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Lindström, Jens

Open this publication in new window or tab >>A unified deep artificial neural network approach to partial differential equations in complex geometries### Berg, Jens

### Nyström, Kaj

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##### Abstract [en]

##### National Category

Computational Mathematics
##### Research subject

Mathematics with specialization in Applied Mathematics
##### Identifiers

urn:nbn:se:uu:diva-362369 (URN)10.1016/j.neucom.2018.06.056 (DOI)000444237900003 ()
#####

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##### Funder

Göran Gustafsson Foundation for Research in Natural Sciences and Medicine
Available from: 2018-08-23 Created: 2018-10-04 Last updated: 2018-11-14Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

In this paper, we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator. The method is based on an ansatz for the solution which requires nothing but feedforward neural networks and an unconstrained gradient based optimization method such as gradient descent or a quasi-Newton method. We show an example where classical mesh based methods cannot be used and neural networks can be seen as an attractive alternative. Finally, we highlight the benefits of deep compared to shallow neural networks and device some other convergence enhancing techniques.

Open this publication in new window or tab >>Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations### Berg, Jens

### Nordström, Jan

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##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:uu:diva-214539 (URN)10.1016/j.jcp.2013.11.031 (DOI)000329506500009 ()
#####

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Available from: 2013-12-03 Created: 2014-01-08 Last updated: 2017-12-06Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.

Open this publication in new window or tab >>Conjugate heat transfer for the unsteady compressible Navier–Stokes equations using a multi-block coupling### Nordström, Jan

### Berg, Jens

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2013 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 72, p. 20-29Article in journal (Refereed) Published
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:uu:diva-187192 (URN)10.1016/j.compfluid.2012.11.018 (DOI)000314442900002 ()
#####

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Available from: 2012-12-19 Created: 2012-12-04 Last updated: 2017-12-07Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.

Open this publication in new window or tab >>Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations### Berg, Jens

### Nordström, Jan

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##### Series

Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2013-013
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:uu:diva-200526 (URN)
#####

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Available from: 2013-05-29 Created: 2013-05-29 Last updated: 2013-05-29Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.

Open this publication in new window or tab >>Duality based boundary treatment for the Euler and Navier-Stokes equations### Berg, Jens

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.### Nordström, Jan

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2013 (English)In: Proc. 21st AIAA CFD Conference, AIAA , 2013Conference paper, Published paper (Refereed)
##### Place, publisher, year, edition, pages

AIAA, 2013
##### Series

Conference Proceeding Series ; 2013-2959
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:uu:diva-207702 (URN)10.2514/6.2013-2959 (DOI)
#####

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Available from: 2013-06-26 Created: 2013-09-17 Last updated: 2013-09-17Bibliographically approved

Open this publication in new window or tab >>On the impact of boundary conditions on dual consistent finite difference discretizations### Berg, Jens

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.### Nordström, Jan

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2013 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 236, p. 41-55Article in journal (Refereed) Published
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:uu:diva-187194 (URN)10.1016/j.jcp.2012.11.019 (DOI)000314801500005 ()
#####

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Available from: 2012-12-12 Created: 2012-12-04 Last updated: 2017-12-07Bibliographically approved

Open this publication in new window or tab >>Stable and High-Order Finite Difference Methods for Multiphysics Flow Problems### Berg, Jens

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2013 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Alternative title[sv]

Stabila finita differensmetoder med hög noggrannhetsordning för multifysik- och flödesproblem
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Acta Universitatis Upsaliensis, 2013. p. 35
##### Series

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1004
##### Keywords

Summation-by-parts, Simultaneous Approximation Term, Stability, High-order accuracy, Finite difference methods, Dual consistency
##### National Category

Computational Mathematics
##### Research subject

Scientific Computing with specialization in Numerical Analysis
##### Identifiers

urn:nbn:se:uu:diva-187204 (URN)978-91-554-8557-3 (ISBN)
##### Public defence

2013-02-01, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (English)
##### Opponent

### Zingg, David

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##### Supervisors

### Nordström, Jan

### Lötstedt, Per

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Available from: 2013-01-11 Created: 2012-12-04 Last updated: 2013-04-02Bibliographically approved

Partial differential equations (PDEs) are used to model various phenomena in nature and society, ranging from the motion of fluids and electromagnetic waves to the stock market and traffic jams. There are many methods for numerically approximating solutions to PDEs. Some of the most commonly used ones are the finite volume method, the finite element method, and the finite difference method. All methods have their strengths and weaknesses, and it is the problem at hand that determines which method that is suitable. In this thesis, we focus on the finite difference method which is conceptually easy to understand, has high-order accuracy, and can be efficiently implemented in computer software.

We use the finite difference method on summation-by-parts (SBP) form, together with a weak implementation of the boundary conditions called the simultaneous approximation term (SAT). Together, SBP and SAT provide a technique for overcoming most of the drawbacks of the finite difference method. The SBP-SAT technique can be used to derive energy stable schemes for any linearly well-posed initial boundary value problem. The stability is not restricted by the order of accuracy, as long as the numerical scheme can be written in SBP form. The weak boundary conditions can be extended to interfaces which are used either in domain decomposition for geometric flexibility, or for coupling of different physics models.

The contributions in this thesis are twofold. The first part, papers I-IV, develops stable boundary and interface procedures for computational fluid dynamics problems, in particular for problems related to the Navier-Stokes equations and conjugate heat transfer. The second part, papers V-VI, utilizes duality to construct numerical schemes which are not only energy stable, but also dual consistent. Dual consistency alone ensures superconvergence of linear integral functionals from the solutions of SBP-SAT discretizations. By simultaneously considering well-posedness of the primal and dual problems, new advanced boundary conditions can be derived. The new duality based boundary conditions are imposed by SATs, which by construction of the continuous boundary conditions ensure energy stability, dual consistency, and functional superconvergence of the SBP-SAT schemes.

University of Toronto, Institute for Aerospace Studies.

Linköping University, Department of Mathematics, Computational Mathematics.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing.

Open this publication in new window or tab >>A stable and dual consistent boundary treatment using finite differences on summation-by-parts form### Berg, Jens

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.### Nordström, Jan

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##### Place, publisher, year, edition, pages

Austria: Tech. Univ. Wien, 2012
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:uu:diva-181234 (URN)978-3-9502481-9-7 (ISBN)
#####

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Available from: 2012-09-13 Created: 2012-09-20 Last updated: 2013-03-02Bibliographically approved

Open this publication in new window or tab >>A stable and dual consistent boundary treatment using finite differences on summation-by-parts form### Berg, Jens

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.### Nordström, Jan

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2012 (English)Report (Other academic)
##### Series

Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-014
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:uu:diva-177317 (URN)
#####

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Available from: 2012-07-02 Created: 2012-07-08Bibliographically approved

Open this publication in new window or tab >>On the impact of boundary conditions on dual consistent finite difference discretizations### Berg, Jens

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.### Nordström, Jan

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##### Series

Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-025
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:uu:diva-180905 (URN)
#####

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_j_idt365",{id:"formSmash:j_idt184:9:j_idt188:j_idt365",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_j_idt365",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_j_idt371",{id:"formSmash:j_idt184:9:j_idt188:j_idt371",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_j_idt371",multiple:true});
#####

Available from: 2012-09-07 Created: 2012-09-12Bibliographically approved