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Error Estimates for Neural Network Solutions of Partial Differential Equations [article]

Piotr Minakowski, Thomas Richter
<span title="2021-07-23">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We develop an error estimator for neural network approximations of PDEs. The proposed approach is based on dual weighted residual estimator (DWR).  ...  It is destined to serve as a stopping criterion that guarantees the accuracy of the solution independently of the design of the neural network training.  ...  Introduction In recent years, the emerging field of (deep) neural networks has reached the numerical approximation of partial differential equations (PDE).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2107.11035v1">arXiv:2107.11035v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/py77j2txtbeqpkg6s5vtcslnda">fatcat:py77j2txtbeqpkg6s5vtcslnda</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210727224759/https://arxiv.org/pdf/2107.11035v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d0/d4/d0d43fadbfe7a81944ae3f9ccf98ff4d5c4e9300.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2107.11035v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Solving Partial Differential Equations with Bernstein Neural Networks [chapter]

Sina Razvarz, Raheleh Jafari, Alexander Gegov
<span title="2018-08-11">2018</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/kax3wwzwmncwhi472pxbzqsjja" style="color: black;">Advances in Intelligent Systems and Computing</a> </i> &nbsp;
In this paper, a neural network-based procedure is suggested to produce estimated solutions (controllers) for the second-order nonlinear partial differential equations (PDEs).  ...  The proposed neural network contains the regularizing parameters (weights and biases), that can be utilized for making the error function least.  ...  Controller design with neural networks approximation Here, we construct a neural network for resolving the strongly degenerate parabolic equation that obtains the solution of differential equations in  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-97982-3_5">doi:10.1007/978-3-319-97982-3_5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3lnvpo6oabeupaoifyhrzuwxuy">fatcat:3lnvpo6oabeupaoifyhrzuwxuy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200320175812/https://researchportal.port.ac.uk/portal/files/11174016/UKCI_2018_PAPER_1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/bd/76/bd7607e9de5bc774c1d1f772d0939de7a8a64c9c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-97982-3_5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Physics-informed ConvNet: Learning Physical Field from a Shallow Neural Network [article]

Pengpeng Shi, Zhi Zeng, Tianshou Liang
<span title="2022-02-07">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The effectiveness of the current development is illustrated through some numerical cases involving the solving (and estimation) of nonlinear physical operator equations and recovering physical information  ...  Improving the generalization ability of neural networks by "teaching" domain knowledge and developing a new generation of models combined with the physical laws have become promising areas of machine learning  ...  Acknowledgments This research is supported by financial support from the National Natural Science Foundation of China (NNSFC) (Grant Nos. 11802225 and 61805185).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.10967v2">arXiv:2201.10967v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/slcve5ul6jgqbevq55u5ui3rqa">fatcat:slcve5ul6jgqbevq55u5ui3rqa</a> </span>
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Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative [article]

Yinlin Ye, Yajing Li, Hongtao Fan, Xinyi Liu, Hongbing Zhang
<span title="2021-08-17">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Physics-informed neural networks (PINNs) show great advantages in solving partial differential equations.  ...  For the inverse problem, we use the data obtained to train the neural network, and the estimation of parameter λ in the equation is elaborated.  ...  The results of using neural networks to solve partial differential equations have sprung up.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2108.07490v1">arXiv:2108.07490v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pqgzktffdfgn5mwlulw74hnszm">fatcat:pqgzktffdfgn5mwlulw74hnszm</a> </span>
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Numerical Solution of Sixth-Order Differential Equations Arising in Astrophysics by Neural Network

