We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral coefficients. In contrast to current machine learning approaches which enforce PDE constraints by minimizing the numerical quadrature of the residuals in the spatiotemporal domain, we leverage Parseval's identity and introduce a new training strategy through a \textit{spectral loss}. Our spectral loss enables more efficient differentiation through the neural network, and substantially reduces training complexity. At inference time, the computational cost of our method remains constant, regardless of the spatiotemporal resolution of the domain. Our experimental results demonstrate that our method significantly outperforms previous machine learning approaches in terms of speed and accuracy by one to two orders of magnitude on multiple different problems. When compared to numerical solvers of the same accuracy, our method demonstrates a $10\times$ increase in performance speed.

翻译：我们提出了一种基于经典谱方法的神经网络谱方法，用于求解参数化偏微分方程（PDE）。该方法利用正交基将PDE解学习为谱系数之间的映射。与当前通过最小化时空域残差数值积分来强制约束PDE的机器学习方法不同，我们利用帕塞瓦尔恒等式提出了一种基于“谱损失”的全新训练策略。该谱损失能够更高效地实现神经网络中的微分运算，并显著降低训练复杂度。在推理阶段，无论时空域分辨率如何变化，我们的方法计算成本始终保持恒定。实验结果表明，在多个不同问题上，我们的方法在速度和精度上均比现有机器学习方法提升一至两个数量级。与相同精度的数值求解器相比，我们的方法实现了10倍的计算性能提升。