaenaPrusinkiewicz et al. model G. Yosiphon, SISL, UCI NKS Washington DC 06/15/06

  8. Example: Galaxy Morphology G. Yosiphon, SISL, UCI NKS Washington DC 06/15/06

  9. Example: ArabidopsisShoot Apical Meristem (SAM) NKS Washington DC 06/15/06

  10. Quantification of growth Co-visualization of raw and extracted nuclei data NKS Washington DC 06/15/06

  11. PIN1-GFP expression Time-lapse imaging over 40 hrs (Marcus Heisler, Caltech) NKS Washington DC 06/15/06

  12. Dynamic Phyllotactic Model Emergence of new extended, interacting objects: floral meristem primordia. DG’s at ≥ 3 scales: - molecular; - cellular; - multicellular. NKS Washington DC 06/15/06 H. Jönnson, M. Heisler, B. Shapiro, E. Meyerowitz, E. Mjolsness - Proc. Nat’l Acad. Sci. 1/06

  13. Model simulation on growing template NKS Washington DC 06/15/06

  14. Spatial Dynamics in Biological Development • Reimplemented weak spring model in 1 page • Applying to 1D stem cell niches with diffusion, in plant and animal tissues NKS Washington DC 06/15/06

  15. Ecology: predator-prey models NKS Washington DC 06/15/06 with Elaine Wong, UCI

  16. Example: Hierarchical Clustering NKS Washington DC 06/15/06

  17. ML example: Hierarchical Clustering NKS Washington DC 06/15/06

  18. Logic Programming • E.g. Horn clauses • Rules • Operators • Project to fixed-point semantics NKS Washington DC 06/15/06

  19. An Operator Algebra for Processes • Composition is by independent parallelism • Create elementary processes from yet more elementary “Basis operators” • Term creation/annihilation operators: for each parm value, • Obeying Heisenberg algebra [ai, cj] = di j or • Yet classical, not quantum, probabilities NKS Washington DC 06/15/06

  20. Operator algebra H1 + H2 H1 * H2 (noncommutative) Informal meaning independent, parallel occurrence instantaneous, serial co-occurrence Basic Operator Algebra Composition Operations: +, * G Syntax • parallel rules • Multiple terms on LHS, RHS NKS Washington DC 06/15/06

  21. Time Evolution Operators • Master equation: d p(t) / dt = H p(t) • where 1·H = 0, e.g. H =P(H’)= H’ - 1· diag(1·H’ ) • H = time evolution operator • can be infinite-dimensional • Formal solution: p(t)=exp(t H) p(0) NKS Washington DC 06/15/06

  22. Discrete-Time Semantics of Stochastic Parameterized Grammars This formulation can also be used as a programming language, expressing algorithms. NKS Washington DC 06/15/06

  23. Algorithm Derivation:Conceptual Map Operator Space (high dim) Time Ordered Product Expansion (c) Trotter Product Formula DG rules (H, etH) Heisenberg Picture Euler’s formula CBH stochastic program (H´, H´n/(1· H´n ·p)) Functional Operator Space (d) NKS Washington DC 06/15/06

  24. C2: Representation as discrete labeled graph structure that can be searched and explored computationally NKS Washington DC 06/15/06

  25. Basic Syntax for a Modeling Language: Stochastic Parameterized Grammars (SPG’s) • G = set of rules • Each rule has: • LHSRHS {keywordexpression}* • Parameterized term instances within LHS and/or RHS • LHS, RHS: sets (of such terms) with Variables • LHS matches subsets of parameterized term instances in the Pool • Keyword clauses specify probability rate, as a product • Keyword: with • Algebraic sublanguage for probability rate functions • rates are independent of # of other matches; oblivious. • Rule/object : verb/noun : reaction/reactant bipartite graphs • … with complex labels NKS Washington DC 06/15/06

  26. t = 3 t = 1 t = 3 t = 1 t = 2 t = 2 t = 3 Graph Meta-Grammar NKS Washington DC 06/15/06

  27. “Plenum” SPG/DG implementation • builds on Cellerator experience • [Shapiro et al., Bioinformatics 19(5):677-678 2003] • computer algebra embedding provides • probability rate language • Symbolic transformations to executability • includes mixed stochastic/continuous sims NKS Washington DC 06/15/06

  28. SPG/DG Expressiveness Subsumes … • Logic programming (w. Horn clauses) • LHS  RHS; all probability rates equal • Hence, any simulation or inference algorithms can in principle be expressed as discrete-time SPG’s • Chemical reaction networks • No parameters; stoichiometry = weighted labeled bipartite graph • Context-free (stochastic) grammars • No parameters; 1 input term/rule • Formally “solvable” with generating functions • Stochastic (finite) Markov processes • No parameters; 1 input/rule, 1 output/rule • “Solvable” with matrices (or queuing theory?) NKS Washington DC 06/15/06

  29. SPG/DG Expressiveness Subsumes … • Bayes Nets • Each variable x gets one rule: Unevaluated-term, {evaluated predecessors(y)}  evaluated-term(x) • MCMC dynamics • Inverse rule pairs satisfying detailed balance • Each rule can itself have the power of a Boltzmann distribution • Probabilistic Object Models • “Frameville”, PRM, … • Petri Nets • Graph grammars • Hence, meta-grammars and grammar transformations • DG’s subsume: ODE’s, SDE’s, PDE’s, SPDE’s • Unification with SPG’s too NKS Washington DC 06/15/06

  30. C3: Self-applicability • Arrow reversal • Arrow reversal graph grammar exercise • Machine learning by statistical inference • e.g. hierarchical clustering (reported) • ? Equilibrium reaction networks for MRF’s • Further possible applications … NKS Washington DC 06/15/06

  31. Template: A-Life Concisely expressed in SPG’s Steady state condition: total influx into g = total outflow from g NKS Washington DC 06/15/06

  32. Applications to Dynamic Grammar Optimizationand a “Grammar Soup” • Map genones to grammars • Map hazards to functionality tests • Map reproduction to crossover or simulation NKS Washington DC 06/15/06

  33. Conclusions • Stochastic process operators as the semantics for a language • A fundamental departure • Specializes to all other dynamics • Deterministic, discrete-time, DE, computational, … • Graph grammars allow meta-processing • Operator algebra leads to novel algorithms • Wide variety of examples at multiple scales • Sciences • Cell, developmental biology; astronomy; geology • multiscale integrated models • AI • Pattern Recognition • Machine learning • Searchable space of simple dynamical system models including computations NKS Washington DC 06/15/06

  34. For More Information • www.ics.uci.edu/~emj  modeling frameworks NKS Washington DC 06/15/06

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