Physics-informed Neural Networks (PINNs) have been widely used to obtain
accurate neural surrogates for a system of Partial Differential Equations
(PDE). One of the major limitations of PINNs is that the neural solutions are
challenging to interpret, and are often treated as black-box solvers. While
Symbolic Regression (SR) has been studied extensively, very few works exist
which generate analytical expressions to directly perform SR for a system of
PDEs. In this work, we introduce an end-to-end framework for obtaining
mathematical expressions for solutions of PDEs. We use a trained PINN to
generate a dataset, upon which we perform SR. We use a Differentiable Program
Architecture (DPA) defined using context-free grammar to describe the space of
symbolic expressions. We improve the interpretability by pruning the DPA in a
depth-first manner using the magnitude of weights as our heuristic. On average,
we observe a 95.3% reduction in parameters of DPA while maintaining accuracy at
par with PINNs. Furthermore, on an average, pruning improves the accuracy of
DPA by 7.81% . We demonstrate our framework outperforms the existing
state-of-the-art SR solvers on systems of complex PDEs like Navier-Stokes:
Kovasznay flow and Taylor-Green Vortex flow. Furthermore, we produce analytical
expressions for a complex industrial use-case of an Air-Preheater, without
suffering from performance loss viz-a-viz PINNs.

该研究介绍了一种终端到端框架，用于获得解决 PDE 的数学表达式，并使用训练有素的 PINN 生成数据集，使用文法来描述符号表达式空间，并使用 DPA 对其进行修剪以提高可解释性。平均而言，修剪使 DPA 的参数减少 95.3％，同时保持与 PINN 相同的准确性，而且平均而言，修剪将 DPA 的准确性提高了 7.81％。此外，该框架在复杂的 PDE 系统上表现优于现有的 SR 解算器。