Kolmogorov n–width and Lagrangian physics-informed neural networks: A causality-conforming manifold for convection-dominated PDEs

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dc.citation.articleNumber115810en_US
dc.citation.journalTitleComputer Methods in Applied Mechanics and Engineeringen_US
dc.citation.volumeNumber404en_US
dc.contributor.authorMojgani, Ramboden_US
dc.contributor.authorBalajewicz, Maciejen_US
dc.contributor.authorHassanzadeh, Pedramen_US
dc.date.accessioned2023-01-10T16:18:18Zen_US
dc.date.available2023-01-10T16:18:18Zen_US
dc.date.issued2023en_US
dc.description.abstractWe make connections between complexity of training of physics-informed neural networks (PINNs) and Kolmogorov n-width of the solution. Leveraging this connection, we then propose Lagrangian PINNs (LPINNs) as a partial differential equation (PDE)-informed solution for convection-dominated problems. PINNs employ neural-networks to find the solutions of PDE-constrained optimization problems with initial conditions and boundary conditions as soft or hard constraints. These soft constraints are often blamed to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the complexity of training (i) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts, and therefore becomes apparent in convection-dominated flows, and (ii) persists even when the boundary conditions are strictly enforced. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum learning, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection–diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain, i.e., on the characteristics. This approach is unique as it reduces the complexity of convection-dominated PINNs at the PDE level, instead of optimization strategies and/or schedulers. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called “complexity” of the problems, i.e., convection, compared to those in the traditional PINNs in the Eulerian framework.en_US
dc.identifier.citationMojgani, Rambod, Balajewicz, Maciej and Hassanzadeh, Pedram. "Kolmogorov n–width and Lagrangian physics-informed neural networks: A causality-conforming manifold for convection-dominated PDEs." <i>Computer Methods in Applied Mechanics and Engineering,</i> 404, (2023) Elsevier: https://doi.org/10.1016/j.cma.2022.115810.en_US
dc.identifier.doihttps://doi.org/10.1016/j.cma.2022.115810en_US
dc.identifier.urihttps://hdl.handle.net/1911/114231en_US
dc.language.isoengen_US
dc.publisherElsevieren_US
dc.rightsThis is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier.en_US
dc.subject.keywordDeep learningen_US
dc.subject.keywordKolmogorov n-widthen_US
dc.subject.keywordPartial differential equationsen_US
dc.subject.keywordMethod of characteristicsen_US
dc.subject.keywordLagrangian frame of referenceen_US
dc.titleKolmogorov n–width and Lagrangian physics-informed neural networks: A causality-conforming manifold for convection-dominated PDEsen_US
dc.typeJournal articleen_US
dc.type.dcmiTexten_US
dc.type.publicationpost-printen_US
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