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A PHYSICS-INFORMED NEURAL NETWORK METHOD FOR THE APPROXIMATION OF SLOW INVARIANT MANIFOLDS FOR THE GENERAL CLASS OF STIFF SYSTEMS OF ODES

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arXiv 2024年

作者： Patsatzis, Dimitrios G. Russo, Lucia Siettos, Constantinos Modelling Engineering Risk and Complexity Scuola Superiore Meridionale Naples80138 Italy Institute of Science and Technology for Energy and Sustainable Mobility Consiglio Nazionale delle Ricerche Naples80125 Italy Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" Università degli Studi di Napoli Federico II Naples80126 Italy

We present a physics-informed neural network (PINN) approach for the discovery of slow invariant manifolds (SIMs), for the most general class of fast/slow dynamical systems of ODEs. In contrast to other machine learning (ML) approaches that construct reduced order black box surrogate models using simple regression, and/or require a priori knowledge of the fast and slow variables per se, our approach, simultaneously decomposes the vector field into fast and slow components and provides a functional of the underlying SIM in a closed form. The decomposition is achieved by finding a transformation of the state variables to the fast and slow ones, which enables the derivation of an explicit, in terms of fast variables, SIM functional. The latter is obtained by solving a PDE corresponding to the invariance equation within the Geometric Singular Perturbation Theory (GSPT) using a single-layer feedforward neural network with symbolic differentiation. The performance of the proposed physics-informed ML framework is assessed via three benchmark problems, namely the Michaelis-Menten, the target mediated drug disposition (TMDD) reaction mechanism and, a fully competitive substrate-inhibitor (fCSI) mechanism. We also provide a comparison with other GPST methods, namely the quasi steady state approximation (QSSA), the partial equilibrium approximation (PEA) and CSP with one and two iterations. We show that the proposed PINN scheme provides SIM approximations, of equivalent or even higher accuracy, than those provided by QSSA, PEA and CSP, especially close to the boundaries of the underlying SIMs. Finally, we note that for the TMDD and fCSI problems, the CSP with two iterations (but also QSSA and PEA for the latter problem) does not provide the SIM approximations in an explicit closed form. In these cases, the computational time required for training the proposed PINN is comparable to the one required for the CSP to numerically approximate the SIM with *** Codes

关键词： Dynamical systems

LEARNING THE LATENT DYNAMICS OF FLUID FLOWS FROM HIGH-FIDELITY NUMERICAL SIMULATIONS USING PARSIMONIOUS DIFFUSION MAPS

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arXiv 2024年

作者： Pia, Alessandro Della Patsatzis, Dimitris Russo, Lucia Siettos, Constantinos Modelling Engineering Risk and Complexity Scuola Superiore Meridionale Naples80138 Italy Institute of Science and Technology for Energy and Sustainable Mobility Consiglio Nazionale delle Ricerche Naples80125 Italy Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" Università degli Studi di Napoli Federico II Naples80126 Italy

We use parsimonious diffusion maps (PDMs) to discover the latent dynamics of high-fidelity Navier-Stokes simulations with a focus on the 2D fluidic pinball problem. By varying the Reynolds number, different flow regimes emerge, ranging from steady symmetric flows to quasi-periodic asymmetric and turbulence. We show, that the proposed non-linear manifold learning scheme, identifies in a crisp manner the expected intrinsic dimension of the underlying emerging dynamics over the parameter space. In particular, PDMs, estimate that the emergent dynamics in the oscillatory regime can be captured by just two variables, while in the chaotic regime, the dominant modes are three as anticipated by the normal form theory. On the other hand, proper orthogonal decomposition (POD)/PCA, most commonly used for dimensionality reduction in fluid mechanics, does not provide such a crisp separation between the dominant modes. To validate the performance of PDMs, we also computed the reconstruction error, by constructing a decoder using Geometric Harmonics. We show that the proposed scheme outperforms the POD/PCA over the whole Reynolds number range. Thus, we believe that the proposed scheme will allow for the development of more accurate reduced order models for high-fidelity fluid dynamics simulators, thus relaxing the curse of dimensionality in numerical analysis tasks such as bifurcation analysis, optimization and *** Codes 65Pxx, 68T20, 76Fxx, 76-10 Copyright © 2024, The Authors. All rights reserved.

