Physics-informed neural networks learn PDE solutions by minimizing residuals. But neural networks have a frequency bias: they learn low-frequency components first. And differential operators have the opposite bias: they amplify high-frequency components. The network wants to start smooth; the physics demands sharpness. These competing biases create an optimization landscape where high-frequency modes are simultaneously the hardest to learn and the most penalized by the loss function.
Chiu et al. precondition the residual with an integral operator — the inverse of the differential operator, approximated by the Green's function of a reference elliptic problem. Applying the integral operator before computing the loss damps the high-frequency amplification of the PDE operator, rebalancing the spectral landscape. The network no longer needs to fight its own frequency bias to satisfy the physics.
The approach integrates directly with existing PINN solvers. The Green's function need not be exact — an approximation suffices because the goal is spectral rebalancing, not exact inversion. The preconditioning substantially improves convergence speed and accuracy on multiscale and variable-coefficient problems.
The through-claim is about diagnosing optimization failures through spectral lenses. The standard interpretation of PINN difficulty is that the loss landscape is rough or the network capacity is insufficient. The spectral interpretation is more specific: the optimization fails because the loss function weights frequency modes in a way that fights the network's inductive bias. The fix is not more capacity or better optimization. It is adjusting what the loss function considers important, bringing the physics and the network into spectral agreement.