The “Seven Pillars” of oxidation catalysis proposed by Robert K. Grasselli represent an early example of phenomenological descriptors in the field of heterogeneous catalysis. Major advances in the theoretical description of catalytic reactions have been achieved in recent years and new catalysts are predicted today by using computational methods. To tackle the immense complexity of high-performance systems in reactions where selectivity is a major issue, analysis of scientific data by artificial intelligence and data science provides new opportunities for achieving improved understanding. Modern data analytics require data of highest quality and sufficient diversity. Existing data, however, frequently do not comply with these constraints. Therefore, new concepts of data generation and management are needed. Herein we present a basic approach in defining best practice procedures of measuring consistent data sets in heterogeneous catalysis using “handbooks”. Selective oxidation of short-chain alkanes over mixed metal oxide catalysts was selected as an example.

A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.

Low-rank tensors are an established framework for the parametrization of multivariate polynomials. We propose to extend this framework by including the concept of block-sparsity to efficiently parametrize homogeneous, multivariate polynomials with low-rank tensors. This provides a representation of general multivariate polynomials as a sum of homogeneous, multivariate polynomials, represented by block-sparse, low-rank tensors. We show that this sum can be concisely represented by a single block-sparse, low-rank tensor.We further prove cases, where low-rank tensors are particularly well suited by showing that for banded symmetric tensors of homogeneous polynomials the block sizes in the block-sparse multivariate polynomial space can be bounded independent of the number of variables.We showcase this format by applying it to high-dimensional least squares regression problems where it demonstrates improved computational resource utilization and sample efficiency.

We consider best approximation problems in a nonlinear subset M of a Banach space of functions (V, • ). The norm is assumed to be a generalization of the L 2 -norm for which only a weighted Monte Carlo estimate • n can be computed. The objective is to obtain an approximation v ∈ M of an unknown function u ∈ V by minimizing the empirical norm u − v n . In the case of linear subspaces M it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples n is too close to the number of parameters m = dim(M). We review this statement for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and we show that n m is sufficient for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds.

We consider best approximation problems in a nonlinear subset  [[EQUATION]] of a Banach space of functions [[EQUATION]] . The norm is assumed to be a generalization of the [[EQUATION]] -norm for which only a weighted Monte Carlo estimate [[EQUATION]] can be computed. The objective is to obtain an approximation [[EQUATION]] of an unknown function [[EQUATION]] by minimizing the empirical norm [[EQUATION]] . We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the nonlinear least squares setting. Several model classes are examined where analytical statements can be made about the RIP and the results are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.