SelectNet: Self-paced Learning for High-dimensional Partial Differential Equations

Abstract

The least squares method with deep neural networks as function parametrization has been applied to solve certain high-dimensional partial differential equations (PDEs) successfully; however, its convergence is slow and might not be guaranteed even within a simple class of PDEs. To improve the convergence of the network-based least squares model, we introduce a novel self-paced learning framework, SelectNet, which quantifies the difficulty of training samples, treats samples equally in the early stage of training, and slowly explores more challenging samples, e.g., samples with larger residual errors, mimicking the human cognitive process for more efficient learning. In particular, a selection network and the PDE solution network are trained simultaneously; the selection network adaptively weighting the training samples of the solution network achieving the goal of self-paced learning. Numerical examples indicate that the proposed SelectNet model outperforms existing models on the convergence speed and the convergence robustness, especially for low-regularity solutions.

23 pages, 27 figures; revised

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• GAS: A Gaussian Mixture Distribution-Based Adaptive Sampling Method for PINNs
• Neural Networks Enforcing Physical Symmetries in Nonlinear Dynamical Lattices: The Case Example of the Ablowitz-Ladik Model
• Structure Probing Neural Network Deflation
• Gradient and Uncertainty Enhanced Sequential Sampling for Global Fit
• Reproducing Activation Function for Deep Learning
• On the Representation of Solutions to Elliptic PDEs in Barron Spaces
• Adaptive Learning on the Grids for Elliptic Hemivariational Inequalities
• Learn bifurcations of nonlinear parametric systems via equation-driven neural networks
• A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs
• Stationary Density Estimation of Itô Diffusions Using Deep Learning