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This paper introduces Modified Spectral-Informed Neural Networks (SINNs), which enhance the accuracy of Physics-Informed Neural Networks (PINNs) in solving high-dimensional partial differential equations (PDEs) by integrating coefficient decay scaling and basis embeddings. The approach addresses the limitations of traditional spectral methods, which struggle with the curse of dimensionality, particularly in high-dimensional settings. Numerical experiments show that Modified SINNs outperform both sparse grid spectral methods in middle dimensions and standard PINNs in high dimensions, demonstrating their effectiveness in approximating unknown spectral coefficients accurately.
Modified SINNs achieve superior accuracy in high-dimensional PDEs, outperforming traditional spectral methods and standard PINNs.
For low-dimensional problems ($d\leq3$), spectral methods can 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, for high-dimensional problems ($d\gg 10$), spectral methods suffer frome the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising approach to overcome this challenge, offering scalability to high dimensions, but often suffer 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 to 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.
