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1. Deep Visual Gravity Vector Detection for Unmanned Aircraft Attitude Estimation

   Ellingson, Gary J.; Wingate, David; and McLain, Tim (September 2017, IEEE/RSJ International Conference on Intelligent Robots and Systems)

   This paper demonstrates a feasible method for using a deep neural network as a sensor to estimate the attitude of a flying vehicle using only flight video. A dataset of still images and associated gravity vectors was collected and used to perform supervised learning. The network builds on a previously trained network and was trained to be able to approximate the attitude of the camera with an average error of about 8 degrees. Flight test video was recorded and processed with a relatively simple visual odometry method. The aircraft attitude is then estimated with the visual odometry as the state propagation and network providing the attitude measurement in an extended Kalman filter. Results show that the proposed method of having the neural network provide a gravity vector attitude measurement from the flight imagery reduces the standard deviation of the attitude error by approximately 12 times compared to a baseline approach. 

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2. Efficient Neural Hybrid System Learning and Interpretable Transition System Abstraction for Dynamical Systems

   https://doi.org/10.1115/1.4066516  

   Yang, Yejiang; Mo, Zihao; Xiang, Weiming (January 2025, ASME Letters in Dynamic Systems and Control)

   Abstract This article proposes a neural network hybrid modeling framework for dynamics learning to promote an interpretable, computationally efficient method of dynamics learning and system identification. First, a low-level model is trained to learn the system dynamics, which utilizes multiple simple neural networks to approximate the local dynamics generated from data-driven partitions. Then, based on the low-level model, a high-level model is trained to abstract the low-level neural hybrid system model into a transition system that allows computational tree logic (CTL) verification to promote model’s ability to handle human interaction and verification efficiency. 

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3. Deep Learning Approaches to Surrogates for Solving the Diffusion Equation for Mechanistic Real-World Simulations

   https://doi.org/10.3389/fphys.2021.667828  

   Toledo-Marín, J. Quetzalcóatl; Fox, Geoffrey; Sluka, James P.; Glazier, James A. (June 2021, Frontiers in Physiology)

   In many mechanistic medical, biological, physical, and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs), especially for diffusion, fluid flow and mechanical relaxation, can make simulations impractically slow. Biological models of tissues and organs often require the simultaneous calculation of the spatial variation of concentration of dozens of diffusing chemical species. One clinical example where rapid calculation of a diffusing field is of use is the estimation of oxygen gradients in the retina, based on imaging of the retinal vasculature, to guide surgical interventions in diabetic retinopathy. Furthermore, the ability to predict blood perfusion and oxygenation may one day guide clinical interventions in diverse settings, i.e., from stent placement in treating heart disease to BOLD fMRI interpretation in evaluating cognitive function (Xie et al., 2019 ; Lee et al., 2020 ). Since the quasi-steady-state solutions required for fast-diffusing chemical species like oxygen are particularly computationally costly, we consider the use of a neural network to provide an approximate solution to the steady-state diffusion equation. Machine learning surrogates, neural networks trained to provide approximate solutions to such complicated numerical problems, can often provide speed-ups of several orders of magnitude compared to direct calculation. Surrogates of PDEs could enable use of larger and more detailed models than are possible with direct calculation and can make including such simulations in real-time or near-real time workflows practical. Creating a surrogate requires running the direct calculation tens of thousands of times to generate training data and then training the neural network, both of which are computationally expensive. Often the practical applications of such models require thousands to millions of replica simulations, for example for parameter identification and uncertainty quantification, each of which gains speed from surrogate use and rapidly recovers the up-front costs of surrogate generation. We use a Convolutional Neural Network to approximate the stationary solution to the diffusion equation in the case of two equal-diameter, circular, constant-value sources located at random positions in a two-dimensional square domain with absorbing boundary conditions. Such a configuration caricatures the chemical concentration field of a fast-diffusing species like oxygen in a tissue with two parallel blood vessels in a cross section perpendicular to the two blood vessels. To improve convergence during training, we apply a training approach that uses roll-back to reject stochastic changes to the network that increase the loss function. The trained neural network approximation is about 1000 times faster than the direct calculation for individual replicas. Because different applications will have different criteria for acceptable approximation accuracy, we discuss a variety of loss functions and accuracy estimators that can help select the best network for a particular application. We briefly discuss some of the issues we encountered with overfitting, mismapping of the field values and the geometrical conditions that lead to large absolute and relative errors in the approximate solution. 

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4. Efficient approximation of deep ReLU networks for functions on low dimensional manifolds

   Chen, Minshuo; Jiang, Haomin; Liao, Wenjing; Zhao, Tuo (December 2019, Conference on Neural Information Processing Systems)

   Deep neural networks have revolutionized many real world applications, due to their flexibility in data fitting and accurate predictions for unseen data. A line of research reveals that neural networks can approximate certain classes of functions with an arbitrary accuracy, while the size of the network scales exponentially with respect to the data dimension. Empirical results, however, suggest that networks of moderate size already yield appealing performance. To explain such a gap, a common belief is that many data sets exhibit low dimensional structures, and can be modeled as samples near a low dimensional manifold. In this paper, we prove that neural networks can efficiently approximate functions supported on low dimensional manifolds. The network size scales exponentially in the approximation error, with an exponent depending on the intrinsic dimension of the data and the smoothness of the function. Our result shows that exploiting low dimensional data structures can greatly enhance the efficiency in function approximation by neural networks. We also implement a sub-network that assigns input data to their corresponding local neighborhoods, which may be of independent interest. 

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5. Efficient Approximation of Deep ReLU Networks for Functions on Low Dimensional Manifolds

   Chen, Minshuo; Jiang, Haoming; Liao, Wenjing; Zhao, Tuo (December 2019, Advances in Neural Information Processing Systems)

   Deep neural networks have revolutionized many real world applications, due to their flexibility in data fitting and accurate predictions for unseen data. A line of research reveals that neural networks can approximate certain classes of functions with an arbitrary accuracy, while the size of the network scales exponentially with respect to the data dimension. Empirical results, however, suggest that networks of moderate size already yield appealing performance. To explain such a gap, a common belief is that many data sets exhibit low dimensional structures, and can be modeled as samples near a low dimensional manifold. In this paper, we prove that neural networks can efficiently approximate functions supported on low dimensional manifolds. The network size scales exponentially in the approximation error, with an exponent depending on the intrinsic dimension of the data and the smoothness of the function. Our result shows that exploiting low dimensional data structures can greatly enhance the efficiency in function approximation by neural networks. We also implement a sub-network that assigns input data to their corresponding local neighborhoods, which may be of independent interest. 

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