Topological Approach for Data Assimilation
This addresses the challenge of updating data-driven models in forecasting for fields like meteorology or climate science, though it appears incremental as it builds on existing data assimilation methods with a novel topological twist.
The paper tackles the problem of data assimilation in dynamical systems where measurement noise statistics are unknown, by introducing a new algorithm based on topological data analysis that minimizes topological differences between measurements and forecasts without requiring noise information, demonstrating its performance on the Lorenz 63 and Lorenz 96 systems.
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 63 system as an example and we also show that the method works on a higher dimensional example with the Lorenz 96 system.