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arXiv 2023年

作者： Woodward, Michael Tian, Yifeng Mohan, Arvind Lin, Yen Ting Hader, Christoph Fasel, Hermann Chertkov, Misha Livescu, Daniel Graduate Interdisciplinary Program in Applied Mathematics Department of Mathematics University of Arizona TucsonAZ85721 United States Computer Computational and Statistical Sciences Division Los Alamos National Laboratory Los AlamosNM87544 United States Department of Aerospace and Mechanical Engineering University of Arizona TucsonAZ85721 United States

In this work, we apply, for the first time to spatially inhomogeneous flows, a recently developed data-driven learning algorithm of Mori-Zwanzig (MZ) operators, which is based on a generalized Koopman’s description of dynamical systems. The MZ formalism provides a mathematically exact procedure for constructing non-Markovian reduced-order models of resolved variables from high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. The algorithm developed in this work applies Mori’s linear projection operator and an SVD based compression to the selection of the resolved variables (equivalently, a low rank approximation of the two time covariance matrices). We show that this MZ decomposition not only identifies the same spatio-temporal structures found by DMD, but it can also be used to extract spatio-temporal structures of the hysteresis effects present in the memory kernels. We perform an analysis of these structures in the context of a laminar-turbulent boundary-layer transition flow over a flared cone at Mach 6, and show the dynamical relevance of the memory kernels. Additionally, by including these memory terms learned in our data-driven MZ approach, we show improvement in prediction accuracy over DMD at the same level of truncation and at a similar computational cost. Furthermore, an analysis of the spatio-temporal structures of the MZ operators shows identifiable structures associated with the nonlinear generation of the so-called "hot" streaks on the surface of the flared cone, which have previously been observed in experiments and direct numerical simulations. Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamic mode decomposition

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Mori–Zwanzig Modal Decomposition

arXiv

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arXiv 2023年

作者： Woodward, Michael Tian, Yifeng Lin, Yen Ting Hader, Christoph Fasel, Hermann Livescu, Daniel Graduate Interdisciplinary Program in Applied Mathematics Department of Mathematics University of Arizona TucsonAZ85721 United States Computer Computational and Statistical Sciences Division Los Alamos National Laboratory Los AlamosNM87544 United States Department of Aerospace and Mechanical Engineering University of Arizona TucsonAZ85721 United States

We introduce the Mori-Zwanzig (MZ) Modal Decomposition (MZMD), a novel technique for performing modal analysis of large scale spatio-temporal structures in complex dynamical systems, and show that it represents an efficient generalization of Dynamic Mode Decomposition (DMD). The MZ formalism provides a mathematical framework for constructing non-Markovian reduced-order models of resolved variables from high-dimensional dynamical systems, incorporating the effects of unresolved dynamics through the memory kernel and orthogonal dynamics. We present a formulation and analysis of the modes and spectrum from MZMD and compare it to DMD when applied to a complex flow: a Direct Numerical Simulation (DNS) data-set of laminar-turbulent boundary-layer transition flow over a flared cone at Mach 6. We show that the addition of memory terms by MZMD improves the resolution of spatio-temporal structures within the transitional/turbulent regime, which contains features that arise due to nonlinear mechanisms, such as the generation of the so-called "hot" streaks on the surface of the flared cone. As a result, compared to DMD, MZMD improves future state prediction accuracy, while requiring nearly the same computational cost. Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamic mode decomposition

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Interacting models for twisted bilayer graphene: A quantum chemistry approach

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Physical Review B 2023年第23期107卷 235123-235123页

作者： Fabian M. Faulstich Kevin D. Stubbs Qinyi Zhu Tomohiro Soejima Rohit Dilip Huanchen Zhai Raehyun Kim Michael P. Zaletel Garnet Kin-Lic Chan Lin Lin Department of Mathematics University of California Berkeley California 94720 USA Department of Physics University of California Berkeley California 94720 USA Division of Engineering and Applied Science California Institute of Technology Pasadena California 91125 United States Division of Chemistry and Chemical Engineering California Institute of Technology Pasadena California 91125 USA Materials Science Division Lawrence Berkeley National Laboratory Berkeley California 94720 USA Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory Berkeley California 94720 USA

