We present Decapodes, a diagrammatic tool for representing, composing, and solving partial differential equations. Decapodes provides an intuitive diagrammatic representation of the relationships between variables in a system of equations, a method for composing systems of partial differential equations using an operad of wiring diagrams, and an algorithm for deriving solvers using hypergraphs and string diagrams. The string diagrams are in turn compiled into executable programs using the techniques of categorical data migration, graph traversal, and the discrete exterior calculus. The generated solvers produce numerical solutions consistent with state-of-the-art open source tools as demonstrated by benchmark comparisons with SU2. These numerical experiments demonstrate the feasibility of this approach to multiphysics simulation and identify areas requiring further development.

翻译：我们提出Decapodes，一种用于表示、组合和求解偏微分方程的图解工具。Decapodes提供了一种直观的图解表示方法来描述方程组中变量之间的关系，一种利用布线图操作子组合偏微分方程组的方法，以及一种基于超图和字符串图推导求解器的算法。该字符串图通过分类数据迁移、图遍历和离散外微分技术编译为可执行程序。所生成的求解器能够产生与开源工具SU2基准测试一致的数值解。这些数值实验证明了该方法在多物理场仿真中的可行性，并指出了需要进一步发展的方向。