M. Khalid, Mariam Sultana, Faheem Zaidi
<span title="2014-12-18">2014</span> <i title="Foundation of Computer Science"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/b637noqf3vhmhjevdfk3h5pdsu" style="color: black;">International Journal of Computer Applications</a> </i> &nbsp;
In the current paper, a neural network method to solve sixth-order differential equations and their boundary conditions has been presented.  ...  The idea this method incorporates is to integrate knowledge about the differential equation and its boundary conditions into neural networks and the training sets.  ...  [34] investigated a class of partial differential equations using multilayer neural network.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5120/18752-0023">doi:10.5120/18752-0023</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/onpm3klqqff4hfaxfjsqvql4xq">fatcat:onpm3klqqff4hfaxfjsqvql4xq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706120419/http://research.ijcaonline.org/volume107/number6/pxc3900023.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/dd/8f/dd8f8e73e248d339fd79f697c88a78b455b49734.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5120/18752-0023"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations [article]

Maziar Raissi, Paris Perdikaris, George Em Karniadakis
<span title="2017-11-28">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential  ...  In this second part of our two-part treatise, we focus on the problem of data-driven discovery of partial differential equations.  ...  Through the lens of different benchmark problems, we highlighted the key features of physics informed neural networks in the context of data-driven solutions of partial differential equations [5, 6] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1711.10566v1">arXiv:1711.10566v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tuutvwlaejbp3ejcfxmsr76lkq">fatcat:tuutvwlaejbp3ejcfxmsr76lkq</a> </span>
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Differential Neural Networks (DNN)

Sergio Ledesma, Dora-Luz Almanza-Ojeda, Mario-Alberto Ibarra-Manzano, Eduardo Cabal-Yepez, Juan-Gabriel Avina-Cervantes, Pascal Fallavollita
<span title="">2020</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/q7qi7j4ckfac7ehf3mjbso4hne" style="color: black;">IEEE Access</a> </i> &nbsp;
ACKNOWLEDGEMENTS We acknowledge DAIP, University of Guanajuato and the University of Ottawa for their sponsorship in the realization of this work.  ...  This work was developed during the sabbatical stay of Sergio Ledesma at the Faculty of Health Sciences in the University of Ottawa, Canada.  ...  In the same context, the authors of [16] analyzed the solution of linear fractional-order ordinary differential equations using artificial neural networks.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/access.2020.3019307">doi:10.1109/access.2020.3019307</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dml5rprddjdadirnj2myupckiy">fatcat:dml5rprddjdadirnj2myupckiy</a> </span>
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Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey

Manoj Kumar, Neha Yadav
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nkrwe4pmozafvnd72yxufztpku" style="color: black;">Computers and Mathematics with Applications</a> </i> &nbsp;
Since neural networks have universal approximation capabilities, therefore it is possible to postulate them as solutions for given differential equations that define unsupervised errors.  ...  In this paper, we present a wide survey and classification of different Multilayer Perceptron (MLP) and Radial Basis Function (RBF) neural network techniques, which are used for solving differential equations  ...  Acknowledgments The authors extend their appreciation to anonymous reviewers for their valuable suggestions in revising this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.camwa.2011.09.028">doi:10.1016/j.camwa.2011.09.028</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/snzxuw2pjngnli3bi35zigspva">fatcat:snzxuw2pjngnli3bi35zigspva</a> </span>
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Hybrid FEM-NN models: Combining artificial neural networks with the finite element method [article]

Sebastian K. Mitusch, Simon W. Funke, Miroslav Kuchta
<span title="2021-01-04">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs).  ...  The approach allows to train neural networks while respecting the PDEs as a strong constraint in the optimisation as apposed to making them part of the loss function.  ...  Karniadakis, Variational Physics-Informed Neural Networks For Solving Partial Differential Equations, arXiv:1912.00873. [19] U.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2101.00962v1">arXiv:2101.00962v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vvarnz6wavhwxfdcmgtgilzncq">fatcat:vvarnz6wavhwxfdcmgtgilzncq</a> </span>
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Hidden physics models: Machine learning of nonlinear partial differential equations