The nature of correlated states in twisted bilayer graphene (TBG) at the magic angle has received intense attention in recent years. We present a numerical study of an interacting Bistritzer-MacDonald (IBM) model of TBG using a suite of methods in quantum chemistry, including Hartree-Fock, coupled cluster singles, doubles (CCSD), and perturbative triples [CCSD(T)], as well as a quantum chemistry formulation of the density matrix renormalization group method (DMRG). Our treatment of TBG is agnostic to gauge choices, and hence we present a new gauge-invariant formulation to detect the spontaneous symmetry breaking in interacting models. To benchmark our approach, we focus on a simplified spinless, valleyless IBM model. At integer filling (ν=0), all numerical methods agree in terms of energy and C2zT symmetry breaking. Additionally, as part of our benchmarking, we explore the impact of different schemes for removing “double-counting” in the IBM model. Our results at integer filling suggest that cross validation of different IBM models may be needed for future studies of the TBG system. After benchmarking our approach at integer filling, we perform the first systematic study of the IBM model near integer filling (for |ν|<0.2). In this regime, we find that the ground state can be in a metallic and C2zT symmetry breaking phase. The ground state appears to have low entropy, and therefore can be relatively well approximated by a single Slater determinant. Furthermore, we observe many low entropy states with energies very close to the ground-state energy in the near integer filling regime.

关键词： Electronic structure First-principles calculations Bilayer graphene Twisted bilayer graphene Approximation methods for many-body systems Electron-correlation calculations Many-body techniques Mean field theory Quantum chemistry methods Second quantization Symmetries

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Equivalent slip length of flow around a super-hydrophobic cylinder

arXiv

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arXiv 2024年

作者： Li, Zhi-Yong Xiao, Ya-Kang Li, Yan-Cheng Yu, Li Peng, Sai Xiong, Yong-Liang Department of Energy and Power Engineering School of Chemical Engineering and Energy Technology Dongguan University of Technology China National Center for Applied Mathematics in Hunan Xiangtan411105 China School of Naval Engineering Wuxi Institute of Communications Technology Wuxi214151 China School of Civil Engineering and Architecture Southwest University of Science and Technology Mianyang621010 China School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Department of Mechanics Huazhong University of Science and Technology Wuhan430074 China

In this research, a two-dimensional numerical simulation is conducted to determine the equivalent wall slip length for flow around a circular cylinder featuring a superhydrophobic surface. The super-hydrophobic surface is modeled as an alternating distribution of slip and no-slip conditions along the cylinder's surface. The smallest unit of this alternating pattern is referred to as a monomer. The study takes into account the Reynolds number and two critical dimensionless parameters: the gas fraction (GF) and the ratio l/a. GF indicates the proportion of the slip length relative to the total length of the monomer, while l/a denotes the ratio of the monomer length (l) to the cylinder's radius (a). The ranges considered for the Reynolds number, GF, and l/a are from 0.2 to 180, 0.1 to 0.99, and π/80 to π/5, respectively. A dimensionless number, the Knudsen number (Kn), is introduced to measure the ratio between the equivalent slip length (λ) and the cylinder's diameter (D). By equating the integral wall friction resistance on the cylinder surface, a quantitative relationship between the equivalent Kn and the parameters (Re, GF, l/a) is established. A meticulous comparison of flow parameters between the equivalent slip length model and the slip-no-slip scenario reveals that the slip length model is an effective approximation for the slip-no-slip alternating model. © 2024, CC BY.

