  Physically informed neural networks (PINNs) are a promising emerging method
for solving differential equations. As in many other deep learning approaches,
the choice of PINN design and training protocol requires careful craftsmanship.
Here, we suggest a comprehensive theoretical framework that sheds light on this
important problem. Leveraging an equivalence between infinitely
over-parameterized neural networks and Gaussian process regression (GPR), we
derive an integro-differential equation that governs PINN prediction in the
large data-set limit -- the neurally-informed equation. This equation augments
the original one by a kernel term reflecting architecture choices and allows
quantifying implicit bias induced by the network via a spectral decomposition
of the source term in the original differential equation.
