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, equation parameters, discretizations, etc., in 1D and 2D.

翻译：偏微分方程（PDE）的数值求解十分困难，至今已有长达一个世纪的研究。近年来，学界开始推动构建神经-数值混合求解器，这顺应了当前向完全端到端学习系统发展的趋势。目前大多数工作仅能泛化至通用求解器所面临的部分属性，包括分辨率、拓扑结构、几何形状、边界条件、域离散正则性、维度等。在本工作中，我们构建了一个满足上述所有属性的求解器，其所有组件均基于神经消息传递机制，将计算图中所有启发式设计的组件替换为基于反向传播优化的神经函数逼近器。我们证明，神经消息传递求解器在表示能力上包含经典方法，如有限差分法、有限体积法和WENO格式。为促进自回归模型训练中的稳定性，我们提出一种基于零稳定性原理的方法，将稳定性问题视为域适应任务。我们在多种流体类流动问题上验证了该方法，展示了其在1D和2D场景中跨越不同域拓扑、方程参数、离散格式等条件下的快速、稳定且准确的性能。