关键词： Reynolds number

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Fast-slow neural networks for learning singularly perturbed dynamical systems

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Journal of computational Physics 2025年 537卷

作者： Serino, Daniel A. Alvarez Loya, Allen Burby, J.W. Kevrekidis, Ioannis G. Tang, Qi Los Alamos National Laboratory Los AlamosNM United States Department of Physics University of Texas at Austin AustinTX United States Department of Chemical and Biomolecular Engineering and Department of Applied Mathematics and Statistics Johns Hopkins University BaltimoreMD United States School of Computational Science and Engineering Georgia Institute of Technology AtlantaGA United States

Singularly perturbed dynamical systems play a crucial role in climate dynamics and plasma physics. A powerful and well-known tool to address these systems is the Fenichel normal form, which significantly simplifies fast dynamics near slow manifolds through a transformation. However, this normal form is difficult to realize in conventional numerical algorithms. In this work, we explore an alternative way of realizing it through structure-preserving machine learning. Specifically, a fast-slow neural network (FSNN) is proposed for learning data-driven models of singularly perturbed dynamical systems with dissipative fast timescale dynamics. Our method enforces the existence of a trainable, attracting invariant slow manifold as a hard constraint. Closed-form representation of the slow manifold enables efficient integration on the slow time scale and significantly improves prediction accuracy beyond the training data. We demonstrate the FSNN on examples including the Grad moment system, two-scale Lorenz96 equations, and Abraham-Lorentz dynamics modeling radiation reaction of electrons. © 2025 Elsevier Inc.

关键词： Plant shutdowns

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Multi-hump Collapsing Solutions in the Nonlinear Schrödinger Problem: Existence, Stability and Dynamics

arXiv

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arXiv 2025年

作者： Chapman, S. Jon Kavousanakis, Mihalis Charalampidis, Efstathios G. Kevrekidis, Ioannis G. Kevrekidis, Panayotis G. Mathematical Institute University of Oxford AWB ROQ Woodstock Road OxfordOX2 6GG United Kingdom School of Chemical Engineering National Technical University of Athens Athens15780 Greece Department of Mathematics and Statistics Computational Science Research Center San Diego State University San DiegoCA92182-7720 United States Department of Chemical and Biomolecular Engineering Department of Applied Mathematics and Statistics Johns Hopkins University BaltimoreMD21218 United States Department of Mathematics and Statistics University of Massachusetts AmherstMA01003-4515 United States Department of Physics University of Massachusetts AmherstMA01003 United States

In the present work we examine multi-hump solutions of the nonlinear Schrödinger equation in the blowup regime of the one-dimensional model with power law nonlinearity, bearing a suitable exponent of σ > 2. We find that families of such solutions exist for arbitrary pulse numbers, with all of them bifurcating from the critical case of σ = 2. Remarkably, all of them involve "bifurcations from infinity", i.e., the pulses come inward from an infinite distance as the exponent σ increases past the critical point. The position of the pulses is quantified and the stability of the waveforms is also systematically examined in the so-called "co-exploding frame". Both the equilibrium distance between the pulse peaks and the point spectrum eigenvalues associated with the multi-hump configurations are obtained as a function of the blowup rate G theoretically, and these findings are supported by detailed numerical computations. Finally, some prototypical dynamical scenarios are explored, and an outlook towards such multi-hump solutions in higher dimensions is provided. Copyright © 2025, The Authors. All rights reserved.

关键词： Schrodinger equation

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Application of Fuzzy Support Vector Regression to Predict the Kovats Retention Indices of Flavors and Fragrances

Application of Fuzzy Support Vector Regression to Predict th...

引用

2022 International Conference on Electrical engineering and Informatics, ICELTICs 2022

作者： Agustia, Mariza Noviandy, Teuku Rizky Maulana, Aga Suhendra, Rivansyah Muslem, Muslem Sasmita, Novi Reandy Idroes, Ghazi Mauer Rahimah, Souvia Afidh, Razief Perucha Fauzie Subianto, Muhammad Irvanizam, Irvanizam Idroes, Rinaldi Universitas Syiah Kuala Faculty of Mathematics and Natural Sciences Department of Informatics Banda Aceh23111 Indonesia Universitas Teuku Umar Faculty of Engineering Department of Information Technology Aceh Barat 23681 Indonesia Universitas Islam Negeri Ar-Raniry Faculty of Science and Technology Department of Chemistry Banda Aceh23111 Indonesia Universitas Syiah Kuala Faculty of Mathematics and Natural Sciences Computational and Applied Statistics Research Group Department of Statistics Banda Aceh23111 Indonesia Universitas Syiah Kuala Faculty of Engineering Department of Chemical Engineering Banda Aceh23111 Indonesia Universitas Padjadjaran Faculty of Agroindustrial Technology Department of Food Industrial Technology Bandung Indonesia

