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1. Discovering Nonlinear Dynamics Through Scientific Machine Learning

   https://doi.org/10.1007/978-3-030-82193-7_17  

   Huang, Lei; Vrinceanu, Daniel; Wang, Yunjiao; Kulathunga, Nalinda; Ranasinghe, Nishath (August 2021, IntelliSys 2021: Intelligent Systems and Applications)

   Arai, Kohei (Ed.)

   Scientific Machine Learning (SciML) is a new multidisciplinary methodology that combines the data-driven machine learning models and the principle-based computational models to improve the simulations of scientific phenomenon and uncover new scientific rules from existing measurements. This article reveals the experience of using the SciML method to discover the nonlinear dynamics that may be hard to model or be unknown in the real-world scenario. The SciML method solves the traditional principle-based differential equations by integrating a neural network to accurately model the nonlinear dynamics while respecting the scientific constraints and principles. The paper discusses the latest SciML models and apply them to the oscillator simulations and experiment. Besides better capacity to simulate, and match with the observation, the results also demonstrate a successful discovery of the hidden physics in the pendulum dynamics using SciML. 

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2. Symbolic-numeric integration of univariate expressions based on sparse regression

   https://doi.org/10.1145/3572867.3572882  

   Iravanian, Shahriar; Martensen, Carl Julius; Cheli, Alessandro; Gowda, Shashi; Jain, Anand; Ma, Yingbo; Rackauckas, Chris (June 2022, ACM Communications in Computer Algebra)

   The majority of computer algebra systems (CAS) support symbolic integration using a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Our method is broadly similar to the Risch-Norman algorithm. The primary motivation for this work is to add symbolic integration functionality to a modern CAS (the symbolic manipulation packages of SciML, the Scientific Machine Learning ecosystem of the Julia programming language), which is designed for numerical and machine learning applications. The symbolic part of our method is based on the combination of candidate terms generation (ansatz generation using a methodology borrowed from the Homotopy operators theory) combined with rule-based expression transformations provided by the underlying CAS. The numeric part uses sparse regression, a component of the Sparse Identification of Nonlinear Dynamics (SINDy) technique, to find the coefficients of the candidate terms. We show that this system can solve a large variety of common integration problems using only a few dozen basic integration rules. 

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3. Accelerating Simulation of Stiff Nonlinear Systems using Continuous-Time Echo State Networks

   Anantharaman, Ranjan; Ma, Yingbo; Gowda, Shashi; Laughman, Chris; Shah, Viral; Edelman, Alan; Rackauckas, Chris (January 2021, Proceedings of the AAAI 2021 Spring Symposium on Combining Artificial Intelligence and Machine Learning with Physical Sciences)

   Lee, Jonghyun; Darve, Eric F.; Kitanidis, Peter K.; Mahoney, Michael W.; Karpatne, Anuj; Farthing, Matthew W.; Hesser, Tyler (Ed.)

   Modern design, control, and optimization often require multiple expensive simulations of highly nonlinear stiff models. These costs can be amortized by training a cheap surrogate of the full model, which can then be used repeatedly. Here we present a general data-driven method, the continuous time echo state network (CTESN), for generating surrogates of nonlinear ordinary differential equations with dynamics at widely separated timescales. We empirically demonstrate the ability to accelerate a physically motivated scalable model of a heating system by 98x while maintaining relative error of within 0.2 %. We showcase the ability for this surrogate to accurately handle highly stiff systems which have been shown to cause training failures with common surrogate methods such as Physics-Informed Neural Networks (PINNs), Long Short Term Memory (LSTM) networks, and discrete echo state networks (ESN). We show that our model captures fast transients as well as slow dynamics, while demonstrating that fixed time step machine learning techniques are unable to adequately capture the multi-rate behavior. Together this provides compelling evidence for the ability of CTESN surrogates to predict and accelerate highly stiff dynamical systems which are unable to be directly handled by previous scientific machine learning techniques. 

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4. Panoramic Mapping of Phonon Transport from Ultrafast Electron Diffraction and Scientific Machine Learning

   https://doi.org/10.1002/adma.202206997  

   Chen, Zhantao; Shen, Xiaozhe; Andrejevic, Nina; Liu, Tongtong; Luo, Duan; Nguyen, Thanh; Drucker, Nathan_C; Kozina, Michael_E; Song, Qichen; Hua, Chengyun; et al (November 2022, Advanced Materials)

   Abstract One central challenge in understanding phonon thermal transport is a lack of experimental tools to investigate frequency‐resolved phonon transport. Although recent advances in computation lead to frequency‐resolved information, it is hindered by unknown defects in bulk regions and at interfaces. Here, a framework that can uncover microscopic phonon transport information in heterostructures is presented, integrating state‐of‐the‐art ultrafast electron diffraction (UED) with advanced scientific machine learning (SciML). Taking advantage of the dual temporal and reciprocal‐space resolution in UED, and the ability of SciML to solve inverse problems involving coupled Boltzmann transport equations, the frequency‐dependent interfacial transmittance and frequency‐dependent relaxation times of the heterostructure from the diffraction patterns are reliably recovered. The framework is applied to experimental Au/Si UED data, and a transport pattern beyond the diffuse mismatch model is revealed, which further enables a direct reconstruction of real‐space, real‐time, frequency‐resolved phonon dynamics across the interface. The work provides a new pathway to probe interfacial phonon transport mechanisms with unprecedented details. 

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5. Physics-Based Regression vs. CFD for Hagen-Poiseuille and Womersley Flows and Uncertainty Quantification

   Li, H.; Islam, M.; Yu, H.; Du, X. (July 2022, Eleventh International Conference on Computational Fluid Dynamics (ICCFD11))

   Brehm, Christoph; Pandya, Shishir (Ed.)

   Computational fluid dynamics (CFD) and its uncertainty quantification are computationally expensive. We use Gaussian Process (GP) methods to demonstrate that machine learning can build efficient and accurate surrogate models to replace CFD simulations with significantly reduced computational cost without compromising the physical accuracy. We also demonstrate that both epistemic uncertainty (machine learning model uncertainty) and aleatory uncertainty (randomness in the inputs of CFD) can be accommodated when the machine learning model is used to reveal fluid dynamics. The demonstration is performed by applying simulation of Hagen-Poiseuille and Womersley flows that involve spatial and spatial-tempo responses, respectively. Training points are generated by using the analytical solutions with evenly discretized spatial or spatial-temporal variables. Then GP surrogate models are built using supervised machine learning regression. The error of the GP model is quantified by the estimated epistemic uncertainty. The results are compared with those from GPU-accelerated volumetric lattice Boltzmann simulations. The results indicate that surrogate models can produce accurate fluid dynamics (without CFD simulations) with quantified uncertainty when both epistemic and aleatory uncertainties exist. 

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