# Loss#

As discussed in Theory, we solve a minimization problem to fit the potential. For physics-motivated potentials, the geodesic Levenberg-Marquardt (geodesicLM) minimization method [transtrum2012geodesicLM] can be used, which has been shown to perform well for potentials in [wen2016potfit]. KLIFF also interacts with SciPy to utilize the zoo of optimization methods there. For machine learning potentials, KLIFF wraps the optimization methods in PyTorch.

KLIFF provides a uniform interface to use all the optimization methods. To carry out optimization, first create a loss object:

from kliff.loss import Loss

calculator = ...  # create a calculator
Loss(calculator, nprocs=1, residual_fn=None, residual_data=None)


calculator (discussed in Calculators) provides predictions calculated using a potential and the corresponding reference data via get_prediction() and get_reference(), respectively, which the optimizer can be used to construct the objective function.

nprocs informs KLIFF the number of cores that KLIFF can use to parallelize over the dataset to evaluate the objective function.

residual_data is a dictionary that will be used by residual_fn to compute the residual. residual_data is optional, and its default is:

residual_data = {'normalize_by_natoms': True}


The meaning of this value is made clear in the below discussion.

residual_fn is a function used to compute the residual. As discussed in Theory, the objective function is a sum of the square of the norm of the residual of each individual configuration, i.e. with the residual in which is a vector of predictions computed using the potential for the -th configuration, and is a vector of the corresponding reference data. The residual is computed using the residual_fn, which should be of the form

def residual_fn(identifier, natoms, weight, prediction, reference, data):
"""A function to compute the residual for a configuration."""

# ... compute u based on p (prediction) and q (reference)
# and it should be a vector
return u


In the above residual function,

• identifier is a (unique) str associated with the configuration, which is specified in Configuration. If it is not provided there, identifier is default to the path to the file that storing the configuration, e.g. Si_training_set/NVT_runs/T300_step100.xyz.

• natoms is an int denoting the number of atoms in the configuration.

• weight is a Weight instance that generates the weights from the configuration (see Weight).

• prediction is a vector of the prediction computed from the potential.

• reference is a vector of the corresponding reference data .

• data is residual_data provided at the initialization of Loss. residual_data is a dictionary, with which the user can provide extra information to residual_fn.

residual_fn is also optional, and it defaults to energy_forces_residual() discussed below.

## Built-in residual function#

KLIFF provides a number of residual functions readily to be plugged into Loss and let the wheel spin. For example, the energy_forces_residual() that constructs the residual using energy and forces is defined as (in a nutshell):

def energy_forces_residual(identifier, natoms, weight, prediction, reference, data):

# extract up the weight information
config_weight = weight.config_weight
energy_weight = weight.energy_weight
forces_weight = weight.forces_weight

# obtain residual and properly normalize it
residual = config_weight * (prediction - reference)
residual *= energy_weight
residual[1:] *= forces_weight

if data["normalize_by_natoms"]:
residual /= natoms

return residual


This residual function retrieves the weights for energy and forces fweight instance and enables the normalization of the residual based on the number of atoms. Normalization by the number of atoms makes each individual configuration in the training set contributes equally to the loss function; otherwise, configurations with more atoms can dominate the loss, which (most of the times) is not what we prefer.

One can provide a residual_data instead of using the default one to tune the loss, for example, if one wants to ignore the normalization by the number of atoms.

from kliff.loss import Loss
from kliff.loss import energy_forces_residual

calculator = ...  # create a calculator

# provide my data
residual_data = {'normalize_by_natoms': False}
Loss(calculator, nprocs=1, residual_fn=energy_forces_residual, residual_data=residual_data)


Warning

Even though residual_fn and residual_data is optional, we strongly recommend the users to explicitly provide them to reminder themselves what they are doing as done above.

See kliff.loss for other built-in residual functions.

## Use your own residual function#

The built-in residual function treats each configuration in the training set, and each atom in a configuration equally important. Sometimes, this may not be what you want. In these cases, you can define and use your own residual_fn.

For example, if you are creating a potential that is going to be used to investigate fracture properties, and your training set include both configurations with cracks and configurations without cracks, then you may want to weigh more for the configurations with cracks.

from kliff.loss import Loss

# define my own residual function
def residual_fn(identifier, natoms, weight, prediction, reference, data):

# extract the weight information
config_weight = weight.config_weight
energy_weight = weight.energy_weight
forces_weight = weight.forces_weight

# larger weight for configuration with cracks
if 'with_cracks' in identifer:
config_weight *= 10

normalize = data["normalize_by_natoms"]
if normalize:
energy_weight /= natoms
forces_weight /= natoms

# obtain residual and properly normalize it
residual = config_weight * (prediction - reference)
residual *= energy_weight
residual[1:] *= forces_weight

return residual

calculator = ...  # create a calculator
loss = Loss(
calculator,
nprocs=1,
residual_fn=residual_fn,
residual_data={"normalize_by_natoms": True}
)


The above code takes advantage of identifier to distinguish configurations with cracks and without cracks, and then weigh more for configurations with cracks.

For configurations with cracks, you may even want to weigh more for the atoms near the creak tip. Then you need to identify which atoms are near the crack tip and manipulate the corresponding components of residual.

Note

If you are using your own residual_fn, its data argument can be completely ignored since it can be directly provided in your own residual_fn.

See Define your weight class for an alternative implementation of this example.

Note

Handling the weight is preferably done using the weight class (see Weight) instead of in the residual function.

wen2016potfit

Wen, M., Li, J., Brommer, P., Elliott, R.S., Sethna, J.P. and Tadmor, E.B., 2016. A KIM-compliant potfit for fitting sloppy interatomic potentials: application to the EDIP model for silicon. Modelling and Simulation in Materials Science and Engineering, 25(1), p.014001.

transtrum2012geodesicLM

Transtrum, M.K., Sethna, J.P., 2012. Improvements to the Levenberg-Marquardt algorithm for nonlinear least-squares minimization. arXiv:1201.5885 [physics].