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：我们提出神经谱方法，这是一种基于经典谱方法求解参数化偏微分方程（PDE）的技术。该方法利用正交基将PDE解学习为谱系数之间的映射。与当前通过在时空域中最小化残差数值求积来施加PDE约束的机器学习方法不同，我们借助帕塞瓦尔恒等式，引入了一种通过**谱损失**进行训练的新策略。我们的谱损失能够更高效地对神经网络进行微分，并大幅降低训练复杂度。在推理阶段，无论时空域的分辨率如何，我们方法的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显著优于以往的机器学习方法，提升幅度达一至两个数量级。当与同等精度的数值求解器相比时，我们的方法在性能速度上实现了$10\times$的提升。