Models based on partial differential equations (PDEs) are powerful for describing a wide range of complex relationships in the natural sciences. Accurately identifying the PDE model, which represents the underlying physical law, is essential for a proper understanding of the problem. This reconstruction typically relies on indirect and noisy measurements of the system's state and, without specifically tailored methods, rarely yields symbolic expressions, thereby hindering interpretability. In this work, we address this issue by considering existing neural network architectures based on rational functions for the symbolic representation of physical laws. These networks leverage the approximation power of rational functions while also benefiting from their flexibility in representing arithmetic operations. Our main contribution is an identifiability result, showing that, in the limit of noiseless, complete measurements, such symbolic networks can uniquely reconstruct the simplest physical law within the PDE model. Specifically, reconstructed laws remain expressible within the symbolic network architecture, with regularization-minimizing parameterizations promoting interpretability and sparsity in case of $L^1$-regularization. In addition, we provide regularity results for symbolic networks. Empirical validation using the ParFam architecture supports these theoretical findings, providing evidence for the practical reconstructibility of physical laws.

翻译：基于偏微分方程（PDE）的模型在描述自然科学中各类复杂关系方面具有强大能力。准确识别代表底层物理规律的PDE模型，对于正确理解问题至关重要。这种重构通常依赖于对系统状态的间接且有噪声的测量，若无专门定制的方法，则很少能获得符号表达式，从而阻碍了可解释性。在本工作中，我们通过考虑基于有理函数的现有神经网络架构来解决此问题，这些架构用于物理规律的符号表示。这些网络利用了有理函数的逼近能力，同时也受益于其在表示算术运算方面的灵活性。我们的主要贡献是一个可辨识性结果，表明在无噪声、完整测量的极限情况下，此类符号网络能够唯一地重构PDE模型中最简单的物理规律。具体而言，重构的规律在符号网络架构内仍可表达，且正则化最小化的参数化在$L^1$正则化情况下促进了可解释性和稀疏性。此外，我们提供了符号网络的正则性结果。使用ParFam架构进行的实证验证支持了这些理论发现，为物理规律的实际可重构性提供了证据。