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  1. Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in Fourier/Wavelet space, so the difficulty for solving the coupled PDEs depends on dealing with the coupled mappings between the functions. Towards this end, we propose a coupled multiwavelets neural operator (CMWNO) learning scheme by decoupling the coupled integral kernels during the multiwavelet decomposition and reconstruction procedures in the Wavelet space. The proposed model achieves significantly higher accuracy compared to previous learning-based solvers in solving the coupled PDEs including Gray-Scott (GS) equations and the non-local mean field game (MFG) problem. According to our experimental results, the proposed model exhibits a 2ˆ „ 4ˆ improvement relative L2 error compared to the best results from the state-of-the-art models. 
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  2. Purpose: To improve the image reconstruction for prospective motion correction (PMC) of simultaneous multislice (SMS) EPI of the brain, an update of receiver phase and resampling of coil sensitivities are proposed and evaluated. Methods: A camera-based system was used to track head motion (3 translations and 3 rotations) and dynamically update the scan position and orientation. We derived the change in receiver phase associated with a shifted field of view (FOV) and applied it in real-time to each k-space line of the EPI readout trains. Second, for the SMS reconstruction, we adapted resampled coil sensitivity profiles reflecting the movement of slices. Single-shot gradient-echo SMS-EPI scans were performed in phantoms and human subjects for validation. Results: Brain SMS-EPI scans in the presence of motion withPMCand no phase correction for scan plane shift showed noticeable artifacts. These artifacts were visually and quantitatively attenuated when corrections were enabled. Correcting misaligned coil sensitivity maps improved the temporal SNR (tSNR) of time series by 24% (p=0.0007) for scans with large movements (up to ∼35mm and 30◦). Correcting the receiver phase improved the tSNR of a scan with minimal head movement by 50% from 50 to 75 for a United Kingdom biobank protocol. Conclusion: Reconstruction-induced motion artifacts in single-shot SMS-EPI scans acquired with PMC can be removed by dynamically adjusting the receiver phase of each line across EPI readout trains and updating coil sensitivity profiles during reconstruction. The method may be a valuable tool for SMS-EPI scans in the presence of subject motion. 
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  3. Bentley, Barry L. (Ed.)
    The nematode Caenorhabditis elegans ( C. elegans ) is a model organism used frequently in developmental biology and neurobiology [White, (1986), Sulston, (1983), Chisholm, (2016) and Rapti, (2020)]. The C. elegans embryo can be used for cell tracking studies to understand how cell movement drives the development of specific embryonic tissues. Analyses in late-stage development are complicated by bouts of rapid twitching motions which invalidate traditional cell tracking approaches. However, the embryo possesses a small set of cells which may be identified, thereby defining the coiled embryo’s posture [Christensen, 2015]. The posture serves as a frame of reference, facilitating cell tracking even in the presence of twitching. Posture identification is nevertheless challenging due to the complete repositioning of the embryo between sampled images. Current approaches to posture identification rely on time-consuming manual efforts by trained users which limits the efficiency of subsequent cell tracking. Here, we cast posture identification as a point-set matching task in which coordinates of seam cell nuclei are identified to jointly recover the posture. Most point-set matching methods comprise coherent point transformations that use low order objective functions [Zhou, (2016) and Zhang, (2019)]. Hypergraphs, an extension of traditional graphs, allow more intricate modeling of relationships between objects, yet existing hypergraphical point-set matching methods are limited to heuristic algorithms which do not easily scale to handle higher degree hypergraphs [Duchenne, (2010), Chertok, (2010) and Lee, (2011)]. Our algorithm, Exact Hypergraph Matching ( EHGM ), adapts the classical branch-and-bound paradigm to dynamically identify a globally optimal correspondence between point-sets under an arbitrarily intricate hypergraphical model. EHGM with hypergraphical models inspired by C. elegans embryo shape identified posture more accurately (56%) than established point-set matching methods (27%), correctly identifying twice as many sampled postures as a leading graphical approach. Posterior region seeding empowered EHGM to correctly identify 78% of postures while reducing runtime, demonstrating the efficacy of the method on a cutting-edge problem in developmental biology. 
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  4. Normalizing flows provide an elegant approach to generative modeling that allows for efficient sampling and exact density evaluation of unknown data distributions. However, current techniques have significant limitations in their expressivity when the data distribution is supported on a lowdimensional manifold or has a non-trivial topology. We introduce a novel statistical framework for learning a mixture of local normalizing flows as “chart maps” over the data manifold. Our framework augments the expressivity of recent approaches while preserving the signature property of normalizing flows, that they admit exact density evaluation. We learn a suitable atlas of charts for the data manifold via a vector quantized autoencoder (VQ-AE) and the distributions over them using a conditional flow. We validate experimentally that our probabilistic framework enables existing approaches to better model data distributions over complex manifolds. 
