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.

本文提出了一种高度可扩展的策略，用于从现有的科学计算中的数值离散化来开发免网格神经符号偏微分方程求解器。该策略可用于有效地训练神经网络代理模型，以保留最先进数值求解器的精度和收敛特性，基于在一组随机配置点上使离散化的微分系统残差最小化来进行神经启动。