r-Adaptive deep learning method for solving partial differential equations
Copyright policy )r-Adaptive deep learning method for solving partial differential equations
We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we build two- or three-dimensional meshes. The method allows the definition of fixed interfaces to design conforming meshes, and enables changes in the topology, i.e., some nodes can jump across fixed interfaces. The method simultaneously optimizes the node locations and the PDE solution values over the resulting mesh. To numerically illustrate the performance of our proposed $r-$adaptive method, we apply it in combination with a collocation method, a Least Squares Method, and a Deep Ritz Method. We focus on the latter to solve one- and two-dimensional problems whose solutions are smooth, singular, and/or exhibit strong gradients.
19 pages
FOS: Computer and information sciences, Numerical optimization and variational techniques, Computer Science - Machine Learning, Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs, 65N50, 68T07, Machine Learning (cs.LG), Second-order elliptic equations, partial differential equations, FOS: Mathematics, Mathematics - Numerical Analysis, Neural and Evolutionary Computing (cs.NE), Artificial neural networks and deep learning, First-order hyperbolic equations, Error bounds for boundary value problems involving PDEs, deep learning, Computer Science - Neural and Evolutionary Computing, Deep learning, Numerical Analysis (math.NA), neural networks, adaptive mesh, Adaptive mesh, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Spectral, collocation and related methods for boundary value problems involving PDEs, Neural networks
FOS: Computer and information sciences, Numerical optimization and variational techniques, Computer Science - Machine Learning, Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs, 65N50, 68T07, Machine Learning (cs.LG), Second-order elliptic equations, partial differential equations, FOS: Mathematics, Mathematics - Numerical Analysis, Neural and Evolutionary Computing (cs.NE), Artificial neural networks and deep learning, First-order hyperbolic equations, Error bounds for boundary value problems involving PDEs, deep learning, Computer Science - Neural and Evolutionary Computing, Deep learning, Numerical Analysis (math.NA), neural networks, adaptive mesh, Adaptive mesh, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Spectral, collocation and related methods for boundary value problems involving PDEs, Neural networks
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