论文标题
深度神经网络,可平滑具有高阶和连续性B型基础功能的物理近似
Deep neural networks for smooth approximation of physics with higher order and continuity B-spline base functions
论文作者
论文摘要
本文讨论了以下重要的研究问题。传统上,神经网络采用与线性算子串联的非线性激活函数来近似给定的物理现象。他们用激活函数和线性算子的串联“填充空间”,并调整其系数以近似物理现象。我们声称,最好用同几年分析所采用的光滑高阶B-Splines底座功能的线性组合“填充空间”,并利用神经网络来调整线性组合的系数。换句话说,使用神经网络近似B-Spline基函数的系数并直接近似溶液的可能性。 Maziar Raissi等人提出了用神经网络求解微分方程。在2017年,通过引入物理知识的神经网络(PINN),该神经网络自然地将基本的物理定律编码为先验信息。使用函数作为输入的系数近似利用神经网络的众所周知的能力是通用函数近似值。本质上,在PINN方法中,网络在给定点上近似给定场的值。我们提出了一种替代方法,其中将物理量近似为平滑B型基函数的线性组合,而神经网络近似于B型的系数。这项研究比较了DNN的结果,该DNN近似B-Spline基函数的线性组合的系数与DNN直接近似溶液。我们表明,在近似光滑的物理场时,我们的方法更便宜,更准确。
This paper deals with the following important research question. Traditionally, the neural network employs non-linear activation functions concatenated with linear operators to approximate a given physical phenomenon. They "fill the space" with the concatenations of the activation functions and linear operators and adjust their coefficients to approximate the physical phenomena. We claim that it is better to "fill the space" with linear combinations of smooth higher-order B-splines base functions as employed by isogeometric analysis and utilize the neural networks to adjust the coefficients of linear combinations. In other words, the possibilities of using neural networks for approximating the B-spline base functions' coefficients and by approximating the solution directly are evaluated. Solving differential equations with neural networks has been proposed by Maziar Raissi et al. in 2017 by introducing Physics-informed Neural Networks (PINN), which naturally encode underlying physical laws as prior information. Approximation of coefficients using a function as an input leverages the well-known capability of neural networks being universal function approximators. In essence, in the PINN approach the network approximates the value of the given field at a given point. We present an alternative approach, where the physcial quantity is approximated as a linear combination of smooth B-spline basis functions, and the neural network approximates the coefficients of B-splines. This research compares results from the DNN approximating the coefficients of the linear combination of B-spline basis functions, with the DNN approximating the solution directly. We show that our approach is cheaper and more accurate when approximating smooth physical fields.