Multi-Fidelity Surrogate Module

This module implements two major bricks required to perform multi-fidelity surrogate-assisted optimization:

  1. mf_models which defines model classes to be used for single- / multi-fidelity and single- / multi-objective prediction,
  2. mf_infill which defines sub-optimization problems to be minimized or maximized in the context of Bayesian or non-Bayesian infill strategies.

All features related to surrogate models come with additional dependencies listed in requirements_sm.txt and recalled below: 

torch       # Pytorch for deep learning based models (MFDNN)
smt         # for single- and multi-fidelity kriging (MfSMT)
GPy         # for co-kriging and other Gaussian processes models
emukit      # for co-kriging and other Gaussian processes models

These dependencies can be added to the user's environment with the following command: 

# from aero-optim
pip install -r requirements_sm.txt

Multi-Fidelity Models

The MfModel abstract class defines how any multi-fidelity surrogate model can be wrapped. It is initialized with the following positional arguments:

  • dim (int) which indicates the dimension of the problem,
  • model_dict (dict) which contains model specific information,
  • outdir (str) the directory where model parameters should be saved,
  • seed (int) the random seed the model will be initialized with,
  • x_lf_DOE (np.ndarray) and y_lf_DOE (np.ndarray) the low-fidelity initial DOE,
  • x_hf_DOE (np.ndarray) and y_hf_DOE (np.ndarray) the high-fidelity initial DOE.

When inherited, the following methods should also be overridden:

  • train() which defines how the model should be trained,
  • evaluate() which defines how the model should be evaluated,
  • evaluate_std() which defines how the model standard deviation should be computed when the model is of Bayesian nature.

In addition, any MfModel based model will have access to two base methods:

  • set_DOE() which updates the model DOEs,
  • get_DOE() which returns the model DOEs.

Except for Neural Network based models, most multi-fidelity models available in mf_models are single-output models, the MultiObjectiveModel class is implemented to turn any MfModel into a multi-objective model. To do so, MfModel objects are built for each objective and passed as a list when MultiObjectiveModel is initialized. This way, the use of MultiObjectiveModel is transparent to the user. The only difference occurs when the model DOEs are updated. For such models, the arrays of objectives must be passed as a list of 1D arrays via the set_DOE() method.

Infill strategies

The mf_infill module implements various sub-optimization problems related to both single- and bi-objective adaptive infill. The single-objective adaptive infill strategies are:

  • the Lower Confidence Bound minimized via minimize_LCB,
  • the Expected Improvement maximized via maximize_EI.

The bi-objective infill strategies are:

  • the Probability of Improvement maximized via maximize_PI_BO,
  • the minimal Probability of Improvement maximized via maximize_MPI_BO.

Note

The multi-fidelity aspect of the surrogate model used to solve one of these sub-optimization problems only explicitly intervenes with bi-objective adaptive strategies which can use the Pareto front based on the current low- or high-fidelity DOE.

Regardless of the number of objectives, the max-min Euclidean Distance sub-optimization problems provides samples that harmonize the coverage of the design space. It is solved via maximize_ED.

Quick Experiments

Surrogate models can be evaluated with 1D or multi-dimensional functions with the scripts contained in scripts/MF-SM. For instance, the SMT multi-fidelity co-kriging, can be evaluated on all four Brevault's 1D functions via the command below: 

# from aero-optim to scripts/MF-SM
cd scripts/MF-SM
python3 main_mf.py -c config/example.json -o mfsmt -m mfsmt -f 1d -n 1
where example.json is a configuration file specifying the parameters of the training and evaluation.

If the configuration file contains a "function nd" entry, the model can be trained and evaluated with Brevault's multi-dimensional functions by simply changing the command line argument -f from 1d to nd: 

python3 main_mf.py -c config/example.json -o mfsmt -m mfsmt -f nd -n 1

Note

For 1D functions, the "parameter" entry of the configuration file indicates Brevault's functions parameters while for multi-dimensional functions, it indicates their dimension.

In addition to this script, two other notebooks designed for prototyping and validation purposes for single- and multi-objective infill strategies are available in scripts/MF-SM.