关键词： Reynolds number

GORINNS: GODUNOV-RIEMANN INFORMED NEURAL NETWORKS FOR LEARNING HYPERBOLIC CONSERVATION LAWS

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arXiv 2024年

作者： Patsatzis, Dimitrios G. di Bernardo, Mario Russo, Lucia Siettos, Constantinos Modelling Engineering Risk and Complexity Scuola Superiore Meridionale Naples Italy Dept. of Electrical Engineering and Information Technology University of Naples Federico II Naples Italy Institute of Science and Technology for Energy and Sustainable Mobility Consiglio Nazionale delle Ricerche Naples Italy Dept. of Mathematics and Applications "Renato Caccioppoli" University of Naples Federico II Naples Italy

We present GoRINNs: numerical analysis-informed neural networks for the solution of inverse problems of non-linear systems of conservation laws. GoRINNs are based on high-resolution Godunov schemes for the solution of the Riemann problem in hyperbolic Partial Differential Equations (PDEs). In contrast to other existing machine learning methods that learn the numerical fluxes of conservative Finite Volume methods, GoRINNs learn the physical flux function per se. Due to their structure, GoRINNs provide interpretable, conservative schemes, that learn the solution operator on the basis of approximate Riemann solvers that satisfy the Rankine-Hugoniot condition. The performance of GoRINNs is assessed via four benchmark problems, namely the Burgers’, the Shallow Water, the Lighthill-Whitham-Richards and the Payne-Whitham traffic flow models. The solution profiles of these PDEs exhibit shock waves, rarefactions and/or contact discontinuities at finite times. We demonstrate that GoRINNs provide a very high accuracy both in the smooth and discontinuous *** Codes 65M08, 65M32, 35L65, 68T07 © 2024, CC BY.

关键词： Partial differential equations

STABILITY ANALYSIS OF PHYSICS-INFORMED NEURAL NETWORKS FOR STIFF LINEAR DIFFERENTIAL EQUATIONS A PREPRINT

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arXiv 2024年

作者： Fabiani, Gianluca Bollt, Erik Siettos, Constantinos Yannacopoulos, Athanasios N. Modelling Engineering Risk and Complexity Scuola Superiore Meridionale Naples80138 Italy Dept. of Electrical and Computer Engineering Clarkson University PotsdamNY United States Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" Università degli Studi di Napoli Federico II Naples80126 Italy Department of Statistics Athens University of Economics and Business Greece

We present a stability analysis of Physics-Informed Neural Networks (PINNs) coupled with random projections, for the numerical solution of (stiff) linear differential equations. For our analysis, we consider systems of linear ODEs, and linear parabolic PDEs. We prove that properly designed PINNs offer consistent and asymptotically stable numerical schemes, thus convergent schemes. In particular, we prove that multi-collocation random projection PINNs guarantee asymptotic stability for very high stiffness and that single-collocation PINNs are A-stable. To assess the performance of the PINNs in terms of both numerical approximation accuracy and computational cost, we compare it with other implicit schemes and in particular backward Euler, the midpoint, trapezoidal (Crank-Nikolson), the 2-stage Gauss scheme and the 2 and 3 stages Radau schemes. We show that the proposed PINNs outperform the above traditional schemes, in both numerical approximation accuracy and importantly computational cost, for a wide range of step *** Codes 65L20, 68T07, 65L04, 37N30 Copyright © 2024, The Authors. All rights reserved.

