The classical hedging problem entails replicating the payoff of a contingent claim under a certain stochastic model. While we can find a complete hedging strategy in a complete market like Black-Scholes, a market is in general incomplete, including jump diffusion, and stochastic volatility models. While there are several hedging approaches in an incomplete market, it is often very difficult to get a closed form solution or even calculate numerically. Even in a complete market like Black-Scholes, there are drawbacks to this strategy in both execution and the theory it is based on. A traditional asset pricing and hedging method assumes frictionless markets, perfect liquidity, and normally distributed returns among many other conditions.

Hall Effect Thrusters (HETs) are electric thrusters that eject heavy ionized gas particles from the spacecraft to generate thrust. Although traditionally they were used for station keeping, recently They have been used for interplanetary space missions due to their high delta-V potential and their operational longevity in contrast to other thrusters, e.g., chemical. However, the operation of HETs involves complex processes such as ionization of gases, strong magnetic fields, and complicated solar panel power supply interactions. Therefore, their operation is extremely difficult to model thus necessitating Data Assimilation (DA) approaches for estimating and predicting their operational states. Because HET's operating environment is often noisy with non-Gaussian sources, this significantly limits applicable DA tools. We describe a topological approach for data assimilation that bypasses these limitations that does not depend on the noise model, and utilize it to forecast spatiotemporal plume field states of HETs. Our approach is a generalization of the Topological Approach for Data Assimilation (TADA) method that allows including different forecast functions. We show how TADA can be combined with the Long Short-Term Memory network for accurate forecasting. We then apply our approach to high-fidelity Hall Effect Thruster (HET) simulation data from the Air Force Research Laboratory (AFRL) rocket propulsion division where we demonstrate the forecast resiliency of TADA on noise contaminated, high-dimensional data.

Many dynamical systems are difficult or impossible to model using high fidelity physics based models. Consequently, researchers are relying more on data driven models to make predictions and forecasts. Based on limited training data, machine learning models often deviate from the true system states over time and need to be continually updated as new measurements are taken using data assimilation. Classical data assimilation algorithms typically require knowledge of the measurement noise statistics which may be unknown. In this paper, we introduce a new data assimilation algorithm with a foundation in topological data analysis. By leveraging the differentiability of functions of persistence, gradient descent optimization is used to minimize topological differences between measurements and forecast predictions by tuning data driven model coefficients without using noise information from the measurements. We describe the method and focus on its capabilities performance using the chaotic Lorenz system as an example.

Data quality is crucial for the successful training, generalization and performance of artificial intelligence models. Furthermore, it is known that the leading approaches in artificial intelligence are notoriously data-hungry. In this paper, we propose the use of small training datasets towards faster training. Specifically, we provide a novel topological method based on morphisms between persistence modules to measure the training data quality with respect to the complete dataset. This way, we can provide an explanation of why the chosen training dataset will lead to poor performance.

Nowadays, Machine Learning and Deep Learning methods have become the state-of-the-art approach to solve data classification tasks. In order to use those methods, it is necessary to acquire and label a considerable amount of data; however, this is not straightforward in some fields, since data annotation is time consuming and might require expert knowledge. This challenge can be tackled by means of semi-supervised learning methods that take advantage of both labelled and unlabelled data. In this work, we present new semi-supervised learning methods based on techniques from Topological Data Analysis (TDA), a field that is gaining importance for analysing large amounts of data with high variety and dimensionality. In particular, we have created two semi-supervised learning methods following two different topological approaches.

We propose a clustering algorithm which, for input, takes data assumed to be sampled from a uniform distribution supported on a metric space $X$, and outputs a clustering of the data based on a topological estimate of the connected components of $X$. The algorithm works by choosing a weighted graph on the samples from a natural one-parameter family of graphs using an error based on the heat operator on the graphs. The estimated connected components of $X$ are identified as the support of the eigenfunctions of the heat operator with eigenvalue $1$, which allows the algorithm to work without requiring the number of expected clusters as input.