IMA Thematic Year on Mathematics of Mathematics of Materials and Materials and Macromolecules Macromolecules Thanks to Local Organizers: Mitch Luskin, Maria Calderer, Dick James Effective Theories for Effective Theories for Materials and Materials and Macromolecules Macromolecules
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Mathematics of Materials and Macromolecules IMA Thematic Year on Mathematics of Materials and Macromolecules Thanks to Local Organizers: Mitch Luskin,
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IMA Thematic Year on Mathematics of Materials and Mathematics of Materials and
MacromoleculesMacromolecules
Thanks to Local Organizers: Mitch Luskin, Maria Calderer,
Dick James
Effective Theories for Materials Effective Theories for Materials and Macromoleculesand Macromolecules
Sloppy Models: Universality in Sloppy Models: Universality in Data FittingData Fitting
Kevin S. Brown, JPS, Rick Cerione, Chris Myers, Kelvin Lee, Josh Waterfall, Fergal Casey, Ryan Gutenkunst, Søren Frederiksen,
radioactive pile, can we recover the identity of the elements and/or
predict future radioactivity? Good fits with bad decay rates!
Fit
DN
i i
iyyC
12
2))(()(
P, S, I3532 125
6 Parameter Fit
PC12 DifferentiationPC12 DifferentiationE
RK
*
Time
10’
ER
K*
Time
10’
EGFREGFR NGFRNGFR
RasRas
SosSos
ERK1/2ERK1/2
MEK1/2MEK1/2
Raf-1Raf-1
+NGF+EGF
Pumps up signal (Mek)Tunes down
signal (Raf-1)
Biologists study which proteins talk to which. Modeling?
48 Parameter Fit
‘‘Sloppy Model’ Errors for AtomsSloppy Model’ Errors for AtomsBayesian Ensemble Approach to Error Estimation of Interatomic Potentials
Søren Frederiksen, Karsten W. Jacobsen, Kevin Brown, JPS
Atomistic potential820,000 Mo atoms(Jacobsen, Schiøtz)
Quantum Electronic
Structure (Si)90 atoms (Mo)
(Arias)
Interatomic Potentials V(r1,r2,…)• Fast to compute• Limit me/M → 0 justified• Guess functional form Pair potential V(ri-rj) poor Bond angle dependence Coordination dependence• Fit to experiment (old)• Fit to forces from electronic structure calculations (new)
17 Parameter Fit
Why the Name Sloppy Model?Why the Name Sloppy Model?Huge Fluctuations around Best Fit
eigen
parameters
bare parameters
Best Fit
Hessian ∂2C/∂2 at Best FitSloppy Directions Small Eigenvalues
Eigenvalues Span Huge Range
Each eigenvalue ~three times next
Ill-conditioned
Stiff 1cm Sloppy~meters,km
Local Collinearity of Parameters
Many alternative fits just as good
Huge ranges of allowed parameters
Eig
enva
lue
Tyson
Brown
Kholodenko
Stiff Sloppy
Sloppy Model EigenvaluesSloppy Model EigenvaluesMany fitting problems are sloppy
Molybdenum Interatomic Potential
Cell Dynamics Lessons:• Sloppy Due to Insufficient Data?
Exploring Parameter SpaceExploring Parameter SpaceRugged? More like Grand Canyon (Josh)
Glasses: Rugged LandscapeMetastable Local ValleysTransition State Passes
Optimization Hell: Golf CourseSloppy Models
Minima: 5 stiff, N-5 sloppySearch: Flat planes with cliffs
Ensemble Fluctuations Along Ensemble Fluctuations Along EigendirectionsEigendirections
log e
flu
ctua
tions
alo
ng
eige
ndire
ctio
n
stiff
sloppy
3x previous
Monte Carlo Fluctuations Suppressed in Soft Directions: Anharmonicity or Convergence?
Work In Progress
Error BarsError BarsStochastic versus Sensitivity
Sensitivity Analysis = Harmonic Approximation for Errors• Yields Much Larger Prediction Fluctuations• Anharmonicity Constrains Soft Modes• Mimic w/ modest prior (fluctuations < 106, one )• Sensitivity w/Prior Fluctuations Now Close to Monte Carlo
Work In Progress
Sloppy Model Universality?Sloppy Model Universality?Why are all these problems so similar?
Work In Progress
Random Matrix GOE Ensemble: many different NxN random symmetric matrices have level repulsion, universal ~Wigner-Dyson spacings as N→ Product ensemble: equally spaced logs! stronger level repulsionFitting exponentials: very strong level repulsion!New random matrix ensemble?