Natalia A. Trayanova

Minglang Yin, Nicolas Charon, Ryan Brody et al.

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 $\{Ω_θ}_θ$, that learns the map from initial/boundary conditions and domain $Ω_θ$ to the solution of the PDE, or to specified functionals thereof. DIMON is based on transporting a given problem (initial/boundary conditions and domain $Ω_θ$) to a problem on a reference domain $Ω_{0}$, where training data from multiple problems is used to learn the map to the solution on $Ω_{0}$, which is then re-mapped to the original domain $Ω_θ$. 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.

Zan Ahmad, Shiyi Chen, Minglang Yin et al.

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

Md Shakil Zaman, Jwala Dhamala, Pradeep Bajracharya et al.

Probabilistic estimation of cardiac electrophysiological model parameters serves an important step towards model personalization and uncertain quantification. The expensive computation associated with these model simulations, however, makes direct Markov Chain Monte Carlo (MCMC) sampling of the posterior probability density function (pdf) of model parameters computationally intensive. Approximated posterior pdfs resulting from replacing the simulation model with a computationally efficient surrogate, on the other hand, have seen limited accuracy. In this paper, we present a Bayesian active learning method to directly approximate the posterior pdf function of cardiac model parameters, in which we intelligently select training points to query the simulation model in order to learn the posterior pdf using a small number of samples. We integrate a generative model into Bayesian active learning to allow approximating posterior pdf of high-dimensional model parameters at the resolution of the cardiac mesh. We further introduce new acquisition functions to focus the selection of training points on better approximating the shape rather than the modes of the posterior pdf of interest. We evaluated the presented method in estimating tissue excitability in a 3D cardiac electrophysiological model in a range of synthetic and real-data experiments. We demonstrated its improved accuracy in approximating the posterior pdf compared to Bayesian active learning using regular acquisition functions, and substantially reduced computational cost in comparison to existing standard or accelerated MCMC sampling.

Haley G. Abramson, Dan M. Popescu, Rebecca Yu et al.

Visualizing disease-induced scarring and fibrosis in the heart on cardiac magnetic resonance (CMR) imaging with contrast enhancement (LGE) is paramount in characterizing disease progression and quantifying pathophysiological substrates of arrhythmias. However, segmentation and scar/fibrosis identification from LGE-CMR is an intensive manual process prone to large inter-observer variability. Here, we present a novel fully-automated anatomically-informed deep learning solution for left ventricle (LV) and scar/fibrosis segmentation and clinical feature extraction from LGE-CMR. The technology involves three cascading convolutional neural networks that segment myocardium and scar/fibrosis from raw LGE-CMR images and constrain these segmentations within anatomical guidelines, thus facilitating seamless derivation of clinically-significant parameters. In addition to available LGE-CMR images, training used "LGE-like" synthetically enhanced cine scans. Results show excellent agreement with those of trained experts in terms of segmentation (balanced accuracy of $96\%$ and $75\%$ for LV and scar segmentation), clinical features ($2\%$ difference in mean scar-to-LV wall volume fraction), and anatomical fidelity. Our segmentation technology is extendable to other computer vision medical applications and to problems requiring guidelines adherence of predicted outputs.