ISBN: (数字)9781665487665

ISBN: (纸本)9781665487665

The Kovats retention index is the most robust retention parameter used to identify analytes in gas chromatography. The Kovats retention index's value in similar compounds tends to have a similar pattern. This pattern allows a close relationship between chemical structure and the retention index value. The Fuzzy Support Vector Regression (FSVR) method is a robust algorithm that has been used to solve various regression problems. This study aimed to evaluate the FSVR algorithm performance to predict the Kovats retention index of flavor and fragrance compounds. The model was built using the Python programming language. The retention index value calculation begins with inputting the molecular descriptor data into the sigmoid fuzzy membership function. The dataset consists of 51 compounds then partitioned into two sets, namely 80% in the training set and 20% in the testing set. The best parameters to build the model were obtained using hyperparameter tuning. The kernels used are linear kernel and Radial Basis Function (RBF) kernel. The model with a linear kernel with parameter values of C = 1000 and epsilon = 10 managed to get the best results compared to the model with an RBF kernel, with R2 = 0.977 and Root Mean Square Error (RMSE) = 37.21 on the testing set. This model also can predict the Kovats retention index on the testing set with an average difference of 2.8%. These results showed that the linear kernel is better than the RBF kernel in predicting the value of the Kovats retention index. It can be concluded that the FSVR method can be used to predict the retention index of flavor and fragrance compounds. © 2022 IEEE.

关键词： Forecasting

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Reliable Extrapolation of Deep Neural Operators Informed by Physics or Sparse Observations

SSRN

引用

SSRN 2023年

作者： Zhu, Min Zhang, Handi Jiao, Anran Karniadakis, George Em Lu, Lu Department of Chemical and Biomolecular Engineering University of Pennsylvania PhiladelphiaPA19104 United States Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PhiladelphiaPA19104 United States Department of Computer and Information Science University of Pennsylvania PhiladelphiaPA19104 United States Division of Applied Mathematics Brown University ProvidenceRI02912 United States School of Engineering Brown University ProvidenceRI02912 United States

Deep neural operators can learn nonlinear mappings between infinite-dimensional function spaces via deep neural networks. As promising surrogate solvers of partial differential equations (PDEs) for real-time prediction, deep neural operators such as deep operator networks (DeepONets) provide a new simulation paradigm in science and engineering. Pure data-driven neural operators and deep learning models, in general, are usually limited to interpolation scenarios, where new predictions utilize inputs within the support of the training set. However, in the inference stage of real-world applications, the input may lie outside the support, i.e., extrapolation is required, which may result to large errors and unavoidable failure of deep learning models. Here, we address this challenge of extrapolation for deep neural operators. First, we systematically investigate the extrapolation behavior of DeepONets by quantifying the extrapolation complexity via the 2-Wasserstein distance between two function spaces and propose a new strategy of bias-variance trade-off for extrapolation with respect to model capacity. Subsequently, we develop a complete workflow, including extrapolation determination, and we propose five reliable learning methods that guarantee a safe prediction under extrapolation by requiring additional information-the governing PDEs of the system or sparse new observations. The proposed methods are based on either fine-tuning a pre-trained DeepONet or multifidelity learning. We demonstrate the effectiveness of the proposed framework for various types of parametric PDEs. Our systematic comparisons provide practical guidelines for selecting a proper extrapolation method depending on the available information, desired accuracy, and required inference speed. © 2023, The Authors. All rights reserved.

