Katarzyna Michalowska: Multi-Resolution Learning with Neural Operators

In real-world applications, collecting large amounts of high-resolution data is rarely practical. The cost of specialized equipment, time-intensive measurements, and expensive high-fidelity simulations often means that high-resolution samples are limited. Meanwhile, low-resolution data are far easier to obtain and often exist in abundance, creating a common scenario: plentiful coarse data but limited fine data. Standard neural networks, which require fixed-resolution inputs, cannot exploit this imbalance and typically perform poorly when generalizing across resolutions.

Deep operator networks, or DeepONets offer a distinct advantage over standard neural networks through a property known as discretization invariance, enabling learning across varying data resolutions. However, DeepONets alone do not effectively capture long-term temporal dependencies, limiting their performance on problems involving long time horizons.

In this talk, we present a framework that addresses both challenges: multi-resolution learning and long-time horizon modelling. We achieve this by extending DeepONets with long short-term memory networks (LSTMs) and introducing a multi-stage training procedure that leverages data at multiple resolutions. This hybrid architecture first learns global dynamics from abundant low-resolution data and then fine-tunes on limited high-resolution samples, capturing both multi-resolution structure and temporal dependencies. In tests on nonlinear dynamical systems, our multi-resolution DON-LSTM achieves lower generalization error and requires fewer high-resolution samples than standard DeepONet or LSTM models. Our results demonstrate that the proposed approach is well-suited for real-world scenarios where high-resolution data are limited, highlighting its potential for practical applications across science and engineering.

Katarzyna Michałowska is a Research Scientist in the Analytics and Artificial Intelligence group in the Department of Mathematics and Cybernetics at SINTEF Digital, Norway. She holds a Ph.D. in physics-informed machine learning from the University of Oslo. Her research focuses on applied machine learning with a focus on physics-informed and robust machine learning solutions. The presented work is a collaboration between SINTEF and the CRUNCH group at the Division of Applied Mathematics at Brown University, USA.