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  5. Image reconstructions involving neural networks (NNs) are generally non-iterative and computationally efficient. However, without analytical expression describing the reconstruction process, the computation of noise propagation becomes difficult. Automated differentiation allows rapid computation of derivatives without an analytical expression. In this work, the feasibility of computing noise propagation with automated differentiation was investigated. The noise propagation of image reconstruction by End-to-end variational-neural-network was estimated using automated differentiation and compared with Monte-Carlo simulation. The root-mean-square error (RMSE) map showed great agreement between automated differentiation and Monte-Carlo simulation over a wide range of SNRs. 
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  6. Time-evolution of partial differential equations is fundamental for modeling several complex dynamical processes and events forecasting, but the operators associated with such problems are non-linear. We propose a Pad´e approximation based exponential neural operator scheme for efficiently learning the map between a given initial condition and the activities at a later time. The multiwavelets bases are used for space discretization. By explicitly embedding the exponential operators in the model, we reduce the training parameters and make it more data-efficient which is essential in dealing with scarce and noisy real-world datasets. The Pad´e exponential operator uses a recurrent structure with shared parameters to model the non-linearity compared to recent neural operators that rely on using multiple linear operator layers in succession. We show theoretically that the gradients associated with the recurrent Pad´e network are bounded across the recurrent horizon. We perform experiments on non-linear systems such as Korteweg-de Vries (KdV) and Kuramoto–Sivashinsky (KS) equations to show that the proposed approach achieves the best performance and at the same time is data-efficient. We also show that urgent real-world problems like epidemic forecasting (for example, COVID- 19) can be formulated as a 2D time-varying operator problem. The proposed Pad´e exponential operators yield better prediction results (53% (52%) better MAE than best neural operator (non-neural operator deep learning model)) compared to state-of-the-art forecasting models. 
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  7. We address the problem of graph regression using graph convolutional neural networks and permutation invariant representation. Many graph neural network algorithms can be abstracted as a series of message passing functions between the nodes, ultimately producing a set of latent features for each node. Processing these latent features to produce a single estimate over the entire graph is dependent on how the nodes are ordered in the graph’s representation. We propose a permutation invariant mapping that produces graph representations that are invariant to any ordering of the nodes. This mapping can serve as a pivotal piece in leveraging graph convolutional networks for graph classification and graph regression problems. We tested out this method and validated our solution on the QM9 dataset. 
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  8. Image reconstructions involving neural networks (NNs) are generally non-iterative and computationally efficient. However, without analytical expression describing the reconstruction process, the computation of noise propagation becomes difficult. Automated differentiation allows rapid computation of derivatives without an analytical expression. In this work, the feasibility of computing noise propagation with automated differentiation was investigated. The noise propagation of image reconstruction by the End-to-end variational-neural-network was estimated using automated differentiation and compared with Monte-Carlo simulation. The root-mean-square error (RMSE) map showed great agreement between automated differentiation and Monte-Carlo simulation over a wide range of SNRs. 
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  9. Time-evolution of partial differential equations is fundamental for modeling several complex dynamical processes and events forecasting, but the operators associated with such problems are non-linear. We propose a Pad´e approximation based exponential neural operator scheme for efficiently learning the map between a given initial condition and the activities at a later time. The multiwavelets bases are used for space discretization. By explicitly embedding the exponential operators in the model, we reduce the training parameters and make it more data-efficient which is essential in dealing with scarce and noisy real-world datasets. The Pad´e exponential operator uses a recurrent structure with shared parameters to model the non-linearity compared to recent neural operators that rely on using multiple linear operator layers in succession. We show theoretically that the gradients associated with the recurrent Pad´e network are bounded across the recurrent horizon. We perform experiments on non-linear systems such as Korteweg-de Vries (KdV) and Kuramoto–Sivashinsky (KS) equations to show that the proposed approach achieves the best performance and at the same time is data-efficient. We also show that urgent real-world problems like epidemic forecasting (for example, COVID- 19) can be formulated as a 2D time-varying operator problem. The proposed Pad´e exponential operators yield better prediction results (53% (52%) better MAE than best neural operator (non-neural operator deep learning model)) compared to state-of-the-art forecasting models. 
    more » « less