Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems

By Cosmin Anitescu, Elena Atroshchenko, Naif Alajlan, Timon Rabczuk. Computers Materials & Continua, 2019

644 citations recorded by the source corpus.

: We present a method for solving partial differential equations using artificial neural networks and an adaptive collocation strategy. In this procedure, a coarse grid of training points is used at the initial training stages, while more points are added at later stages based on the value of the residual at a larger set of evaluation points. This method increases the robustness of the neural network approximation and can result in significant computational savings, particularly when the solution is non-smooth. Numerical results are presented for benchmark problems for scalar-valued PDEs, namely Poisson and Helmholtz equations, as well as for an inverse acoustics problem.

Subfields: Electrical and Electronic Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8ecb8048c2422b5ced1f949282915926648d0c8d

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Citations in the corpus

Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture (2020, methodology): “Due to these advantages, many variations of the PINN approach have been successfully used to solve PDEs in a wide range of engineering applications [24, 29–37].”
Prediction and Sensitivity Analysis of Bubble Dissolution Time in 3D Selective Laser Sintering Using Ensemble Decision Trees (2019, background): “This technique exhibits a lot of advantages not found in traditional computational simulation [25], especially for predicting material behaviors, for instance, rheological properties of materials [26], permeability of polymers [27], composite materials [28], structure under compression [29], or conc”
Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates (2020, methodology): “Due to the limitations of analytical methods, a variety of numerical methods have been developed including the finite element method [2, 3], boundary element method [4, 5], meshfree method [6, 7], isogeometric analysis (IGA) formulations [8], numerical manifold method [9, 10, 11] and recently deep l”
Weak Adversarial Networks for High-dimensional Partial Differential Equations (2019, methodology): “More recently, an adaptive collocation strategy is presented for a method in [27].”
Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks (2020, background): “In general, we can define the neural network as [13] a[1]= x∈Rn1 , a[k]=σ ( W [k]a[k−1]+b[k] ) ∈Rnk , for k=2,3,··· ,L, (1.”