Neural-network methods for boundary value problems with irregular boundaries

By Isaac E. Lagaris, Aristidis C. Likas, Dimitris G. Papageorgiou. 2000

547 citations recorded by the source corpus.

Subfields: Materials Chemistry, Mechanics of Materials, Statistical and Nonlinear Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Materials Chemistry

In Mechanics of Materials

In Statistical and Nonlinear Physics

Citations in the corpus

Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges (2021, background): “[168, 169] explored the idea of using neural networks to solve initial and boundary value problems.”
Physics informed deep learning for computational elastodynamics without labeled data (2020, background): “ why DNN is well qualied as the solution approximator includes: (1) its extraordinary approximation capability proven by the universal approximation theorem [49], (2) its innite continuity property [14] with proper activation function, (3) the derivatives that appears in the PDEs can be calculated e”
DGM: A deep learning algorithm for solving partial differential equations (2017, positive): “[29], [30], [46], [31], and [35] propose to use neural networks to solve PDEs and ODEs.”
Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates (2020, unknown): “They were also adopted to solve partial diﬀerential equations (PDEs) [32, 33, 34] but shallow ANNs are unable to learn the complex nonlinear patterns eﬀectively.”
Self-adaptive loss balanced Physics-informed neural networks (2021, methodology): “Due to the breakthrough approximation capabilities of neural networks [10,11], there have been several remarkable results in solving forward and inverse problems for fluid simulation [12,13,14], instead of using the classical numerical schemes.”
Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks (2019, methodology): “In this part, we brieﬂy review using DNNs to solve the deterministic di ﬀ erential equations [26, 27, 28], and its generalization for solving deterministic inverse problems in [6].”
DeepXDE: A Deep Learning Library for Solving Differential Equations (2019, methodology): “Alternatively, one could use the PDE in strong form directly [15, 52, 30, 31, 7, 50, 47]; in this form, automatic”
Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems (2018, methodology): “In this part, we brieﬂy review using DNNs to solve deterministic diﬀer-ential equations [8, 9, 11], and its generalization for solving deterministic inverse problems in [18].”
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, methodology): “[3] used a RBF network in the first term to satisfy the boundary conditions at a collection of discrete points on the boundary.”
Digital Twin: Values, Challenges and Enablers From a Modeling Perspective (2019, unknown): “[330] I.”
Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data (2019, background): “This idea of physics-constrained learning is not new and was proposed back in the late ’90s in the context of solving classic diﬀerential equations [35– 37].”
Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations (2021, methodology): “This formulation has since been extended to irregular domains [57, 58], and general schemes for constructing suitable ansatze have been proposed [59].”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “Lagaris et al [90] extends the method to problems with irregular borders.”