关键词： Asymptotic stability

SLOW INVARIANT MANIFOLDS OF SINGULARLY PERTURBED SYSTEMS VIA PHYSICS-INFORMED MACHINE LEARNING

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arXiv 2023年

作者： Patsatzis, Dimitrios G. Fabiani, Gianluca Russo, Lucia Siettos, Constantinos Modelling Engineering Risk and Complexity Scuola Superiore Meridionale Naples80138 Italy Institute of Science and Technology for Energy and Sustainable Mobility Consiglio Nazionale delle Ricerche Naples80125 Italy Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" Università degli Studi di Napoli Federico II Naples80126 Italy

We present a physics-informed machine-learning (PIML) approach for the approximation of slow invariant manifolds (SIMs) of singularly perturbed systems, providing functionals in an explicit form that facilitate the construction and numerical integration of reduced order models (ROMs). The proposed scheme solves the partial differential equation corresponding to the invariance equation (IE) within the Geometric Singular Perturbation Theory (GSPT) framework. For the solution of the IE, we used two neural network structures, namely feedforward neural networks (FNNs), and random projection neural networks (RPNNs), with symbolic differentiation for the computation of the gradients required for the learning process. The efficiency of our PIML method is assessed via three benchmark problems, namely the Michaelis-Menten, the target mediated drug disposition reaction mechanism, and the 3D Sel’kov model. We show that the proposed PIML scheme provides approximations, of equivalent or even higher accuracy, than those provided by other traditional GSPT-based methods, and importantly, for any practical purposes, it is not affected by the magnitude of the perturbation parameter. This is of particular importance, as there are many systems for which the gap between the fast and slow timescales is not that big, but still ROMs can still be constructed. A comparison of the computational costs between symbolic, automatic and numerical approximation of the required derivatives in the learning process is also *** Codes 65L11, 65P99, 34C45, 37N30, 68T05 Copyright © 2023, The Authors. All rights reserved.

关键词： Machine learning

TIMESCALE DYNAMICS OF COVID-19 PANDEMIC WAVES: THE CASE OF GREECE

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arXiv 2023年

作者： Manias, Dimitris M. Patsatzis, Dimitris G. Goussis, Dimitris A. Department of Mathematics Khalifa University of Science and Technology Abu Dhabi127788 United Arab Emirates Modelling Engineering Risk and Complexity Scuola Superiore Meridionale Naples80138 Italy Department of Mechanical Engineering Khalifa University of Science and Technology Abu Dhabi127788 United Arab Emirates Research and Innovation Center on CO2 and H2 Khalifa University of Science and Technology Abu Dhabi127788 United Arab Emirates

The results of an alternative methodology for making predictions about the COVID-19 pandemic in Greece are presented. Instead of focusing on the various population profiles (subjected to instabilities introduced by the fitting process), this methodology focuses on the time scale that characterises the intensity and duration of the outbreak phase. Therefore, instead of predicting the peak of active cases, here their inflection point is predicted (the point where the increase of active cases stops accelerating and starts decelerating). Since the inflection point precedes the peak, this methodology can serve as an early warning of the peak. In addition, the paths between the various populations (healthy, exposed, infected, etc) that contribute the most to the outbreak phase are identified. © 2023, CC BY.

关键词： COVID-19

NONLINEAR DISCRETE-TIME OBSERVERS WITH PHYSICS-INFORMED NEURAL NETWORKS

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arXiv 2024年

作者： Alvarez, Hector Vargas Fabiani, Gianluca Kevrekidis, Ioannis G. Kazantzis, Nikolaos Siettos, Constantinos Modelling Engineering Risk and Complexity Scuola Superiore Meridionale Naples80138 Italy Department of Chemical and Biomolecular Engineering Johns Hopkins University BaltimoreMD21218 United States Department of Chemical Engineering Worcester Polytechnic Institute WorcesterMA01609 United States Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" Università degli Studi di Napoli Federico II Naples80126 Italy Department of Applied Mathematics and Statistics Johns Hopkins University BaltimoreMD21218 United States Department of Urology School of Medicine Johns Hopkins University BaltimoreMD21218 United States

We use Physics-Informed Neural Networks (PINNs) to solve the discrete-time nonlinear observer state estimation problem. Integrated within a single-step exact observer linearization framework, the proposed PINN approach aims at learning a nonlinear state transformation map by solving a system of inhomogeneous functional equations. The performance of the proposed PINN approach is assessed via two illustrative case studies for which the observer linearizing transformation map can be derived analytically. We also perform an uncertainty quantification analysis for the proposed PINN scheme and we compare it with conventional power-series numerical implementations, which rely on the computation of a power series *** Codes 37N30, 68T05, 93C55, 65D15 Copyright © 2024, The Authors. All rights reserved.

