论文标题
神经符号偏微分方程求解器
Neuro-symbolic partial differential equation solver
论文作者
论文摘要
我们提出了一种高度可扩展的策略，用于从科学计算中发现的现有数值离散化中开发无网状神经符号偏微分方程求解器。该策略的独特之处在于它可用于有效地训练解决方案功能和差异操作员的神经网络替代模型，同时保留最先进的数值求解器的准确性和收敛性。这种神经自举方法基于在一组随机搭配点上与神经网络的可训练参数上离散的差分系统的残差最小化，从而实现了前所未有的分辨率和最佳缩放量表，以求解物理和生物系统。
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.
