We present a highly scalable strategy for developing mesh-free neuro-symbolic partial differential equation solvers from existing numerical discretizations found in scientific computing. This strategy is unique in that it can be used to efficiently train neural network surrogate models for the solution functions and the differential operators, while retaining the accuracy and convergence properties of state-of-the-art numerical solvers. This neural bootstrapping method is based on minimizing residuals of discretized differential systems on a set of random collocation points with respect to the trainable parameters of the neural network, achieving unprecedented resolution and optimal scaling for solving physical and biological systems.

翻译：我们提出了一个高度可扩展的战略,从科学计算中发现的现有数字分化中开发出无网状神经 -- -- ylmbolic部分偏差方程式,这一战略具有独特性,因为它可用于高效地培训神经网络模型,以替代解决方案功能和不同操作者,同时保留最先进的数字解答器的准确性和趋同性。 这种神经靴式方法的基础是将离散差异系统的残渣减少到一套随机合用点,即神经网络的可训练参数,实现前所未有的解析和最佳规模,以解决物理和生物系统。