Bayesian Model Calibration with Integrated Discrepancy: Addressing Inexact Dislocation Dynamics Models

Abstract

In this work, a novel approach to Bayesian model calibration routines is developed which reinterprets the traditional definition of model discrepancy as defined by Kennedy and O'Hagan (KOH). The novelty lies in the integration of $δ_θ(x_i)$ GPs within the simulator, which is approximated as a GP surrogate model to ensure computational tractability. This approach assumes that the utilized simulator sufficiently predicts observed trends when calibrated with respect to the application domain, and that all model-form errors can be attributed to uncertainty in the input parameters. In contrast, the KOH method assumes discrepancy to be inherently decoupled from the simulator, acting as a 'catch-all' for various sources of model error. The new method is applied to Molecular Dynamics observations of the critical stress to drive dislocation dipoles, and equivalent predictions using a Discrete Dislocation Dynamics simulator whose coarse-grained physical interpretation of the underlying physical mechanisms requires calibration against MD observations. We present an overview of similar state-aware calibration routines; differentiate the provided approach through redefining the commonly used discrepancy Gaussian process and benchmark against KOH. A philosophical argument as to when application of the proposed method is appropriate is provided, and future directions for expanding upon this methodology are proposed.

Figures (5)

• Figure 1: MD (left) and DDD (right) equivalent simulation cells. Dislocation dipoles dissociate into Shockley partials in FCC creating an intrinsic stacking fault colored by the atoms in MD and the shaded field in DDD. The dipole height is the number of glide planes separating dislocation defects.
• Figure 2: Plot of $\tau^{CRSS}$ in both MD (Observed Data) and DDD computed for a simple dipole microstructure. Colorbars indicate the sampled input parameters for each simulation. The x-axis indicates the vertical separation distance between the glide planes of each segment of the dipole.
• Figure 3: Predictions of the $\eta(x,\theta^*)$ emulator with coupled uncertainty.
• Figure 4: Parameter drift as a function of the normalized application domain. The opaque line and shaded region denote the posterior mean and standard deviation of each discrepancy field, while the translucent lines denote sampled $\delta_\theta(x_i)$ functions.
• Figure 5: Model emulator and learned discrepancy from GPMSA gattiker_gaussian_2015