发表机构
Applied AI Institute（应用人工智能研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究高维偏微分方程问题，核心方法是将谱方法与PINNs结合成改进的SINNs，通过系数衰减缩放和基嵌入提升精度。主要贡献是在中维问题上优于稀疏网格谱方法，在高维问题上比PINNs精度更高。
AI 中文摘要
对于低维问题（\(d\leq3\)），谱方法能实现高精度。中维问题（\(4 \leq d \lesssim 10\)）可通过稀疏网格等技术解决。但高维问题（\(d\gg 10\)）中谱方法受维数诅咒。物理信息神经网络（PINNs）可克服此挑战，但精度和效率有限。新提出的谱信息神经网络（SINNs）结合谱方法与PINNs，直接在谱域操作。本文引入改进的SINNs，通过系数衰减缩放和基嵌入提高高维精度。实验表明，改进的SINNs在中维问题上优于稀疏网格谱方法，在高维问题上比PINNs精度更高。
英文摘要
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.
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