In low-dimensional problems ($d \leq 3$), spectral methods achieve exceptionally high accuracy. For middle-dimensional problems ($4 \leq d \lesssim 10$), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, in high-dimensional problems ($d \gg 10$), spectral methods suffer from the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising approach to overcome this challenge, offering scalability to high dimensions, yet often suffering from limited accuracy and efficiency.
Recently proposed spectral-informed neural networks (SINNs) combine spectral methods with PINNs, operating directly in the spectral domain to avoid spatial derivative computations and reduce memory consumption. In this work, we introduce Modified SINNs, which integrate coefficient decay scaling and basis embeddings motivated by harmonic analysis to enhance accuracy in high-dimensional problems and enable accurate approximation of unknown spectral coefficients. Numerical experiments on steady and time-dependent partial differential equations demonstrate that Modified SINNs outperform sparse grid spectral methods on middle-dimensional problems with incomplete spectral information and achieve superior accuracy compared to PINNs on high-dimensional problems.
Blogger's Review: The Modified Spectral-Informed Neural Networks (SINNs) offer a compelling solution to tackle high-dimensional partial differential equations by merging spectral methods with PINNs. This approach not only enhances computational accuracy but also reduces memory consumption, paving the way for new possibilities in solving high-dimensional problems that warrant further exploration and application.