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1. 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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2. Multi-grade Deep Learning

   https://doi.org/10.1007/s42967-024-00474-y  

   Xu, Yuesheng (March 2025, Communications on Applied Mathematics and Computation)

   Abstract Deep learning requires solving a nonconvex optimization problem of a large size to learn a deep neural network (DNN). The current deep learning model is of asingle-grade, that is, it trains a DNN end-to-end, by solving a single nonconvex optimization problem. When the layer number of the neural network is large, it is computationally challenging to carry out such a task efficiently. The complexity of the task comes from learning all weight matrices and bias vectors from one single nonconvex optimization problem of a large size. Inspired by the human education process which arranges learning in grades, we propose a multi-grade learning model: instead of solving one single optimization problem of a large size, we successively solve a number of optimization problems of small sizes, which are organized in grades, to learn a shallow neural network (a network having a few hidden layers) for each grade. Specifically, the current grade is to learn the leftover from the previous grade. In each of the grades, we learn a shallow neural network stacked on the top of the neural network, learned in the previous grades, whose parameters remain unchanged in training of the current and future grades. By dividing the task of learning a DDN into learning several shallow neural networks, one can alleviate the severity of the nonconvexity of the original optimization problem of a large size. When all grades of the learning are completed, the final neural network learned is astair-shapeneural network, which is thesuperpositionof networks learned from all grades. Such a model enables us to learn a DDN much more effectively and efficiently. Moreover, multi-grade learning naturally leads to adaptive learning. We prove that in the context of function approximation if the neural network generated by a new grade is nontrivial, the optimal error of a new grade is strictly reduced from the optimal error of the previous grade. Furthermore, we provide numerical examples which confirm that the proposed multi-grade model outperforms significantly the standard single-grade model and is much more robust to noise than the single-grade model. They include three proof-of-concept examples, classification on two benchmark data sets MNIST and Fashion MNIST with two noise rates, which is to find classifiers, functions of 784 dimensions, and as well as numerical solutions of the one-dimensional Helmholtz equation. 

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3. Optimization of the Variable Inertia Rotational Mechanism using Machine Learning

   Sarkar, Anika T.; Wierschem, Nicholas E. (January 2022, Proceedings of the 8th World Conference on Structural Control and Monitoring)

   Rotational inertial mechanisms can produce mass amplification effects with only a small physical mass by converting translation to the rotation of a fly-wheel, which makes them attractive for structural control applications. A variable inertia rotational mechanism (VIRM) is a nonlinear mechanism in which masses in the flywheel can move radially, causing variable inertia. The performance of the VIRM depends on its parameters and the objectives considered. This paper presents the optimum parameters of the VIRM in a single-degree-of-freedom (SDOF) system using an artificial neural network (ANN) model. Optimum VIRM values of several sets of SDOF systems are used to train the ANN model. These values are determined using numerical simulations, and the RMS amplitude of total energy in the system is considered the optimization objective. Numerical simulations of VIRM systems are presented to demonstrate the effective-ness and examine the ANN-based machine learning optimization process's performance. 

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4. Neural network approaches for parameterized optimal control

   https://doi.org/10.3934/fods.2024042  

   Verma, Deepanshu; Winovich, Nick; Ruthotto, Lars; van_Bloemen_Waanders, Bart (January 2025, Foundations of Data Science)

   We consider numerical approaches for deterministic, finite-dimensional optimal control problems whose dynamics depend on unknown or uncertain parameters. We seek to amortize the solution over a set of relevant parameters in an offline stage to enable rapid decision-making and be able to react to changes in the parameter in the online stage. To tackle the curse of dimensionality arising when the state and/or parameter are high-dimensional, we represent the policy using neural networks. We compare two training paradigms: First, our model-based approach leverages the dynamics and definition of the objective function to learn the value function of the parameterized optimal control problem and obtain the policy using a feedback form. Second, we use actor-critic reinforcement learning to approximate the policy in a data-driven way. Using an example involving a two-dimensional convection-diffusion equation, which features high-dimensional state and parameter spaces, we investigate the accuracy and efficiency of both training paradigms. While both paradigms lead to a reasonable approximation of the policy, the model-based approach is more accurate and considerably reduces the number of PDE solves. 

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5. A Network Parameter Database False Data Injection Correction Physics-Based Model: A Machine Learning Synthetic Measurement-Based Approach

   https://doi.org/10.3390/app11178074  

   Zou, Tierui; Aljohani, Nader; Nagaraj, Keerthiraj; Zou, Sheng; Ruben, Cody; Bretas, Arturo; Zare, Alina; McNair, Janise (September 2021, Applied Sciences)

   null (Ed.)

   Concerning power systems, real-time monitoring of cyber–physical security, false data injection attacks on wide-area measurements are of major concern. However, the database of the network parameters is just as crucial to the state estimation process. Maintaining the accuracy of the system model is the other part of the equation, since almost all applications in power systems heavily depend on the state estimator outputs. While much effort has been given to measurements of false data injection attacks, seldom reported work is found on the broad theme of false data injection on the database of network parameters. State-of-the-art physics-based model solutions correct false data injection on network parameter database considering only available wide-area measurements. In addition, deterministic models are used for correction. In this paper, an overdetermined physics-based parameter false data injection correction model is presented. The overdetermined model uses a parameter database correction Jacobian matrix and a Taylor series expansion approximation. The method further applies the concept of synthetic measurements, which refers to measurements that do not exist in the real-life system. A machine learning linear regression-based model for measurement prediction is integrated in the framework through deriving weights for synthetic measurements creation. Validation of the presented model is performed on the IEEE 118-bus system. Numerical results show that the approximation error is lower than the state-of-the-art, while providing robustness to the correction process. Easy-to-implement model on the classical weighted-least-squares solution, highlights real-life implementation potential aspects. 

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