Deep learning methods for stochastic Galerkin approximations of elliptic random PDEs

Fabio Musco, Andrea Barth
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Abstract

This work considers stochastic Galerkin approximations of linear elliptic partial differential equations with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and infinity. A traditional numerical method for solving the resulting high-dimensional coupled system of partial differential equations (PDEs) is replaced by deep learning techniques. In order to achieve this, physics-informed neural networks (PINNs), which typically operate on the strong residual of the PDE and can therefore be applied in a wide range of settings, are considered. As a second approach, the Deep Ritz method, which is a neural network that minimizes the Ritz energy functional to find the weak solution, is employed. While the second approach only works in special cases, it overcomes the necessity of testing in variational problems while maintaining mathematical rigor and ensuring the existence of a unique solution. Furthermore, the residual is of a lower differentiation order, reducing the training cost considerably. The efficiency of the method is demonstrated on several model problems.
椭圆随机 PDE 随机 Galerkin 近似的深度学习方法
这项研究考虑了线性椭圆偏微分方程的随机 Galerkin 近似,该方程具有随机强迫项和随机扩散系数,无法均匀地远离零和无限。深度学习技术取代了求解由此产生的高维耦合偏微分方程(PDEs)系统的传统数值方法。为了实现这一目标,我们考虑了物理信息神经网络(PINNs),这种网络通常在偏微分方程的强残差上运行,因此可以应用于多种场合。作为第二种方法,我们采用了深度里兹法(Deep Ritz method),这是一种最小化里兹能量函数以找到弱解的神经网络。虽然第二种方法只适用于特殊情况,但它克服了在变分问题中进行测试的必要性,同时保持了数学上的合理性,并确保了唯一解的存在。此外,该方法的微分阶数较低,大大降低了训练成本。在几个模型问题上证明了该方法的效率。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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