Optimally weighted loss functions for solving PDEs with neural networks

DOI10.1016/j.cam.2021.113887OpenAlexW3215332512WikidataQ114201957 ScholiaQ114201957MaRDI QIDQ2068635

Remco van der Meer, Anastasia Borovykh, Cornelis W. Oosterlee

Publication date: 20 January 2022

Published in: Journal of Computational and Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2002.06269

zbMATH Keywords

convection-diffusion equationneural networkpartial differential equationPoisson equationhigh-dimensional problemsloss functional

Mathematics Subject Classification ID

Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs (65N75) Artificial neural networks and deep learning (68T07) Probabilistic models, generic numerical methods in probability and statistics (65C20) Numerical optimization and variational techniques (65K10) Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Diffusion (76R50) Second-order elliptic equations (35J15) Numerical methods for partial differential equations, boundary value problems (65N99)

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Uses Software

• LBFGS-B
• Adam
• DGM

Cites Work

• Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures
• Machine learning of linear differential equations using Gaussian processes
• Multilayer feedforward networks are universal approximators
• DGM: a deep learning algorithm for solving partial differential equations
• Transfer learning based multi-fidelity physics informed deep neural network
• Machine learning for semi linear PDEs
• Brittleness of Bayesian inference under finite information in a continuous world
• Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations
• A Limited Memory Algorithm for Bound Constrained Optimization
• Solving high-dimensional partial differential equations using deep learning
• Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black--Scholes Partial Differential Equations
• Bayesian Numerical Homogenization
• Breaking the Curse of Dimensionality with Convex Neural Networks
• Unnamed Item

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