Separated-variable spectral neural networks with characteristic-frequency multi-level initialization for high-frequency partial differential equations

Physics-Informed Neural Networks (PINNs) have emerged as a mesh-free paradigm for solving Partial Differential Equations (PDEs), yet a spectral bias severely limits their ability to resolve high-frequency components of oscillatory problems, and conventional Fourier-feature remedies scale exponentially with dimension. To overcome these limitations, we propose Separated-Variable Spectral Neural Networks (SV-SNN), a physics-informed framework with three core innovations. First, a separated-variable spectral representation expresses multivariate solutions as sums of N separable modes, each a product of univariate adaptive Fourier spectral factors with K frequencies per direction; this unites the dimensional scalability of variable separation with the frequency control of spectral methods, reducing the number of trainable parameters from order K d of tensor-product spectral bases to order d N K , linear in the dimension d , while admitting closed-form spatial derivatives through a hybrid differentiation strategy. Second, a characteristic-frequency multi-level sampling strategy allocates spectral resources across low-, characteristic-, and high-frequency bands, thereby targeting spectral bias and reducing sensitivity to frequency-estimation errors, with the characteristic frequency estimated automatically when not available a priori. Third, an effective-rank diagnostic derived from the singular value decomposition of the residual Jacobian quantifies spectral-bias severity a posteriori, explaining why standard architectures fail on oscillatory problems and guiding architecture selection. Across nine benchmarks—high-wavenumber heat and Helmholtz equations, nonlinear elliptic and Poisson problems on complex domains, and incompressible Navier–Stokes flows—SV-SNN achieves lower error than every competing baseline in matched-budget comparisons, by factors of 1.9 to 94, and remains more accurate even when baselines are afforded 15 to 200 times more parameters. Training completes within 11 to 34 s per problem on a single graphics processing unit. SV-SNN thus provides an accurate and parameter-efficient framework for wave propagation, heat conduction, and incompressible flow simulation. The source code is available at https://github.com/xgxgnpu/Physics-informed-vibe-coding/tree/main/SV-SNN .