Bayesian strategies for uncertainty quantification of the thermodynamic properties of materials

Abstract

Reliable models of the thermodynamic properties of materials are critical for industrially relevant applications that require a good understanding of equilibrium phase diagrams, thermal and chemical transport, and microstructure evolution. The goal of thermodynamic models is to capture data from both experimental and computational studies and then make reliable predictions when extrapolating to new regions of parameter space. These predictions will be impacted by artifacts present in real data sets such as outliers, systematics errors and unreliable or missing uncertainty bounds. Such issues increase the probability of the thermodynamic model producing erroneous predictions. We present a Bayesian framework for the selection, calibration and quantification of uncertainty of thermodynamic property models. The modular framework addresses numerous concerns regarding thermodynamic models including thermodynamic consistency, robustness to outliers and systematic errors by the use of hyperparameter weightings and robust Likelihood and Prior distribution choices. Furthermore, the framework's inherent transparency (e.g. our choice of probability functions and associated parameters) enables insights into the complex process of thermodynamic assessment. We introduce these concepts through examples where the true property model is known. In addition, we demonstrate the utility of the framework through the creation of a property model from a large set of experimental specific heat and enthalpy measurements of Hafnium metal from 0 to 4900K.

Thermodynamic Modeling of the Uranium–Tellurium System: Estimation of the Uncertainties by a Bayesian Approach

“…In the context of the CALPHAD modeling, the integration of the uncertainty study is still at its early stage. However we can quote the works of [5][6][7]33,34] in which the authors propose different methods to deal with uncertainty management using the Bayesian approach. A wider list of references can be found in ( [35] Section III.3.3) as well as a more global description of the methods used in these works.…”

“…Nevertheless, the propagation of these fitting data uncertainties onto the thermodynamic parameters featured in the Gibbs energy functions in the CALPHAD models are rarely estimated. A few works have been published on this topic, see [5][6][7] for the most recent analyses. Moreover the propagation of these uncertainties on the calculated phase diagram and thermodynamic data are almost never determined.…”

Uncertainty Quantification in Atomistic Modeling of Metals and Its Effect on Mesoscale and Continuum Modeling: A Review

“…Each of the preceding studies propagate the uncertainty in atomistic data forward to the CALPHAD predictions but provide no mechanism to estimate the error contribution from each data set. A possible path forward can be found in a 2019 paper from Paulson et al, wherein Bayesian inference was employed to assess and calibrate models for the thermodynamic properties of elemental hafnium and rescale the reported errors for the included data sets [84].This Bayesian approach was additionally compared to a frequentist approach in [85]. In this approach, the reported variances corresponding to each dataset served as a first guess for the variance in the likelihood.…”

Quantified uncertainty in thermodynamic modeling for materials design

“…Bayesian inference for the CALPHAD model parameters is performed using the open source ESPEI [5] and pycalphad [19] Python packages. We assume that the reader has a working knowledge of Bayesian inference, and recommend our previous work alongside standard texts for an introduction and further detail [20], [21]. The selected model forms are identical to the authors' recent study highlighting the use of ESPEI for CALPHAD optimization [5].…”