  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.
