We present a scalable strategy for development of mesh-free hybrid neuro-symbolic partial differential equation solvers based on existing mesh-based numerical discretization methods. Particularly, this strategy can be used to efficiently train neural network surrogate models of partial differential equations by (i) leveraging the accuracy and convergence properties of advanced numerical methods, solvers, and preconditioners, as well as (ii) better scalability to higher order PDEs by strictly limiting optimization to first order automatic differentiation. The presented neural bootstrapping method (hereby dubbed NBM) is based on evaluation of the finite discretization residuals of the PDE system obtained on implicit Cartesian cells centered on a set of random collocation points with respect to trainable parameters of the neural network. Importantly, the conservation laws and symmetries present in the bootstrapped finite discretization equations inform the neural network about solution regularities within local neighborhoods of training points. We apply NBM to the important class of elliptic problems with jump conditions across irregular interfaces in three spatial dimensions. We show the method is convergent such that model accuracy improves by increasing number of collocation points in the domain and predonditioning the residuals. We show NBM is competitive in terms of memory and training speed with other PINN-type frameworks. The algorithms presented here are implemented using \texttt{JAX} in a software package named \texttt{JAX-DIPS} (https://github.com/JAX-DIPS/JAX-DIPS), standing for differentiable interfacial PDE solver. We open sourced \texttt{JAX-DIPS} to facilitate research into use of differentiable algorithms for developing hybrid PDE solvers.

翻译：我们提出了一种可扩展的策略，用于基于现存的基于网格的数值离散化方法开发无网格混合神经符号偏微分方程求解器。特别地，该策略通过以下方式高效训练偏微分方程的神经网络替代模型：（i）利用先进数值方法、求解器和预条件子的精度与收敛性，以及（ii）通过严格限制优化于一阶自动微分，实现对高阶偏微分方程更好的可扩展性。所提出的神经自举方法（此处简称为NBM）基于在随机配点集中心的隐式笛卡尔单元上获得的偏微分方程系统的有限离散化残差相对于神经网络可训练参数的评估。重要的是，被自举的有限离散化方程中的守恒律和对称性将训练点局部邻域内的解正则性信息传递给神经网络。我们将NBM应用于三维空间中不规则界面处具有跳跃条件的重要椭圆问题类别。我们证明了该方法是收敛的，即模型精度通过增加域内配点数量和对残差进行预条件而提高。我们展示了NBM在内存和训练速度方面与其他PINN型框架具有竞争力。本文提出的算法使用\texttt{JAX}在名为\texttt{JAX-DIPS}（https://github.com/JAX-DIPS/JAX-DIPS）的软件包中实现，该软件包代表可微界面偏微分方程求解器。我们开源了\texttt{JAX-DIPS}，以促进利用可微算法开发混合偏微分方程求解器的研究。