Compressed Sensing MRI reconstructs images of the body’s internal anatomy from undersampled measurements, thereby reducing scan time—the time subjects need to remain still. Recently, deep learning has shown great potential for reconstructing high-fidelity images from highly undersampled measurements. However, one needs to train multiple models for different undersampling patterns and desired output image resolutions, since most networks operate on a fixed discretization. Such approaches are highly impractical in clinical settings, where undersampling patterns and image resolutions are frequently changed to accommodate different real-time imaging and diagnostic requirements.We propose a unified MRI reconstruction model robust to various measurement undersampling patterns and image resolutions. Our approach uses neural operators—a discretization-agnostic architecture applied in both image and measurement spaces—to capture local and global features. Empirically, our model improves SSIM by 11% and PSNR by 4 dB over a state-of-the-art CNN (End-to-End VarNet), with 600× faster inference than diffusion methods. The resolution-agnostic design also enables zero-shot super-resolution and extended field-of-view reconstruction, offering a versatile and efficient solution for clinical MR imaging. Our unified model offers a versatile solution for MRI, adapting seamlessly to various measurement undersampling and imaging resolutions, making it highly effective for flexible and reliable clinical imaging. Our code is available at https://armeet.ca/nomri.

Neural operators learn mappings between function spaces, which is practical for learning solution operators of PDEs and other scientific modeling applications. Among them, the Fourier neural operator (FNO) is a popular architecture that performs global convolutions in the Fourier space. However, such global operations are often prone to over-smoothing and may fail to capture local details. In contrast, convolutional neural networks (CNN) can capture local features but are limited to training and inference at a single resolution. In this work, we present a principled approach to operator learning that can capture local features under two frameworks by learning differential operators and integral operators with locally supported kernels. Specifically, inspired by stencil methods, we prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions. Both these approaches preserve the properties of operator learning and, hence, the ability to predict at any resolution. Adding our layers to FNOs significantly improves their performance, reducing the relative L2-error by 34-72% in our experiments, which include a turbulent 2D Navier-Stokes and the spherical shallow water equations.

Chaotic systems are notoriously challenging to predict because of their sensitivity to perturbations and errors due to time stepping. Despite this unpredictable behavior, for many dissipative systems the statistics of the long term trajectories are governed by an invariant measure supported on a set, known as the global attractor; for many problems this set is finite dimensional, even if the state space is infinite dimensional. For Markovian systems, the statistical properties of long-term trajectories are uniquely determined by the solution operator that maps the evolution of the system over arbitrary positive time increments. In this work, we propose a machine learning framework to learn the underlying solution operator for dissipative chaotic systems, showing that the resulting learned operator accurately captures short-time trajectories and long-time statistical behavior. Using this framework, we are able to predict various statistics of the invariant measure for the turbulent Kolmogorov Flow dynamics with Reynolds numbers up to $5000$.

Identifying the location of the ice-bedrock interface of glaciers and ice sheets is crucial for a wide range of geophysical applications, such as searching for liquid water in basal regions and computing ice thickness to quantify ice sheet and glacier mass balance. Simple, record-by-record, approaches to detecting the bottom of the ice echo may be affected by spurious off-nadir noise that requires significant manual interaction to correct. In this paper, we propose a deep learning model based on convolutional neural networks (CNNs) and continuous conditional random fields (CCRFs) to automate ice bed identification and better capture fine-grained basal detail. We deploy this approach on HiCARS radargrams, the first time deep learning methods have been applied to this dataset. Intuitively, our CNN captures the global geometry of the ice bed, while the CCRF adjusts initial CNN outputs to better incorporate fine-scale spatial information into the final prediction. We also develop a coherent geophysical framework using three echo characters (along-track continuity, relative delay, and signal coherency) to compare our model’s outputs with those of a manually targeted approach. Our analysis suggests that our CNN + CCRF model is as suitable as the manual approach for radiometric applications, and it outperforms the manual technique in identifying the first continuous return, which is most often the near-nadir reflection. Thus, our approach is more universal than the current manual labeling methodology in the range of geophysical applications where it can be used, and it provides better confidence regarding the source of the basal return detected.