Maziar Raissi, George Em Karniadakis
<span title="">2018</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/f7kkigxhzjakref5b42jt4uoya" style="color: black;">Journal of Computational Physics</a> </i> &nbsp;
In this paper, we present a new paradigm of learning partial differential equations from small data.  ...  The proposed methodology may be applied to the problem of learning, system identification, or data-driven discovery of partial differential equations.  ...  The neural networks u and N are trained by minimizing the sum of squared errors loss of equation (3) .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jcp.2017.11.039">doi:10.1016/j.jcp.2017.11.039</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5fcctj6ayzeutdzn6r3clq373q">fatcat:5fcctj6ayzeutdzn6r3clq373q</a> </span>
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Matrix Lie Maps and Neural Networks for Solving Differential Equations [article]

Andrei Ivanov, Sergei Andrianov
<span title="2019-08-16">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also demonstrate the building of the neural network that describes the solution of Burgers' equation that is a fundamental partial differential equation.  ...  If the differential equation is provided, training a neural network is unnecessary. The weights of the network can be directly calculated from the equation.  ...  The limitations of the data-driven approach for large-scale systems, optimal network configuration, and noisy data consideration should be examined in further research.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1908.06088v1">arXiv:1908.06088v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dedpj5s7dfertkuq42xlwyk43i">fatcat:dedpj5s7dfertkuq42xlwyk43i</a> </span>
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Learning To Solve Differential Equations Across Initial Conditions [article]

Shehryar Malik, Usman Anwar, Ali Ahmed, Alireza Aghasi
<span title="2020-04-19">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Recently, there has been a lot of interest in using neural networks for solving partial differential equations.  ...  In this work, we posit the problem of approximating the solution of a fixed partial differential equation for any arbitrary initial conditions as learning a conditional probability distribution.  ...  INTRODUCTION Partial differential equations (PDEs) are of great importance in various fields such as science, engineering and economics.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2003.12159v2">arXiv:2003.12159v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/imzo2g2d2vba3ofpiji4i5a6zm">fatcat:imzo2g2d2vba3ofpiji4i5a6zm</a> </span>
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Multigoal-oriented dual-weighted-residual error estimation using deep neural networks [article]

Ayan Chakraborty, Thomas Wick, Xiaoying Zhuang, Timon Rabczuk
<span title="2021-12-22">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our approach is based on a posteriori error estimation in which the adjoint problem is solved for the error localization to formulate an error estimator within the framework of neural network.  ...  computation of both primal and adjoint solutions using the neural network.  ...  Carlier, Deep relaxation: partial differential equations for optimizing deep neural networks, Research in the Mathematical Sciences 5 (3) (2018) 30. [20] R. Becker, R.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2112.11360v2">arXiv:2112.11360v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/35pmqlj4z5fsbd7wj6lyeqcyqi">fatcat:35pmqlj4z5fsbd7wj6lyeqcyqi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220103220638/https://arxiv.org/pdf/2112.11360v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/99/28/9928ffba2f9280a9ba9be28af5562d682f539fba.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2112.11360v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Application of ANNs approach for wave-like and heat-like equations

Ahmad Jafarian, Dumitru Baleanu
<span title="2017-12-29">2017</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/r4ecuox7yrdy7h75vf7dx5nbau" style="color: black;">Open Physics</a> </i> &nbsp;
To the numerical solution of these equations, a combination of the power series method and artificial neural networks approach, is used to seek an appropriate bivariate polynomial solution of the mentioned  ...  In this study, we intend to duplicate an efficient iterative method to the numerical solution of two famous partial differential equations, namely the wave-like and heat-like problems.  ...  partial differential equations.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/phys-2017-0135">doi:10.1515/phys-2017-0135</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ehcezgih2jdqzgkeyruu6r6nfy">fatcat:ehcezgih2jdqzgkeyruu6r6nfy</a> </span>
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Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations [article]

Maziar Raissi, Paris Perdikaris, George Em Karniadakis
<span title="2017-11-28">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential  ...  In this two part treatise, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations.  ...  Note that this error is about two orders of magnitude lower than the one reported in our previous work on data-driven solution of partial differential equation using Gaussian processes [9] .  ... 
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