relentless.optimize.RelativeEntropy#

class relentless.optimize.RelativeEntropy(target, simulation, potentials, thermo, extensive=False)#

Relative entropy.

The relative entropy \(S_{\rm rel}\) (or Kullback-Leibler divergence) quantifies the overlap of two probability distributions. For a known target statistical mechanical ensemble having distribution \(p_0\) and a simulated model ensemble having distribution \(p\) and parametrized on a set of design variables \(\mathbf{x}\), the relative entropy from the model to the target is:

\[S_{\rm rel} = -\int d\Gamma p_0(\Gamma)\ln\left(\frac{p(\Gamma)}{p_0(\Gamma)}\right)\]

where \(\Gamma\) is an element of phase space. The relative entropy is zero when the two ensembles overlap completely, and it is positive otherwise. Hence, minimization of \(S_{\rm rel}\) can be used to find parameters of a model that reproduce a known target ensemble.

The value of the relative entropy is not readily determined in molecular simulations, so this ObjectiveFunction does not return a value. However, the gradient of the relative entropy with respect to the design variables \(\mathbf{x}\) is much easier to compute as ensemble averages. Currently, the RelativeEntropy objective function supports only PairPotential interactions. These interactions are characterized by \(g_{ij}(r)\), an RDF for each pair of interacting types \((i,j)\) in each Ensemble. The gradient of \(S_{\rm rel}\) is then:

\[\nabla_\mathbf{x} S_{\rm rel} = -\frac{1}{2} \sum_{i,j} \int dr 4\pi r^2 \left[\frac{\beta N_i N_j}{V} g_{ij}(r) \right. \left. -\frac{\beta_0 N_{i,0} N_{j,0}}{V_0} g_{ij,0}(r)\right] \nabla_\mathbf{x} u_{ij}(r)}\]

where \(\beta=1/(k_{\rm B}T)\), \(N_i\) is the number of particles of type \(i\), \(V\) is the extent, and \(u_{ij}(r)\) is the pair potential in the model ensemble. The corresponding properties of the target ensemble are denoted with subscript \(0\).

\(S_{\rm rel}\) is extensive as written, meaning that it depends on the size of the system. This can be undesirable for optimization because it means certain hyperparameters are system-dependent, so the default behavior is to normalize \(s_{\rm rel}=S_{\rm rel}/V_0\). To use the extensive relative entropy set extensive=True.

Parameters:

  • target (Ensemble) – The target ensemble (must have specified V and N).

  • simulation (Simulation) – The simulation engine to use, with specified simulation operations.

  • potentials (Potentials) – The pair potentials to use in the simulations.

  • thermo (SimulationOperation) – The thermodynamic analyzer operation for the simulation ensemble and rdf (usually AddEnsembleAnalyzer). The model ensemble will be extracted from this operation.

  • extensive (bool) – Specification of whether the relative entropy is extensive (defaults to False).

Methods

compute(variables[, directory])

Evaluate the value and gradient of the relative entropy function.

compute_gradient(ensemble, variables)

Computes the relative entropy gradient for an ensemble.

Attributes

target

The target ensemble.
compute(variables, directory=None)#

Evaluate the value and gradient of the relative entropy function.

The value of the relative entropy is not computed, but the gradient is. Calculating the gradient requires running a simulation, which may be computationally expensive.

Optionally, a directory can be specified to write the simulation output as defined in run(), namely the values of the pair potential design variables, which are written to pair_potential.i.json for the \(i\)th pair potential, and the ObjectiveFunctionResult, which is written to result.json.

Parameters:

  • variables (Variable or tuple) – Variables with respect to which to compute gradient.

  • directory (str or Directory) – The output directory

Returns:

The result, which has unknown value None and known gradient.

Return type:

ObjectiveFunctionResult

compute_gradient(ensemble, variables)#

Computes the relative entropy gradient for an ensemble.

Parameters:

  • ensemble (Ensemble) – The ensemble for which to evaluate the gradient.

  • variables (Variable or tuple) – Variables with respect to which to compute gradient.

Returns:

The gradient, keyed on the variables.

Return type:

KeyedArray

property target#

The target ensemble. Must have both V and N parameters set.

Type:

Ensemble