The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, discretization, etc. in 1D and 2D. Our model outperforms state-of-the-art numerical solvers in the low resolution regime in terms of speed and accuracy.
翻译:部分差异方程式( PDEs) 的数值解决方案是困难的, 导致了一个世纪的研究。 最近, 出现了建立神经- 数字混合求解器的推力, 将现代完全端对端学习的系统趋势搭载起来。 多数迄今为止的工作只能对一个通用求解器可能面临的一系列属性进行概括化, 包括: 分辨率、 地形学、 几何学、 边界条件、 域分解规律、 维度等。 在这项工作中, 我们建立了一个解决方案, 满足了这些属性, 所有组件都以神经信息传递为基础, 将计算图中的所有超常设计构件替换为反向偏偏向式的神经功能代谢器。 我们显示, 神经信息传递解答器包含一些典型方法, 如: 分辨率的有限差异、 数量有限 和 WENO 计划等。 为了鼓励在培训自制偏向型模型中保持稳定, 我们提出了一种基于零可性原则的方法, 将稳定性作为域适应问题的域模型。 我们验证了我们的方法, 以各种离差的精确性、 直径、 直径、 直径、 直径、 直径、 直径、 直径、 直径、 度 直径的域的系统、 解、 解、 解、 解、 解、 解、 直径直径直径直径向、 直径向、 直径直径向、 等的系统 等的系统 的系统 的系统 解的系统 解的系统的系统的系统的系统 。