A Local Deep Learning Method for Solving High Order Partial Differential Equations

Abstract

At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly, and high order derivatives lack robustness for training purposes. We propose a novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variables to rewrite the PDEs into a system of low order differential equations as what is done in the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neural network. By taking the residual of the system as a loss function, we can optimize the network parameters to approximate the solution. The whole process relies on low order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularly well-suited for high-dimensional PDEs with high order derivatives.

Error Analysis of the Mixed Residual Method for Elliptic Equations

“…In recent years, deep learning-based methods [7] have emerged as a promising approach for tackling high-dimensional PDEs. Notable examples of these methods include the deep Ritz method (DRM) [4,8,10], the deep Galerkin method (DGM) [20], the physics informed neural network method [19], the weak adversarial network method [22], the deep least-squares methods [3], least-squares ReLU neural network method [2] and the local deep learning method [21]. These techniques have shown significant potential for solving highdimensional PDEs, offering an attractive alternative to traditional methods.…”

PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equations

“…In recent years, the use of deep learning to solve fundamental partial differential equations (PDEs) has gained considerable attention [10,24,26], thanks to the high expressiveness of neural networks and the rapid growth of computing hardware. Among them, physics-informed neural networks (PINNs) [5,11,16,17,19,23,[27][28][29]31] are a particularly interesting approach. PINNs incorporate physical knowledge as soft constraints in the empirical loss function and employ machine learning methodologies like automatic differentiation and stochastic optimization to train the model.…”