Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are well studied: here we show how similar workflows can be used to recover functions from data. Specifically, we embed neural networks into the PDE and show how, as they are trained on data, they can approximate unknown functions with arbitrary accuracy. Using nonlocal aggregation-diffusion equations as a case study, we recover interaction kernels and external potentials from steady state data. Specifically, we investigate how a wide range of factors, such as the number of available solutions, their properties, sampling density, and measurement noise, affect our ability to successfully recover functions. Our approach is advantageous because it can utilise standard parameter-fitting workflows, and in that the trained PDE can be treated as a normal PDE for purposes such as generating system predictions.

翻译：偏微分方程中常包含难以或无法直接测量的未知函数，这阻碍了我们从模型中推导预测的能力。从数据中恢复标量偏微分方程参数的工作流程已得到充分研究：本文展示了如何利用类似的工作流程从数据中恢复函数。具体而言，我们将神经网络嵌入偏微分方程，并证明其在数据训练过程中能以任意精度逼近未知函数。以非局部聚集-扩散方程为例，我们从稳态数据中恢复了相互作用核与外部势场。我们重点探究了多种因素（如可用解的数量、解的性质、采样密度及测量噪声）如何影响我们成功恢复函数的能力。该方法的优势在于能够利用标准的参数拟合工作流程，且训练后的偏微分方程可被视为常规方程，用于生成系统预测等目的。