Diffeomorphic Latent Neural Operators for Data-Efficient Learning of Solutions to Partial Differential Equations

A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these solution generators after training on high-fidelity ground truth data (e.g. numerical simulations). However, in order to generalize well to unseen spatial domains, neural operators must be trained on an extensive amount of geometrically varying data samples that may not be feasible to acquire or simulate in certain contexts (e.g., patient-specific medical data, large-scale computationally intensive simulations.) We propose that in order to learn a PDE solution operator that can generalize across multiple domains without needing to sample enough data expressive enough for all possible geometries, we can train instead a latent neural operator on just a few ground truth solution fields diffeomorphically mapped from different geometric/spatial domains to a fixed reference configuration. Furthermore, the form of the solutions is dependent on the choice of mapping to and from the reference domain. We emphasize that preserving properties of the differential operator when constructing these mappings can significantly reduce the data requirement for achieving an accurate model due to the regularity of the solution fields that the latent neural operator is training on. We provide motivating numerical experimentation that demonstrates an extreme case of this consideration by exploiting the conformal invariance of the Laplacian

Diffeomorphic Latent Neural Operator Learning for Data-Efficient Predictions of Solutions to Partial Differential Equations

A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these solution generators after training on high-fidelity ground truth data (e.g. numerical simulations). However, in order to generalize well to unseen spatial domains, neural operators must be trained on an extensive amount of geometrically varying data samples that may not be feasible to acquire or simulate in certain contexts (i.e., patient-specific medical data, large-scale computationally intensive simulations.) We propose that in order to learn a PDE solution operator that can generalize across multiple domains without needing to sample enough data expressive enough for all possible geometries, we can train instead a latent neural operator on just a few ground truth solution fields diffeomorphically mapped from different geometric/spatial domains to a fixed reference configuration. Furthermore, the form of the solutions is dependent on the choice of mapping to and from the reference domain. We emphasize that preserving properties of the differential operator when constructing these mappings can significantly reduce the data requirement for achieving an accurate model due to the regularity of the solution fields that the latent neural operator is training on. We provide motivating numerical experimentation that demonstrates an extreme case of this consideration by exploiting the conformal invariance of the Laplacian

A novel open-source ultrasound dataset with deep learning benchmarks for spinal cord injury localization and anatomical segmentation

While deep learning has catalyzed breakthroughs across numerous domains, its broader adoption in clinical settings is inhibited by the costly and time-intensive nature of data acquisition and annotation. To further facilitate medical machine learning, we present an ultrasound dataset of 10,223 Brightness-mode (B-mode) images consisting of sagittal slices of porcine spinal cords (N=25) before and after a contusion injury. We additionally benchmark the performance metrics of several state-of-the-art object detection algorithms to localize the site of injury and semantic segmentation models to label the anatomy for comparison and creation of task-specific architectures. Finally, we evaluate the zero-shot generalization capabilities of the segmentation models on human ultrasound spinal cord images to determine whether training on our porcine dataset is sufficient for accurately interpreting human data. Our results show that the YOLOv8 detection model outperforms all evaluated models for injury localization, achieving a mean Average Precision (mAP50-95) score of 0.606. Segmentation metrics indicate that the DeepLabv3 segmentation model achieves the highest accuracy on unseen porcine anatomy, with a Mean Dice score of 0.587, while SAMed achieves the highest Mean Dice score generalizing to human anatomy (0.445). To the best of our knowledge, this is the largest annotated dataset of spinal cord ultrasound images made publicly available to researchers and medical professionals, as well as the first public report of object detection and segmentation architectures to assess anatomical markers in the spinal cord for methodology development and clinical applications.

Unidirectional brain-computer interface: Artificial neural network encoding natural images to fMRI response in the visual cortex

While significant advancements in artificial intelligence (AI) have catalyzed progress across various domains, its full potential in understanding visual perception remains underexplored. We propose an artificial neural network dubbed VISION, an acronym for "Visual Interface System for Imaging Output of Neural activity," to mimic the human brain and show how it can foster neuroscientific inquiries. Using visual and contextual inputs, this multimodal model predicts the brain's functional magnetic resonance imaging (fMRI) scan response to natural images. VISION successfully predicts human hemodynamic responses as fMRI voxel values to visual inputs with an accuracy exceeding state-of-the-art performance by 45%. We further probe the trained networks to reveal representational biases in different visual areas, generate experimentally testable hypotheses, and formulate an interpretable metric to associate these hypotheses with cortical functions. With both a model and evaluation metric, the cost and time burdens associated with designing and implementing functional analysis on the visual cortex could be reduced. Our work suggests that the evolution of computational models may shed light on our fundamental understanding of the visual cortex and provide a viable approach toward reliable brain-machine interfaces.