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

The solution of a PDE over varying initial/boundary conditions on multiple domains is needed in a wide variety of applications, but it is computationally expensive if the solution is computed de novo whenever the initial/boundary conditions of the domain change. We introduce a general operator learning framework, called DIffeomorphic Mapping Operator learNing (DIMON) to learn approximate PDE solutions over a family of domains $\{\Omega_{\theta}}_\theta$, that learns the map from initial/boundary conditions and domain $\Omega_\theta$ to the solution of the PDE, or to specified functionals thereof. DIMON is based on transporting a given problem (initial/boundary conditions and domain $\Omega_{\theta}$) to a problem on a reference domain $\Omega_{0}$, where training data from multiple problems is used to learn the map to the solution on $\Omega_{0}$, which is then re-mapped to the original domain $\Omega_{\theta}$. We consider several problems to demonstrate the performance of the framework in learning both static and time-dependent PDEs on non-rigid geometries; these include solving the Laplace equation, reaction-diffusion equations, and a multiscale PDE that characterizes the electrical propagation on the left ventricle. This work paves the way toward the fast prediction of PDE solutions on a family of domains and the application of neural operators in engineering and precision medicine.

Diffusion MRI (dMRI) provides the ability to reconstruct neuronal fibers in the brain, $\textit{in vivo}$, by measuring water diffusion along angular gradient directions in $q$-space. High angular resolution diffusion imaging (HARDI) can produce better estimates of fiber orientation than the popularly used diffusion tensor imaging, but the high number of samples needed to estimate diffusivity requires lengthy patient scan times. To accelerate dMRI, compressed sensing (CS) has been utilized by exploiting a sparse dictionary representation of the data, discovered through sparse coding. The sparser the representation, the fewer samples are needed to reconstruct a high resolution signal with limited information loss, and so an important area of research has focused on finding the sparsest possible representation of dMRI. Current reconstruction methods however, rely on an angular representation $\textit{per voxel}$ with added spatial regularization, and so, the global level of sparsity can be no less than the number of voxels. Therefore, state-of-the-art dMRI CS frameworks may have a fundamental limit to the rate acceleration that can be achieved. In contrast, we propose a joint spatial-angular representation of dMRI that will allow us to achieve levels of global sparsity that are below the number of voxels. A major challenge, however, is the computational complexity of solving a global sparse coding problem over large-scale dMRI. In this work, we present novel adaptations of popular sparse coding algorithms that become better suited for solving large-scale problems by exploiting spatial-angular separability. Our experiments show that our method achieves significantly sparser representations of HARDI than the state-of-the-art which has the potential to increase HARDI acceleration to new levels.