Digital Twins’ design frequently incorporates machine learning and, more recently, deep reinforcement learning techniques in order to interpret data and forecast future outcomes based on incoming data. However, because neural networks are typically considered a ”black box” model, doubts over the reliability of the output from deep learning models continue. In our work, we developed crop rotation policies using explainable tabular reinforcement learning techniques. We created five-step rotations, or a succession of five successive crops, to compare these policies to those produced by a deep Q-learning technique. The purpose of the rotations is to maximize crop yields while upholding prescribed planting guidelines and preserving a healthy amount of nitrogen in the soil. We incorporated phenomena related to weather and price fluctuations into the reward signal to account for the potential impact of external factors on crop production. The deployed explainable tabular reinforcement learning methods outperform the deep Q-learning approach regarding collected rewards when the rewards are not perturbed. For the perturbed case, robust tabular reinforcement learning methods outperform the deep learning approach even more while maintaining human interpretable policies. By consulting with farmers and crop rotation experts, we demonstrate that the derived policies are reasonable to use and more resilient towards external perturbations. Furthermore, the use of interpretable and explainable reinforcement learning techniques increases confidence in resulting policies, thereby increasing the likelihood that farmers will adopt the suggested policies. Digital Twins are often intertwined with machine learning and, more recently, deep reinforcement learning methods in their architecture to process data and predict future outcomes based on input data. However, concerns about the trustworthiness of the output from deep learning models persist due to neural networks generally being regarded as a black box model. In our work, we developed crop rotation policies using explainable tabular reinforcement learning techniques. We compared these policies to those generated by a deep Q-learning approach by generating five-step rotations, i.e. producing a series of five consecutive crops. The aim of the rotations is to maximise crop yields while maintaining a healthy nitrogen level in the soil and adhering to established planting rules. Crop yields may vary due to external factors such as weather patterns or changes in market prices, so perturbations have been added to the reward signal to account for those influences. The deployed explainable tabular reinforcement learning methods collect, on average, at least as much reward over 100 crop rotation plans when randomly starting with any crop compared to the deep learning model. For the perturbed case, robust tabular reinforcement learning methods collect similar amounts of reward across 100 crop rotation plans compared to the non-random reward setting, whereas the deep reinforcement learning agent collects even fewer rewards compared to learning on non-perturbed rewards. By consulting with farmers and crop rotation experts, we demonstrate that the derived policies are reasonable to use and more resilient towards external perturbations. Furthermore, the use of interpretable and explainable reinforcement learning techniques increases confidence in resulting policies, thereby increasing the likelihood that farmers will adopt the suggested policies.

Fractional calculus gained a lot of attention in the last couple of years. Researchers discovered that processes in various fields follow rather fractional dynamics than ordinary integer-ordered dynamics, meaning the corresponding differential equations feature non-integer valued derivatives. There are several arguments for why this is the case, one of them being that fractional derivatives’ inherit spatiotemporal memory and/or the ability to express complex naturally occurring phenomena. Another popular topic nowadays is machine learning, i.e., learning behavior and patterns from historical data. In our ever-changing world with ever-increasing amounts of data, machine learning is a powerful tool for data analysis, problem-solving, modeling, and prediction. It further provides many insights and discoveries in various scientific disciplines. As these two modern-day topics provide a lot of potential for combined approaches to describe complex dynamics, this article reviews combined approaches of fractional derivatives and machine learning from the past, puts them into context, and thus provides a list of possible combined approaches and the corresponding techniques. Note, however, that this article does not deal with neural networks, as there already is profound literature on neural networks and fractional calculus. We sorted past combined approaches from the literature into three categories, i.e., preprocessing, machine learning & fractional dynamics, and optimization. The contributions of fractional derivatives to machine learning are manifold as they provide powerful preprocessing and feature augmentation techniques, can improve physically informed machine learning, and are capable of improving hyperparameter optimization. Thus, this article serves to motivate researchers dealing with data-based problems, to be specific machine learning practitioners, to adopt new tools and enhance their existing approaches.

We present a novel method for interpolating univariate time series data. The proposed method combines multi-point fractional Brownian bridges, a genetic algorithm, and Takens’ theorem for reconstructing a phase space from univariate time series data. The basic idea is to first generate a population of different stochastically interpolated time series data, and secondly, to use a genetic algorithm to find the pieces in the population which generate the smoothest reconstructed phase space trajectory. A smooth trajectory curve is hereby found to have a low variance of second derivatives along the curve. For simplicity, we refer to the developed method as PhaSpaSto-interpolation, which is an abbreviation for phase-space-trajectory-smoothing stochastic interpolation. The proposed approach is tested and validated with a univariate time series of the Lorenz system, five non-model data sets and tested against a cubic spline interpolation and a linear linear interpolation. We find that the criterion for smoothness guarantees low errors on known model and non-model data. Finally, we interpolate the discussed non-model data sets, and show the corresponding improved phase space portraits. The proposed method is useful for interpolating low-sampled time series data sets for, e.g., machine learning, regression analysis, or time series prediction approaches.