Accurate identification of basins of attraction in jammed and glassy systems

Praharsh Suryadevara

Mathias Casiulis

Stefano Martiniani

 

Understanding Energy Landscapes

Energy as a function of conformational angles of a molecule

Energy Landscapes in one dimension

\(V(x)\)

\(x\)

Energy Landscapes in one dimension

\(V(x)\)

\(x\)

\(x_0\)

Energy Landscapes in one dimension

\(V(x)\)

\(x\)

$$ \frac{\text{d} x}{\text{d} t} = - \nabla V(x)$$

Steepest Descent

\(x_0\)

Energy Landscapes in one dimension

\(V(x)\)

\(x\)

$$ \frac{\text{d} x}{\text{d} t} = - \nabla V(x)$$

Steepest Descent

\(x_0\)

Energy Landscapes in one dimension

\(V(x)\)

\(x\)

$$ \frac{\text{d} x}{\text{d} t} = - \nabla V(x)$$

Steepest Descent

\(x_0\)

Energy Landscapes in one dimension

\(V(x)\)

\(x\)

$$ \frac{\text{d} x}{\text{d} t} = - \nabla V(x)$$

Steepest Descent

\(x_0\)

\(x_{min}\)

Energy Landscapes in one dimension

\(V(x)\)

\(x\)

$$ \frac{\text{d} x}{\text{d} t} = - \nabla V(x)$$

Steepest Descent

Energy Landscapes in one dimension

\(V(x)\)

\(x\)

basin 2

Basin tiling (what we're interested in)

basin 1

basin 3

$$ \frac{\text{d} x}{\text{d} t} = - \nabla V(x)$$

Steepest Descent

Systems

Bidisperse Hertzian soft spheres

$$  V_{ij}(r) =\begin{cases}  \frac{\epsilon}{p} \left( 1 - \frac{r}{r_{i} + r_{j}} \right) ^{p} &  r < r_{i} + r_{j} \\  0 & r > r_{i} + r_{j}\end{cases}$$ with \(p\) = 2.5 and packing fraction \(\phi=0.9\) 

Lennard Jones

$$  V_{ij}(r) = \epsilon \left( \left( \frac{\sigma}{r} \right) ^{12} - \left( \frac{\sigma}{r}  \right)^{6} \right) $$

with \(\epsilon=1\) and \(\sigma=1\) 

landscape dimension \(d_l \sim (N\times d)\)

Slicing the energy landscape

Each color is a basin

Imagine cubic basins in a regular grid

Slicing the energy landscape

Each color is a basin

Imagine cubic basins in a regular grid

Wales, David. Energy landscapes: Applications to clusters, biomolecules and glasses. Cambridge University Press, 2003.

Basin identification in \(d_l = 3\)

Optimizers give nearly accurate basins in low dimensions !!

LBFGS

FIRE

Steepest Descent

16 particle basin slice with FIRE

16 particle basin slice with FIRE

\(d_l \sim 30\)

\(d_l \sim 30\)

ODE Solver Comparison (1024 particles)

Rackauckas, Christopher,  et, al., Journal of open research software 5.1 (2017).

Local error

Time (s)

CVODE (Best)

QNDF

Adaptive Implicit Runge Kutta

Adaptive Implicit Euler

Better!

ODE Solver Comparison (1024 particles)

Rackauckas, Christopher,  et, al., Journal of open research software 5.1 (2017).

Local error

Time (s)

CVODE (Best)

QNDF

Adaptive Implicit Runge Kutta

Adaptive Implicit Euler

\(\times 10^3\)

16 particle basin slice with CVODE

16 particle basin slice with CVODE

\(d_l \sim 30\)

\(d_l \sim 30\)

Accuracy

FIRE

LBFGS

FIRE

CVODE

cut around the global minimum: LJ13

\( 8 \times \)equilibrum pairwise distance

Global minimum

\(d_l = 33\)

cut around the global minimum: LJ13

\( 8 \times \)equilibrum pairwise distance

Global minimum

\(d_l = 33\)

Can we do better?: Mixed Descent

 $$ V(x) \approx x^T H x $$

Can we do better?: Mixed Descent

  1. Start with the CVODE solver

Can we do better?: Mixed Descent

  1. Start with the CVODE solver
  2. Perform a convexity check every N iterations

Can we do better?: Mixed Descent

  1. Start with the CVODE solver
  2. Perform a convexity check every N iterations
  3. If convexity check passes, switch to using Newton's method

Can we do better?: Mixed Descent

  1. Start with the CVODE solver
  2. Perform a convexity check every N iterations
  3. If convexity check passes, switch to using Newton's method

Skips unnecessary trajectory information!

A fast and accurate algorithm

CVODE

Mixed Descent

FIRE

LBFGS

LBFGS

Mixed Descent

CVODE

FIRE

Acknowledgments

Martiniani Lab

Stefano Martiniani

Mathias Casiulis

Energy Landscapes in one dimension

\(V(x)\)

\(x\)

$$ \frac{\text{d} x}{\text{d} t} = - \nabla V(x)$$

steepest descent

basin 2

Basin Tiling (What we're interested in)

basin 1

basin 3

Basin identification

$$ \frac{\text{d} x}{\text{d} t} = - \nabla V(x)$$

steepest descent

  1. Solving steepest descent in \(nD\) for interacting potentials is hard.
  2. a large volume of previous work mostly relies on  optimization algorithms as proxies for finding basins

Method

  1. Use CVODE to solve the steepest descent equations 
  2. CVODE is much faster than Gradient descent
  3. CVODE is much faster than scipy implemented stiff ODE methods
  4. Identify minima/optimize with pele (https://github.com/martiniani-lab/pele). Hard fork of Wales group code with additional functionality implemented
  5. Code implemented in basinerror (https://github.com/martiniani-lab/basinerror)

Goals for the talk

  1. I want to show you how to get basins of attraction in high dimensions accurately
  2. I want to show you how you can get them fast

Accuracy curves: Harmonic potential (2d)

CVODE: Results

How to get Global Accuracy

  1. Generate \(N\) steepest descent trajectories to the minima at very low \(rtol\) from random initial conditions
  2. verify that the minima  don't change when you increase  \(rtol\) $$\frac{N_\text{incorrect}}{N} < 0.01  $$

\(N\)

\(rtol\)

95 percent accuracy

Accuracy curves: Harmonic potential (2d)

Random cut around the "global maximum"

\( 8 \times \)equilibrum pairwise distance

Energy Landscapes

Glasses

Hierarchical Landscape

Charbonneau, Patrick, et al , Nat Commun 5, 3725 (2014)

Random cut around a local minimum

Random cut around a local minimum LJ 13

\( 8 \times \)equilibrum pairwise distance

\( 8 \times \)equilibrum pairwise distance

Colors not matched!!

Counting by sampling

\(d\)

$$ N_{buildings} = \frac{D}{\langle\mathcal{d}\rangle} $$