关键词： Extrapolation

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Relative magnetic field and slipping effect on Casson dusty fluid of two phase fluctuating flow over inclined parallel plate

引用

South African Journal of chemical engineering 2023年 44卷 135-146页

作者： khan, Dolat Kumam, Poom Kumam, Wiyada Suttiarporn, Panawan Rehman, Ali Fixed Point Research Laboratory Fixed Point Theory and Applications Research Group Center of Excellence in Theoretical and Computational Science (TaCS-CoE) Faculty of Science King Mongkut's University of Technology Thonburi (KMUTT) 126 Pracha Uthit Rd. Bang Mod Bangkok Thung Khru 10140 Thailand Center of Excellence in Theoretical and Computational Science (TaCS-CoE) Faculty of Science King Mongkut's University of Technology Thonburi (KMUTT) 126 Pracha Uthit Rd. Bang Mod Bangkok Thung Khru 10140 Thailand Applied Mathematics for Science and Engineering Research Unit (AMSERU) Program in Applied Statistics Department of Mathematics and Computer Science Faculty of Science and Technology Rajamangala University of Technology Thanyaburi (RMUTT) Pathum Thani 12110 Thailand Faculty of Science Energy and Environment King Mongkut's University of Technology North Bangkok Rayong Campus Rayong 21120 Thailand Special Interest Group for Modelling and Data Analytics Faculty of Ocean Engineering Technology and Informatics Universiti Malaysia Terengganu Terengganu Kuala Nerus 21030 Malaysia

Due to its significant applications in magnetic devices for cell separation, magnetic drugs for cancer tumor treatment, blood flow adjustment during surgery, magnetic endoscopy, and fluid pumping in industrial and engineering processes, magnetohydrodynamics (MHD) has a wide range of applications. Therefore, the aim of this study to investigate the Relative Magnetic Field and slipping effect on Casson dusty fluid of two-phase fluctuating flow over inclined parallel plate. More exactly, the relative magnetic phenomena are fixed relatively to the plate (MFFRP) or fixed relatively to the fluid (MFFRF). The mathematical modeling was established as the set of partial differential equations for dusty Casson fluid. Buckingham's pi theorem is used to find out the dimensionless variables, to make our system dimensionless. The solution in perturb form is to find out by incorporating Poincare–Lighthill perturbation technique (PLPT). To know in-depth different parameters the graphical results for both velocities are plotted with the help of software Mathcad- 15. It is clearly detected that the relative magnetic field performs a vital part in the fluid as well as particle motion. Moreover, the relative magnetic field play a vital role in the decreasing of skin friction of both plates. © 2023

关键词： Brinkman fluid Dust particles Fluctuating vertical plate Inclined relative magnetic field PLPT

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Identifying Equivalent Training Dynamics

arXiv

引用

arXiv 2023年

作者： Redman, William T. Bello-Rivas, Juan Fonoberova, Maria Mohr, Ryan Kevrekidis, Yannis Mezić, Igor AIMdyn Inc. United States Interdepartmental Program in Dynamical Neuroscience UC Santa Barbara United States Department of Chemical and Biomolecular Engineering Whiting School of Engineering Johns Hopkins University United States Departments of Applied Mathematics and Statistics Johns Hopkins University United States Department of Mechanical Engineering UC Santa Barbara United States Department of Mathematics UC Santa Barbara United States

Study of the nonlinear evolution deep neural network (DNN) parameters undergo during training has uncovered regimes of distinct dynamical behavior. While a detailed understanding of these phenomena has the potential to advance improvements in training efficiency and robustness, the lack of methods for identifying when DNN models have equivalent dynamics limits the insight that can be gained from prior work. Topological conjugacy, a notion from dynamical systems theory, provides a precise definition of dynamical equivalence, offering a possible route to address this need. However, topological conjugacies have historically been challenging to compute. By leveraging advances in Koopman operator theory, we develop a framework for identifying conjugate and non-conjugate training dynamics. To validate our approach, we demonstrate that comparing Koopman eigenvalues can correctly identify a known equivalence between online mirror descent and online gradient descent. We then utilize our approach to: (a) identify non-conjugate training dynamics between shallow and wide fully connected neural networks;(b) characterize the early phase of training dynamics in convolutional neural networks;(c) uncover non-conjugate training dynamics in Transformers that do and do not undergo grokking. Our results, across a range of DNN architectures, illustrate the flexibility of our framework and highlight its potential for shedding new light on training dynamics. © 2023, CC BY.

关键词： Topology

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