Constraint Reasoning and Kernel Clustering for Pattern Decomposition with Scaling

Abstract

Abstract. Motivated by an important and challenging task encountered in material discovery, we consider the problem of finding K basis patterns of numbers that jointly compose N observed patterns while enforcing additional spatial and scaling constraints. We propose a Constraint Programming (CP) model which captures the exact problem structure yet fails to scale in the presence of noisy data about the patterns. We alleviate this issue by employing Machine Learning (ML) techniques, namely kernel methods and clustering, to decompose the problem into smaller ones based on a global data-driven view, and then stitch the partial solutions together using a global CP model. Combining the complementary strengths of CP and ML techniques yields a more accurate and scalable method than the few found in the literature for this complex problem.

SMT-Aided Combinatorial Materials Discovery

“…We show that using our novel encoding, state-of-the-art SMT solvers can automatically analyze large synthetic datasets, and generate interpretations that are physically meaningful and very accurate, even in the presence of artificially added noise. Moreover, our approach scales to realistic-sized problem instances, outperforming a previous approach based on Constraint Programming and a set-variables encoding [11]. Further, we show that SMT solving outperforms both Constraint Programming and Mixed Integer Programming translations of our SMT formulation.…”

“…While these approaches are quite effective at extracting information from large amounts of noisy data, their major limitation is that it is hard to enforce the physical constraints of the problem at the same time. As a result, the interpretations obtained with these techniques are often not physically meaningful, for instance because regions corresponding to some basis patterns are not connected [11].…”

“…To address these limitations, in [11] they used a Constraint Programming approach to enforce the constraints on the phase maps, defining a new problem called Pattern Decomposition with Scaling. They propose an encoding based on set variables, but the main limitation of their work is that current state-of-the-art CP solvers cannot scale to realistic size instances (e.g., with at least 40 sample points).…”

“…Our approach is similar to the one proposed in [11], but in this work we introduce a novel SMT encoding based on arithmetic to formulate the phase map identification problem. The SMT formalism nicely captures the hybrid nature of the problem, which involves discrete decision variables and continuous measurement data at the same time.…”

“…However, this approach requires too many variables to obtain a sufficiently fine-grained description of the diffraction patterns, and ultimately leads to instances that cannot be solved using current state-of-the art solvers. We therefore use the same approach presented in [11], and we preprocess the diffraction patterns D 0 , · · · , D P −1 using a peak detection algorithm, extracting the locations of the peaks Q(p) in the x-ray diffraction pattern of each point p (see Figure 1). This is justified by the nature of the diffraction patterns, as constructive interference of the scattered x-rays occurs at specific angles (thus creating peaks of intensities) that characterize the underlying crystal.…”