Benchmark showcase: ZDT1 with 10 parameters

This section showcases the ten-parameter ZDT1 benchmark supplied in example/ZDTS/ZDT1_10/. ZDT1 is a two-objective problem and it does not require an external CFD solver.

ZDT1 definition

The function is defined in src/testFunctions.py. Its parameters are

$$ x_i \in [0,1],\quad i = 1,\ldots,n, $$

where $n = 10$ is the number of parameters.

Its objective values are

$$ \begin{aligned} f_1(\mathbf{x}) &= x_1,\ f_2(\mathbf{x}) &= g(\mathbf{x}) \left[1-\sqrt{\frac{x_1}{g(\mathbf{x})}}\right],\ g(\mathbf{x}) &= 1 + 9 \left(\sum_{i=2}^{n}x_i\right)/(n-1). \end{aligned} $$

Both objectives are minimized. The Pareto-optimal set is

$$ x_1 \in [0,1],\quad x_i = 0,\quad i = 2,\ldots,n, $$

and the Pareto-optimal front is

$$ f_1 \in [0,1],\quad f_2 = 1-\sqrt{f_1}. $$

Running the example

From the repository root:

cd example/ZDTS/ZDT1_10
python Allrun.py

Allrun.py settings

Allrun.py configures a CFDNNetAdaptV3 run as follows:

• nPars = 10 -- number of decision variables
• nObjs = 2 -- number of objectives
• nOuts = 2 -- number of surrogate outputs (optional)
• nSam = 10 -- initial sample count
• deltaNSams = [10, 30, 50, 100, 300, 500, 1000] -- sample count increments over iterations
• nNN = 4 -- number of neural networks generated in each iteration
• minN = 2 -- minimum number of neurons in a hidden layer
• maxN = 20 -- maximum number of neurons in a hidden layer
• nHidLay = 3 -- number of hidden layers
• dRN = 1 -- decrement of variability in hidden layer size
• tol = 1e-5 -- optimization tolerance
• iMax = 200 -- maximum number of iterations
• nComps = 16 -- number points chosen for verification
• nSeeds = 4 -- number of seeds for DNN training
• trainPro = 75 -- training percentage split
• valPro = 15 -- validation percentage split
• testPro = 10 -- testing percentage split
• kMax = 10000 -- maximum number of iterations in DNN training
• rEStop = 1e-5 -- relative error for stopping DNN training
• popSize = 100 -- population size of MOEA algorithm
• offSize = 100 -- offspring size of MOEA algorithm
• nGens = 250 -- number of MOEA generations

Outputs

Allrun.py saves data to 01_algoRuns in a dedicated run_* folder including a log.out file and folders for each optimization step step_*.

The log.out file stores data about:

• used settings;
• architectures of all tested DNNs;
• architecture of the best-found DNN;
• points chosen for verification with DNN-predicted values and values calculated by the model (CFD if required); and
• verification errors.

Each step_* folder includes:

• weights and metadata of each trained DNN; and
• MOEA results from each DNN-based optimization.

Indicators

computeIndicators.py reads the optimization log, locates the best DNNs found in each iteration, and reads their optimization results. It then plots the convergence of inverted generational distance and hypervolume difference as the number of samples increases.

Inverted generational distance

The script uses 100 points from the analytical Pareto-optimal set $\mathcal{P}_\mathrm{opt}$ and the Pareto-optimal set predicted by DNN $\mathcal{P}_\mathrm{dnn}$ to calculate:

$$ \mathrm{IGD} = \frac{1}{\lVert\mathcal{P}\mathrm{opt}\rVert} \sum{\mathbf{x}\in\mathcal{P}\mathrm{opt}} \min{\mathbf{y}\in\mathcal{P}_\mathrm{dnn}} \lVert\mathbf{x}-\mathbf{y}\rVert. $$

Hypervolume difference

The script uses 3000 points from the analytical Pareto-optimal front $\mathcal{O}_\mathrm{opt}$ and the Pareto-optimal front predicted by DNN but recalculated by the model $\mathcal{O}_\mathrm{dnn}$ to calculate:

$$ \Delta\mathrm{HV} = \frac{\mathrm{HV}(\mathcal{O}\mathrm{opt}) -\mathrm{HV}(\mathcal{O}\mathrm{dnn})} {\mathrm{HV}(\min(\mathcal{O}_\mathrm{opt}))}. $$

The hypervolume (HV) is calculated using the Platypus hypervolume implementation and the data-derived objective bounds.

The resulting figure indicators.png is saved to the run folder.

Visualization in objective space

postProcess.py creates the final objective-space comparison. It plots:

• the analytical Pareto-optimal front;
• the NSGA-II sample data;
• the DNN-predicted objectives from the best DNNs listed in log.out; and
• recomputed best-DNN objectives by the model.

The resulting figure objSpacePlot.png is saved to the run folder.