Gaussian processes with linear operator inequality constraints

This paper presents an approach for constrained Gaussian Process (GP) regression where we assume that a set of linear transformations of the process are bounded. It is motivated by machine learning applications for high-consequence engineering systems, where this kind of information is often made available from phenomenological knowledge. We consider a GP $f$ over functions on $\mathcal{X} \subset \mathbb{R}^{n}$ taking values in $\mathbb{R}$, where the process $\mathcal{L}f$ is still Gaussian when $\mathcal{L}$ is a linear operator. Our goal is to model $f$ under the constraint that realizations of $\mathcal{L}f$ are confined to a convex set of functions. In particular, we require that $a \leq \mathcal{L}f \leq b$, given two functions $a$ and $b$ where $a < b$ pointwise. This formulation provides a consistent way of encoding multiple linear constraints, such as shape-constraints based on e.g. boundedness, monotonicity or convexity. We adopt the approach of using a sufficiently dense set of virtual observation locations where the constraint is required to hold, and derive the exact posterior for a conjugate likelihood. The results needed for stable numerical implementation are derived, together with an efficient sampling scheme for estimating the posterior process.Comment: Published in JMLR: http://jmlr.org/papers/volume20/19-065/19-065.pd

Experiment-Based Validation and Uncertainty Quantification of Partitioned Models: Improving Predictive Capability of Multi-Scale Plasticity Models

Partitioned analysis involves coupling of constituent models that resolve their own scales or physics by exchanging inputs and outputs in an iterative manner. Through partitioning, simulations of complex physical systems are becoming evermore present in scientific modeling, making Verification and Validation of partitioned models for the purpose of quantifying the predictive capability of their simulations increasingly important. Parameterization of the constituent models as well as the coupling interface requires a significant amount of information about the system, which is often imprecisely known. Consequently, uncertainties as well as biases in constituent models and their interface lead to concerns about the accumulation and compensation of these uncertainties and errors during the iterative procedures of partitioned analysis. Furthermore, partitioned analysis relies on the availability of reliable constituent models for each component of a system. When a constituent is unavailable, assumptions must be made to represent the coupling relationship, often through uncertain parameters that are then calibrated. This dissertation contributes to the field of computational modeling by presenting novel methods that take advantage of the transparency of partitioned analysis to compare constituent models with separate-effect experiments (measurements contained to the constituent domain) and coupled models with integral-effect experiments (measurements capturing behavior of the full system). The methods developed herein focus on these two types of experiments seeking to maximize the information that can be gained from each, thus progressing our capability to assess and improve the predictive capability of partitioned models through inverse analysis. The importance of this study stems from the need to make coupled models available for widespread use for predicting the behavior of complex systems with confidence to support decision-making in high-risk scenarios. Methods proposed herein address the challenges currently limiting the predictive capability of coupled models through a focused analysis with available experiments. Bias-corrected partitioned analysis takes advantage of separate-effect experiments to reduce parametric uncertainty and quantify systematic bias at the constituent level followed by an integration of bias-correction to the coupling framework, thus ‘correcting’ the constituent model during coupling iterations and preventing the accumulation of errors due to the final predictions. Model bias is the result of assumptions made in the modeling process, often due to lack of understanding of the underlying physics. Such is the case when a constituent model of a system component is entirely unavailable and cannot be developed due to lack of knowledge. However, if this constituent model were to be available and coupled to existing models of the other system components, bias in the coupled system would be reduced. This dissertation proposes a novel statistical inference method for developing empirical constituent models where integral-effect experiments are used to infer relationships missing from system models. Thus, the proposed inverse analysis may be implemented to infer underlying coupled relationships, not only improving the predictive capability of models by producing empirical constituents to allow for coupling, but also advancing our fundamental understanding of dependencies in the coupled system. Throughout this dissertation, the applicability and feasibility of the proposed methods are demonstrated with advanced multi-scale and multi-physics material models simulating complex material behaviors under extreme loading conditions, thus specifically contributing advancements to the material modeling community