关键词： Uncertainty analysis

TASKS MAKYTH MODELS: MACHINE LEARNING ASSISTED SURROGATES FOR TIPPING POINTS

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arXiv 2023年

作者： Fabiani, Gianluca Evangelou, Nikolaos Cui, Tianqi Bello-Rivas, Juan M. Martin-Linares, Cristina Siettos, Constantinos Kevrekidis, Ioannis G. Modelling Engineering Risk and Complexity Scuola Superiore Meridionale Naples80138 Italy Dept. of Chemical and Biomolecular Engineering Johns Hopkins University BaltimoreMD21218 United States Dept. of Mechanical Engineering Johns Hopkins University BaltimoreMD21218 United States Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" Università degli Studi di Napoli Federico II Naples80126 Italy Dept. of Applied Mathematics and Statistics Johns Hopkins University BaltimoreMD21218 United States School of Medicine’s Dept. of Urology Johns Hopkins University BaltimoreMD21218 United States

We present a machine learning (ML)-assisted framework bridging manifold learning, neural networks, Gaussian processes, and Equation-Free multiscale modeling, for (a) detecting tipping points in the emergent behavior of complex systems, and (b) characterizing probabilities of rare events (here, catastrophic shifts) near them. Our illustrative example is an event-driven, stochastic agent-based model (ABM) describing the mimetic behavior of traders in a simple financial market. Given high-dimensional spatiotemporal data –generated by the stochastic ABM– we construct reduced-order models for the emergent dynamics at different scales: (a) mesoscopic Integro-Partial Differential Equations (IPDEs);and (b) mean-field-type Stochastic Differential Equations (SDEs) embedded in a low-dimensional latent space, targeted to the neighborhood of the tipping point. We contrast the uses of the different models and the effort involved in learning them. Copyright © 2023, The Authors. All rights reserved.

关键词： Machine learning

The overlapping burden of the three leading causes of disability and death in sub-Saharan African children

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Nature Communications 2022年第1期13卷 1-14页

