An immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains

DOI10.3934/mbe.2021002zbMath1493.65246OpenAlexW3105450412MaRDI QIDQ1980025

Francisco Hernandez-Lopez, Miguel Uh Zapata, Reymundo Itzá Balam, Joel Antonio Trejo-Sánchez

Publication date: 3 September 2021

Published in: Mathematical Biosciences and Engineering (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.3934/mbe.2021002

zbMATH Keywords

neural networkscontinuous time modelinterfacetwo-phase flowelliptic equationimmersed boundary methoddeep learningsingular forces

Mathematics Subject Classification ID

Artificial neural networks and deep learning (68T07) Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Finite difference methods for boundary value problems involving PDEs (65N06) Higher-order elliptic equations (35J30) PDEs on graphs and networks (ramified or polygonal spaces) (35R02) Liquid-liquid two component flows (76T06)

Related Items

A fourth-order compact implicit immersed interface method for 2D Poisson interface problems, A shallow Ritz method for elliptic problems with singular sources, A compact sixth-order implicit immersed interface method to solve 2D Poisson equations with discontinuities

Uses Software

• LBFGS-B
• ADMB
• AD Model Builder
• na14
• TensorFlow
• DiffSharp
• PyDEns
• DeepXDE

Cites Work

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• A Brownian dynamics model of kinesin in three dimensions incorporating the force-extension profile of the coiled-coil cargo tether
• Weak adversarial networks for high-dimensional partial differential equations
• Numerical simulation of bubble rising in viscous liquid
• Implicit second-order immersed boundary methods with boundary mass
• Numerical analysis of blood flow in the heart
• The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
• Remarks around 50 lines of Matlab: short finite element implementation
• DGM: a deep learning algorithm for solving partial differential equations
• Physics-informed neural networks for high-speed flows
• Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data
• Sequential quadratic programming method for nonlinear least squares estimation and its application
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Flow patterns around heart valves: A numerical method
• Immersed Boundary Method for Variable Viscosity and Variable Density Problems Using Fast Constant-Coefficient Linear Solvers I: Numerical Method and Results
• AD Model Builder: using automatic differentiation for statistical inference of highly parameterized complex nonlinear models
• IMMERSED BOUNDARY METHODS
• The immersed boundary method
• Penalty immersed boundary method for an elastic boundary with mass
• Solution of the Poisson equation by differential quadrature
• The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources
• A Limited Memory Algorithm for Bound Constrained Optimization
• Solving high-dimensional partial differential equations using deep learning
• Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
• DeepXDE: A Deep Learning Library for Solving Differential Equations
• Dynamics of a Closed Rod with Twist and Bend in Fluid
• The Immersed Interface Method for Simulating Two-Fluid Flows
• 2–D Parachute Simulation by the Immersed Boundary Method