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策略，用于基于现有网格数值离散方法开发无网格混合神经符号偏微分方程求解器。具体而言，该策略通过以下方式高效训练偏微分方程的神经网络代理模型：(i) 利用先进数值方法、求解器和预条件器的精度与收敛性，以及 (ii) 将优化严格限制为一阶自动微分，以提升对高阶偏微分方程的扩展能力。本文提出的神经自助法（以下简称NBM）基于评估偏微分方程在隐式笛卡尔网格单元上的有限离散残差，这些单元以随机配位点为中心，且残差相对于神经网络的可训练参数进行计算。重要的是，被自助的有限离散方程中蕴含的守恒律与对称性，为神经网络提供了训练点局部邻域内解的正则性信息。我们将NBM应用于三维空间中跨非规则界面的跳跃条件椭圆问题这一重要类别。研究证明该方法具有收敛性，即模型精度随域内配位点数量增加及残差预条件处理而提升。实验表明，NBM在内存与训练速度方面与其他PINN类框架具有竞争力。本文所述算法采用JAX实现，并封装于名为JAX-DIPS（https://github.com/JAX-DIPS/JAX-DIPS）的软件包中，其全称为可微界面偏微分方程求解器。我们已开源JAX-DIPS，以促进利用可微算法开发混合偏微分方程求解器的相关研究。