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We analyze the training dynamics of Fourier Feature PINNs in the Neural Tangent Kernel regime to address spectral bias, deriving an explicit frequency-domain residual error evolution equation. This equation reveals that convergence rates are governed by the product of the differential operator's symbol and initialization spectral density. We then propose an operator-informed initialization strategy that tailors weight distributions to the PDE, balancing frequency convergence and improving prediction accuracy without added training cost.
Operator-informed initialization for Fourier Feature PINNs directly combats spectral bias by tailoring weight distributions to the PDE's differential operator, leading to balanced frequency convergence and improved accuracy.
Physics-Informed Neural Networks (PINNs) typically exhibit spectral bias, where some frequencies of the target function converge more slowly than others. In this work, we analyze the training dynamics of Fourier Feature PINNs in the Neural Tangent Kernel regime to address this limitation. We derive an explicit evolution equation to estimate the residual error in the frequency domain, demonstrating that the convergence rate of specific frequencies is primarily governed by the product of the differential operator's symbol and the spectral density of the initialization weights. Leveraging this theoretical insight, we propose an informative initialization strategy that tailors the initial weight distribution to the specific PDE being solved. With this method, we can diminish the operator-induced spectral bias, balancing the convergence rates across the frequency spectrum and achieving better prediction accuracy. Numerical experiments on linear and nonlinear partial differential equations confirm that this initialization strategy improves learning dynamics and approximation accuracy across frequencies compared to standard initialization methods, with no additional training cost.