Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky partical differential equation as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy, we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.

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In materials science, the discovery of recipes that yield nanomaterials with defined optical properties is costly and time-consuming. In this study, we present a two-step framework for a machine learning-driven high-throughput microfluidic platform to rapidly produce silver nanoparticles with the desired absorbance spectrum. Combining a Gaussian process-based Bayesian optimization (BO) with a deep neural network (DNN), the algorithmic framework is able to converge towards the target spectrum after sampling 120 conditions. Once the dataset is large enough to train the DNN with sufficient accuracy in the region of the target spectrum, the DNN is used to predict the colour palette accessible with the reaction synthesis. While remaining interpretable by humans, the proposed framework efficiently optimizes the nanomaterial synthesis and can extract fundamental knowledge of the relationship between chemical composition and optical properties, such as the role of each reactant on the shape and amplitude of the absorbance spectrum.