Data mining for faster, interpretable solutions to inverse problems: A case study using additive manufacturing

Abstract

Abstract Solving inverse problems, where we find the input values that result in desired values of outputs, can be challenging. The solution process is often computationally expensive and it can be difficult to interpret the solution in high-dimensional input spaces. In this paper, we use a problem from additive manufacturing to address these two issues with the intent of making it easier to solve inverse problems and exploit their results. First, focusing on Gaussian process surrogates that are used to solve inverse problems, we describe how a simple modification to the idea of tapering can substantially speed up the surrogate without losing accuracy in prediction. Unlike block tapering, which approximates the covariance matrix by diagonal blocks, our approach divides the data itself into blocks. Both approaches reduce the computational cost by replacing the Cholesky decomposition of the full matrix by the decomposition of multiple smaller matrices, but our approach gives accurate predictions despite the approximation as we identify hyperparameters optimal for each block. Second, we demonstrate that Kohonen self-organizing maps can be used to visualize and interpret the solution to the inverse problem in the high-dimensional input space. For our data set, as not all input dimensions are equally important, we show that using weighted distances results in a better organized map that not only makes the relationships among the inputs obvious, but also indicates the location of the solution in the input space so an additive manufacturing engineer can control the inputs appropriately for a desired output.