Print Email Facebook Twitter Solving Partial Differential Equations using Physics-Informed Neural Networks Title Solving Partial Differential Equations using Physics-Informed Neural Networks Author Popa, Vlad (TU Delft Electrical Engineering, Mathematics and Computer Science; TU Delft Applied Sciences) Contributor Thijssen, J.M. (mentor) Möller, M. (mentor) Tao, Q. (graduation committee) Dubbeldam, J.L.A. (graduation committee) Degree granting institution Delft University of Technology Programme Applied Mathematics | Applied Physics Date 2022-08-31 Abstract In an attempt to find alternatives for solving partial differential equations (PDEs)with traditional numerical methods, a new field has emerged which incorporatesthe residual of a PDE into the loss function of an Artificial Neural Network. Thismethod is called Physics-Informed Neural Network (PINN). In this thesis, we study dense neural networks (DNNs), including codes developed in the context of this bachelor project. We derive the backpropagation equations necessary for training and use different configurations in a DNN to test its interpolating accuracy. We distinguish between a-PINNs which use automatic differentiation to evaluate a PDE, and n-PINNs which approximate differential operators in a PDE with numerical differentiation. We compare both PINNs on the harmonic oscillator, the 1D heat equation and the 1-soliton and 2-soliton solutions of the Korteweg-De Vries (KdV) equation. Both PINNs could accurately converge to the solution, except to the 2-soliton solution, where the a-PINN outperformed the n-PINN. Furthermore, we tested a highly nonlinear problem of the KdV equation, which can be described by a train of solitons. We observed that PINNs are inaccurate if insufficient training samples are used for training. Adding training samples on the interior from a numerical solution leads to a good qualitative agreement, though more effort is required to find a better network configuration to obtain more accurate predictions.Additionally, PINNs were used for inverse problems to derive an unknown coefficient in a PDE and proved to be highly accurate for noiseless data. When wegenerated training samples with 10% noise from a uniform distribution, the PINNresults’ relative error stayed within a margin of under 2%. However, inverse PINNs are much more inefficient compared to nonlinear least squares methods like the Levenberg–Marquardt algorithm.As of now, PINNs are still very early in development and stand no match againsttraditional numerical methods to a known PDE. They may, however, provide auseful alternative in the future as they are constantly being improved. Subject PINNphysics-informed neural networksMachine LearningPDEKorteweg-De vriesHarmonic oscillatorheat equation To reference this document use: http://resolver.tudelft.nl/uuid:daab64c0-d36e-4fc4-8702-cbf8cc0fd708 Part of collection Student theses Document type bachelor thesis Rights © 2022 Vlad Popa Files PDF BEP_Vlad_Popa.pdf 20.01 MB Close viewer /islandora/object/uuid:daab64c0-d36e-4fc4-8702-cbf8cc0fd708/datastream/OBJ/view