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 (NIE). 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.

翻译：物理信息神经网络（PINNs）是一种用于求解微分方程的新兴有前途方法。与许多其他深度学习方法类似，PINN的设计与训练协议需要精细的技巧。在此，我们提出了一个全面的理论框架，以阐明这一重要问题。利用无限过参数化神经网络与高斯过程回归（GPR）之间的等价性，我们推导出一个在大数据集极限下支配PINN预测的积分-微分方程——神经信息方程（NIE）。该方程通过反映架构选择的核项对原始方程进行扩充，并允许通过原始微分方程中源项的频谱分解来量化网络引入的隐式偏差。