Table 2 Details about the surrogate model (Gaussian process regressor) and the acquisition function (expected hypervolume improvement)

From: Active learning framework to optimize process parameters for additive-manufactured Ti-6Al-4V with high strength and ductility

Gaussian process regressor

GPR(\(\mu (x)\), \(k({x}^{\left(1\right)},{x}^{\left(2\right)})\))

\(\mu (x)\): constant mean function

\(k\left({x}^{\left(1\right)},{x}^{\left(2\right)}\right)={\sigma }^{2}\left(1+\frac{\sqrt{5}d}{l}+\frac{5{d}^{2}}{3{l}^{2}}\right)\exp \left(-\frac{\sqrt{5}d}{l}\right)\) (1)

Expected hypervolume improvement

\({\rm{EHVI}}=\mathop{\sum }\limits_{i=1}^{N}\varTheta ({l}_{1}^{(i)},{u}_{1}^{(i)},{\mu }_{1},{\sigma }_{1})\cdot \varPsi ({l}_{2}^{(i)},{l}_{2}^{(i)},{\mu }_{2},{\sigma }_{2})+\mathop{\sum }\limits_{i=1}^{N}(\varPsi ({l}_{1}^{(i)},{l}_{1}^{(i)},{\mu }_{1},{\sigma }_{1})-\varPsi ({l}_{1}^{(i)},{u}_{1}^{(i)},{\mu }_{1},{\sigma }_{1}))\cdot \varPsi ({l}_{2}^{(i)},{l}_{2}^{(i)},{\mu }_{2},{\sigma }_{2})\)(2)

\(\varTheta (x,y,z,w)=(y-x)\left[1-\varPhi \left(\frac{y-z}{w}\right)\right]\) (3)

\(\varPsi (x,y,z,w)=w\phi \left(\frac{y-z}{w}\right)+(z-x)\left[1-\varPhi \left(\frac{y-z}{w}\right)\right]\) (4)

  1. \({x}^{\left(1\right)},{x}^{\left(2\right)}\): feature vectors
  2. \(d\): distance between two vectors \({x}^{\left(1\right)}\) and \({x}^{\left(2\right)}\)
  3. \(l\): length scale where \(l \sim {\rm{Gamma}}(3.0,6.0)\)
  4. \(\sigma\): output scale where\(\sigma \sim {\rm{Gamma}}(2.0,0.15)\)
  5. \({l}_{1}^{(i)},{l}_{2}^{(i)}\): lower bounds of partitioning integration region
  6. \({\mu }_{1},{\mu }_{2}\): mean of predicted values for each objective
  7. \({u}_{1}^{(i)},{u}_{2}^{(i)}\): upper bounds of partitioning integration region
  8. \({\sigma }_{1},{\sigma }_{2}\): standard deviation of predictions for each objective
  9. \(\varPhi\): standard cumulative probability distribution function
  10. \(\phi\): standard probability density function
  11. \(N\): number of integration region by partitioning
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