作者： Reiner, Robert C. Welgan, Catherine A. Troeger, Christopher E. Baumann, Mathew M. Weiss, Daniel J. Deshpande, Aniruddha Blacker, Brigette F. Miller-Petrie, Molly K. Earl, Lucas Bhatt, Samir Abolhassani, Hassan Abosetugn, Akine Eshete Abu-Gharbieh, Eman Adekanmbi, Victor Adetokunboh, Olatunji O. Aghaali, Mohammad Aji, Budi Alahdab, Fares Al-Aly, Ziyad Alhassan, Robert Kaba Ali, Saqib Alizade, Hesam Aljunid, Syed Mohamed Almasi-Hashiani, Amir Al-Mekhlafi, Hesham M. Altirkawi, Khalid A. Alvis-Guzman, Nelson Amare, Azmeraw T. Amini, Saeed Amugsi, Dickson A. Ancuceanu, Robert Andrei, Catalina Liliana Ansari, Fereshteh Anvari, Davood Appiah, Seth Christopher Yaw Arabloo, Jalal Aremu, Olatunde Atout, Maha Moh’d Wahbi Ausloos, Marcel Ausloos, Floriane Ayanore, Martin Amogre Aynalem, Yared Asmare Azene, Zelalem Nigussie Badawi, Alaa Baig, Atif Amin Banach, Maciej Bedi, Neeraj Bhagavathula, Akshaya Srikanth Bhandari, Dinesh Bhardwaj, Nikha Bhardwaj, Pankaj Bhattacharyya, Krittika Bhutta, Zulfiqar A. Bijani, Ali Birhanu, Tesega Tesega Mengistu Bitew, Zebenay Workneh Boloor, Archith Brady, Oliver J. Butt, Zahid A. Car, Josip Carvalho, Felix Casey, Daniel C. Chattu, Vijay Kumar Chowdhury, Mohiuddin Ahsanul Kabir Chu, Dinh-Toi Coelho, Camila H. Cook, Aubrey J. Damiani, Giovanni Daoud, Farah Gela, Jiregna Darega Darwish, Amira Hamed Daryani, Ahmad Das, Jai K. Davis Weaver, Nicole Deribe, Kebede Desalew, Assefa Dharmaratne, Samath Dhamminda Dianatinasab, Mostafa Diaz, Daniel Djalalinia, Shirin Dorostkar, Fariba Dubljanin, Eleonora Duko, Bereket Dwyer-Lindgren, Laura Effiong, Andem El Sayed Zaki, Maysaa El Tantawi, Maha Enany, Shymaa Fattahi, Nazir Feigin, Valery L. Fernandes, Eduarda Ferrara, Pietro Fischer, Florian Foigt, Nataliya A. Folayan, Morenike Oluwatoyin Foroutan, Masoud Frostad, Joseph Jon Fukumoto, Takeshi Gaidhane, Abhay Motiramji Gebrekrstos, Hailemikael Gebrekidan G. K. Gebremeskel, Leake Gebreslassie, Assefa Ayalew Gething, Peter W. Gezae, Kebede Embaye Ghadiri, Keyghobad Ghashghaee, Ahmad Golechha, Mahaveer Gubar Institute for Health Metrics and Evaluation University of Washington Seattle WA United States Department of Health Metrics Sciences School of Medicine University of Washington Seattle WA United States Malaria Atlas Project University of Oxford Oxford United Kingdom Imperial College London London United Kingdom Department of Laboratory Medicine Karolinska University Hospital Huddinge Sweden Research Center for Immunodeficiencies Tehran University of Medical Sciences Tehran Iran Department of Public Health Debre Berhan University Debre Berhan Ethiopia Department of Clinical Sciences University of Sharjah Sharjah United Arab Emirates Population Health Sciences King’s College London London United Kingdom Centre of Excellence for Epidemiological Modelling and Analysis Stellenbosch University Stellenbosch South Africa Department of Global Health Stellenbosch University Cape Town South Africa Department of Epidemiology and Biostatistics Qom University of Medical Sciences Qom Iran Faculty of Medicine and Public Health Jenderal Soedirman University Purwokerto Indonesia Mayo Evidence-based Practice Center Mayo Clinic Foundation for Medical Education and Research Rochester MN United States John T. Milliken Department of Internal Medicine Washington University in St. Louis St. Louis MO United States Clinical Epidemiology Center Department of Veterans Affairs St Louis MO United States Institute of Health Research University of Health and Allied Sciences Ho Ghana Department of Information Systems College of Economics and Political Science Sultan Qaboos University Muscat Oman Infectious and Tropical Disease Research Center Hormozgan University of Medical Sciences Bandar Abbas Iran Department of Health Policy and Management Kuwait University Safat Kuwait International Centre for Casemix and Clinical Coding National University of Malaysia Bandar Tun Razak Malaysia Department of Epidemiology Arak University of Medical Sciences Arak Iran Medical Research Center

Despite substantial declines since 2000, lower respiratory infections (LRIs), diarrhoeal diseases, and malaria remain among the leading causes of nonfatal and fatal disease burden for children under 5 years of age (under 5), primarily in sub-Saharan Africa (SSA). The spatial burden of each of these diseases has been estimated subnationally across SSA, yet no prior analyses have examined the pattern of their combined burden. Here we synthesise subnational estimates of the burden of LRIs, diarrhoea, and malaria in children under-5 from 2000 to 2017 for 43 sub-Saharan countries. Some units faced a relatively equal burden from each of the three diseases, while others had one or two dominant sources of unit-level burden, with no consistent pattern geographically across the entire subcontinent. Using a subnational counterfactual analysis, we show that nearly 300 million DALYs could have been averted since 2000 by raising all units to their national average. Our findings are directly relevant for decision-makers in determining which and targeting where the most appropriate interventions are for increasing child survival. © 2022, The Author(s).

关键词： Biogeography Epidemiology Infectious diseases Malaria