We propose an explainable method for solving Partial Differential Equations by using a contextual scheme called PDExplain. During the training phase, our method is fed with data collected from an operator-defined family of PDEs accompanied by the general form of this family. In the inference phase, a minimal sample collected from a phenomenon is provided, where the sample is related to the PDE family but not necessarily to the set of specific PDEs seen in the training phase. We show how our algorithm can predict the PDE solution for future timesteps. Moreover, our method provides an explainable form of the PDE, a trait that can assist in modelling phenomena based on data in physical sciences. To verify our method, we conduct extensive experimentation, examining its quality both in terms of prediction error and explainability.

翻译：我们提出了一种可解释方法，名为PDExplain，通过情境化方案求解偏微分方程。在训练阶段，我们的方法输入由算子定义的偏微分方程族数据，并辅以该族的通式表示。在推理阶段，仅需提供从物理现象中采集的最小样本——该样本虽与偏微分方程族相关，但未必属于训练阶段所见的具体偏微分方程集合。我们展示了算法如何预测未来时间步的偏微分方程解。此外，该方法能呈现偏微分方程的可解释形式，这一特性可辅助物理科学领域基于数据建立现象模型。为验证方法有效性，我们从预测误差与可解释性两个维度开展了系统的实验评估。