Adaptive Multi-Frequency Representations for Physics-Informed Neural Networks with Structured Variable Separation

Physics-informed neural networks (PINNs) often struggle with multiscale, high-dimensional partial differential equations (PDEs) because of spectral bias and inefficient representations of interactions among variables. We propose adaptive multi-frequency representations for PINNs with structured variable separation (MFAR-SVS). Each neuron combines multiple sinusoidal branches with jointly learnable frequencies and weights, allowing its spectral representation to adapt to the target solution's frequency content rather than relying on fixed or manually selected frequencies. To exploit structured dependencies in high-dimensional solutions, we introduce a bipartite low-rank factorization that preserves unrestricted interactions within each variable group while imposing low-rank coupling only across groups. This provides an intermediate representation between fully coupled networks and dimension-wise tensor decompositions. We derive a closed-form neural tangent kernel (NTK) for the resulting architecture and characterize its infinite-width and infinite-rank limits. Theoretical analysis shows that adaptive frequency learning introduces a positive-semidefinite correction to the fixed-frequency kernel, implying non-decreasing ordered eigenvalues, including the minimum eigenvalue. Controlled experiments further show that adaptive frequencies progressively capture higher-frequency components and yield larger minimum NTK eigenvalues. Across four PDE benchmarks, MFAR-SVS achieves consistent accuracy and convergence gains, with the largest improvements on multi-scale and strongly coupled problems.