AL	Active Learning
AM	Additive Manufacturing
BO	Bayesian Optimization
DoE	Design of Experiments
EI	Expected Improvement
FEM	Finite Element Modeling
GP	Gaussian Process
HF	High Fidelity
L-BFGS	Limited Memory Broyden-Fletcher-Goldfarb-Shanno
L-DED	Laser-Directed Energy Deposition
LF	Low Fidelity
LHS	Latin Hypercube Sampling
MF	Multi Fidelity
MFGP	Multi Fidelity Gaussian Process
MFGP-BO	Multi Fidelity Gaussian Process—Bayesian Optimization
MI	Mutual Information
NLML	Negative Log Marginal Likelihood
SF	Single Fidelity
SFGP	Single Fidelity Gaussian Process
SFGP-BO	Single Fidelity Gaussian Process—Bayesian Optimization
UQ	Uncertainty Quantification
Nomenclature
${\alpha }_L$	Absorptivity of laser beam
$\delta \left(\mathbf{x}\right)$	Discrepancy function of $\mathbf{x}$
$\rho$	Correlation function
${\theta }_{\mathbf{1}},{\theta }_{\mathbf{2}}$	Hyperparameters of the covariance
q	Heat flux
$\mu \left(\mathbf{x}\right)$	Mean function of $\mathbf{x}$
${\chi }_{\mathrm{budget}}$	Fraction of the optimization budget that is consumed
cholesky(A)	Cholesky decomposition: L is a lower triangular matrix
	such that $L{L}^T=A$
${\rho }_p$	Material density
$ϵ$	Pre-defined tolerance in optimized value
$\sigma \left(\mathbf{x}\right)$	Variance function of $\mathbf{x}$
${\sigma }_L$	Distribution parameter
${\sigma }_{pred}$	Standard deviation of the posterior predictive Gaussian distribution
	of the melt pool depth
${{\sigma }_{ϵ}}^2$, ${{\sigma }_{{ϵ}_1}}^2$, ${{\sigma }_{{ϵ}_2}}^2$	Noise variance
$a_p$	Thermal diffusivity of material
$c_p$	Specific heat of material
d	Melt pool depth
$d_i$	Melt pool depth closest to desired depth at ith initialization
	for which optimization routine is executed
$d^{*}$	Desired melt pool depth
$d_L,d_H$	Predicted melt pool depth from LF and HF models, respectively
$d_{true}$	True melt pool depth
$d{}_{pred}{}_{\mu }$	Mean of the posterior predictive
	Gaussian distribution of the melt pool depth
$E\left(\mathbf{y}\right)$	Expectation of $\mathbf{y}$
$f\left(x\right)$	Gaussian process function values, $f=(f\left(x_1\right),\cdots ,f\left(x_n\right))$
$f_{x^{*}}$	Gaussian process (posterior) prediction (random variable)
${\widehat{f}}_{x^{*}}$	Gaussian process posterior mean
$h_{\mathrm{eff}}$	Effective heat transfer coefficient
$h_{\mathrm{forced}}$	Forced convection heat transfer coefficient
$h_{\mathrm{free}}$	Free convection heat transfer coefficient
$h_{\mathrm{radiation}}$	Radiation convection heat transfer coefficient
J	Objective function for BO
$k,k(\mathbf{x},{\mathbf{x}}^{\prime })$	Kernel functions of GPs
$k_2,k_1$	Kernel functions of HF and LF GPs
$\mathbf{K}$	Covariance matrix
$k_p$	Thermal conductivity of material
$N_L,N_H$	Number of LF points, Number of HF points
$N_{\mathrm{opt}}$	Maximum allowable optimization iterations
$N_T$	Number of initializations for which the optimization
	routine is executed
$N^{*}$	Sum of optimization iteration numbers at which the
	obtained depth is closest to $d^{*}$ in absolute norm
$N_{d_i\sim d^{*}}$	Optimization iteration number at which
	the obtained depth $d_i$ is closest to $d^{*}$ in absolute norm
P	Laser power
$P^{trn},v^{trn}$	Training input data: Laser power, velocity
$P^{tst},v^{tst}$	Test input data: Laser power, velocity
$QI$	Quality Improvement
$RMSE$	Root mean square error
$RMS{E}_{avg}$	RMSE averaged over all initializations
$RMS{E}_{MFGP},RMS{E}_{MFGP}$	RMSE calculated for MFGP-BO,RMSE calculated for MFGP-BO
$R^2$	Coefficient of Determination
t	Time
$t^{\prime }$	Dummy integration variable
$t_{LF},t_{HF}$	Time taken by LF model (s), Time taken by HF model (s)
$T_0$	Initial temperature
$\Delta t$	Time step
$T_s$	Surface temperature
$T(x_c,y_c,z_c,t$)	Temperature as a function of coordinates ($x_c,y_c,z_c$) and time (t)
$T_{\infty }$	Ambient temperature
$T_L$	Liquidus temperature
v	Laser scan velocity
${\Sigma }^{tst}$	Covariance of the posterior Gaussian distribution at test input
${\mu }^{tst}$	Mean of the posterior Gaussian distribution at test input
$\mathbf{x},{\mathbf{x}}^{\prime }$	Input variables
$\widehat{\mathbf{x}}$	Value of $\mathbf{x}$ that maximizes an objective function
x${}^{trn}$, x${}^{tst}$	Training input, Test input
${\mathbf{x}}_2,{\mathbf{x}}_1$	Inputs to HF and LF models
$\mathbf{X}$	Combined input to a GP consisting of ${\mathbf{x}}_2,{\mathbf{x}}_1$
$X_{Space}$	Process parameter space
$X_{Test}$	Test space
$X^{*}$	Search space for optimization
$\mathbf{y},{\mathbf{y}}^{\prime }$	Output of a GP
$\mathbf{Y}$	Combined output of a GP consisting of ${\mathbf{y}}_2,{\mathbf{y}}_1$
${\mathbf{y}}_2,{\mathbf{y}}_1$	Outputs from HF and LF models
y${}^{trn}$, y${}^{tst}$	Training output, Test output
$\mathbb{V}$	Variance
$\mathcal{N}$	Gaussian distribution
$\alpha ,\mathbf{\Psi },{\psi }_1{\psi }_2,\beta$	Intermediate parameters in Algorithm 1