New Method Solves High-Frequency PDEs with Physics-Informed Machine Learning

Solving partial differential equations (PDEs) that exhibit high-frequency solutions poses a significant challenge for physics-informed machine learning (PIML) models. Neural networks typically suffer from spectral bias, meaning they preferentially learn the lower-frequency components of a solution, leading to inaccuracies when high-frequency phenomena are present. This new research proposes a solution called the Frequency Shift Physics-Informed Extreme Learning Machine (FS-PIELM). The FS-PIELM framework tackles spectral bias through an innovative additive mechanism for initializing network weights. Instead of simply scaling random weights, which can amplify variance, this method shifts the mean of the Gaussian weight distribution while keeping the variance constant. This approach ensures that the frequency variance remains bounded, even for high target frequencies, a crucial improvement over conventional methods where variance grows quadratically. Two variants, FS-PIELM-L and FS-PIELM-G, were developed, with the linear variant (FS-PIELM-L) demonstrating superior accuracy. Experiments across seven benchmark problems, including Helmholtz and wave equations, showed that FS-PIELM-L achieved improvements of one to nearly five orders of magnitude compared to existing PIELM variants. Crucially, the method retains the computational efficiency characteristic of extreme learning machines, requiring only a single linear solve.