论文标题
使用神经网络求解PDE的深度傅立叶残留方法
A Deep Fourier Residual Method for solving PDEs using Neural Networks
论文作者
论文摘要
当将神经网络用作试验函数来求解PDE时，要做出的关键选择是要最小化的损失函数，理想情况下应与误差的规范相对应。在多个问题中，此错误规范与 - 或等同于 - 残差的$ h^{ - 1} $ - norm;但是，通常很难准确地计算它。这项工作假设矩形域，并提出使用离散的正弦/余弦变换来准确有效地计算$ h^{ - 1} $ norm。产生的深傅里叶残差（DFR）方法有效，准确地将解决方案近似于PDE。当解决方案缺乏$ h^{2} $规律性和涉及PDE失败强大表述的方法时，这一点尤其有用。我们观察到，$ h^1 $ -Error与训练过程中离散的损失高度相关，这允许通过损失进行准确的误差估算。
When using Neural Networks as trial functions to numerically solve PDEs, a key choice to be made is the loss function to be minimised, which should ideally correspond to a norm of the error. In multiple problems, this error norm coincides with--or is equivalent to--the $H^{-1}$-norm of the residual; however, it is often difficult to accurately compute it. This work assumes rectangular domains and proposes the use of a Discrete Sine/Cosine Transform to accurately and efficiently compute the $H^{-1}$ norm. The resulting Deep Fourier-based Residual (DFR) method efficiently and accurately approximate solutions to PDEs. This is particularly useful when solutions lack $H^{2}$ regularity and methods involving strong formulations of the PDE fail. We observe that the $H^1$-error is highly correlated with the discretised loss during training, which permits accurate error estimation via the loss.
