Documentation¶
The FUNCLE module
Functional UNcertainty Constrained by Law and Experiment
User’s Guide¶
Definitions¶
Physics Process
“A physics process is a relation between two or more physical quantities, e.g. the rate at which plasma ion and electron temperatures equilibrate as a function of the individual temperatures, particle masses, and number densities. Note that a physics process need not be time dependent, e.g. the shear modulus as a function of temperature and density.”:cite:`VaughanPUBS2014`
Physics Model
“A physics model is an approximate mathematical representation of a physics process. As an example of a physics model, consider the plastic constitutive relation of a metal. This relates the basic quantities governing plastic deformation, e.g. the flow stress, plastic strain, strain rate, temperature, and density. It is an approximate representation of the very complicated mesoscale physics responsible for the flow of a solid, and is therefore a physics model.”:cite:`VaughanPUBS2014`
Degree of Freedom
The mathematical representation of a physics model is dependent on a set of variables whose values can alter the behavior of the model. These variables are referred to as the degrees of freedom or DOF
Experiment
An experiment represents data measured from a real physical experiments. Experiments used in F_UNCLE are chosen so that their results are dominated by a single physics process which can be represented by a physics model Simulation
Simulation
A simulation which represents the physical system from which the experimental data were obtained. The simulation is dependent on one or more physics models and acts as a map between the model degree of freedom space and the experimental result space
F_UNCLE Parent Classes¶
These concepts are implemented in the following three fundamental F_UNCLE classes
Physics Model¶
F_UNCLE.Utils.PhysicsModel.PhysicsModel. This class
represents a physics model. The degrees of freedom of a PhysicsModel
are immutable. Setting the degrees of freedom of a PhysicsModel will
return a independent copy of itself with updated DOF.
F_UNCLE.Utils.PhysicsModel.PhysicsModel.get_dof(): Returns a list of the model degrees of freedomF_UNCLE.Utils.PhysicsModel.PhysicsModel.set_dof(): Generates a new Physics Model with the specified degrees of freedomF_UNCLE.Utils.PhyscisModel.PhysicsModel.get_sigma(): Returns a square matrix representing the co-variance of the model degrees of freedomF_UNCLE.Utils.PhyscisModel.PhysicsModel.shape(): Returns an integer representing the number of model degrees of freedom
Experiment¶
F_UNCLE.Utils.Experiment.Experiment. This class represents
the results for a single experiment performed under a single set of
conditions.
F_UNCLE.Utils.Experiment.Experiment.__call__(): Returns a tuple with the following elements:
- The independent variable of the simulation (i.e. time)
- List of arrays of dependent variables. Element zero is the element for comparison, the last element is the labels for the previous elements
- Dictionary of additional simulation result attributes. Contains at least the key mean_fn which is a functional representation of the relationship between the independent variable and the dependent variable for comparison
F_UNCLE.Utils.Experiment.Experiment.get_sigma(): Returns an nxn co-variance matrix for the experimental dataF_UNCLE.Utils.Experiment.Experiment.shape(): Returns an integer representing the number of experimental data pointsF_UNCLE.Utils.Experiment.Experiment.get_fisher_matrix(): This returns the fisher information matrix of the experiment given the simulation’s sensitivity matrix
The Experiment object provides some internal routines to ease comparison of simulations and experiments
F_UNCLE.Utils.Experiment.Experiment.align(): This takes a set of simulation data and returns a copy of it with the simulation data aligned so it is evaluated at each independent data value of the experiment. This ‘aligned’ tuple of simulation data has a new key tau which is the shift in independent variable (likely time) required to align the simulation to the experiment. In addition, the mean_fn attribute has been modified so it returns values aligned to the experimental data.
- :py:fun:`F_UNCLE.Utils.Experiment.Experiment.compare`: This takes a set of simulation data and returns the error between the experiment and the simulation. The simulation can either be aligned or not. The return value is the experimental value less the aligned simulation value
Simulation¶
F_UNCLE.Utils.Simulation.Simulation. This class wraps some
computer simulation which is dependent on a PhysicsModel
F_UNCLE.Utils.Simulation.Simulation.compare(): Compares two simulations results and returns the difference
F_UNCLE.Utils.Simulations.Simulations.__call__(): Returns the results of the simulation as a tuple
- The independent variable of the simulation (i.e. time)
- List of arrays of dependent variables. Element zero is the element for comparison, the last element is the labels for the previous elements
- Dictionary of additional simulation result attributes. Contains at least the key mean_fn which is a functional representation of the relationship between the independent variable and the dependent variable for comparison
F_UNCLE.Utils.Simulations.Simulations.get_sens(): This returns a sensitivity matrix of the simulation response to changes in the required model degrees of freedom. The response is evaluated at the independent data points of the provided initial_data and the deltas for each finite difference step are evaluated using the compare method.
Optimization Methods¶
The goal of F_UNCLE is to determine the set of model degrees of freedom which minimizes the error between a set of Experiments and associated Simulations.