Deep Multi-Resolution Operator Networks (DMON): Exploring Novel Data-Driven Strategies for Chaotic Inverse Problems

Inverse problems, foundational in applied sciences, involve deducing system inputs from specific output observations. These problems find applications in diverse domains such as aerospace engineering, weather prediction, and oceanography. However, their solution often requires complex numerical simulations and substantial computational resources. Modern machine learning based approaches have emerged as an alternative and flexible methodology for solving these types of problems, however their generalization power often comes at the cost of working with large descriptive datasets, a requirement that many applications cannot afford. This thesis proposes and explores the novel Deep Multi-resolution Operator Network (DMON), inspired by the recently developed DeepONet architecture. The DMON model is designed to solve inverse problems related to chaotic non-linear systems with low-resolution data through intelligently utilizing high-resolution data from a similar system. Performance of the DMON model and the proposed selection mechanisms are evaluated on two chaotic systems, a double pendulum and turbulent flow around a cylinder, with improvements observed under idealized scenarios whereby high and low-resolution inputs are manually paired, along with minor improvements when this pairing is conducted through the proposed the latent space comparison selection mechanism.