Models based on partial differential equations (PDEs) are powerful for describing a wide range of complex phenomena 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 limiting interpretability. In this work, we address this limitation by considering neural network architectures based on rational functions for the symbolic representation of physical laws. These networks combine the approximation power of rational functions with the flexibility to represent arithmetic operations, and generalize ParFam and EQL-type architectures used in symbolic regression for physical law learning. We further establish regularity results for these symbolic networks. Our main contribution is a reconstruction result showing that, if there exists an admissible physical law that is expressible within the symbolic network architecture, then in the limit of noiseless and complete measurements, symbolic networks recover a physical law within the PDE model that is representable by the architecture. Moreover, the recovered law corresponds to a regularization-minimizing parameterization, promoting interpretability and sparsity in case of $L^1$-regularization. Under an additional identifiability condition, the unique true physical law is recovered. These reconstruction and regularity results are derived at the continuous level prior to discretization due to a formulation in function space. Empirical results using the ParFam architecture are consistent with the theoretical findings and suggest the feasibility of reconstructing interpretable physical laws in practice.

翻译：基于偏微分方程（PDE）的模型是描述自然科学中广泛复杂现象的强有力工具。准确识别代表潜在物理定律的PDE模型对于正确理解问题至关重要。这种重建通常依赖于对系统状态的间接且有噪声的测量，且在不借助专门定制方法的情况下，很少能生成符号表达式，从而限制了可解释性。在本工作中，我们通过考虑基于有理函数的神经网络架构用于物理定律的符号表示，来解决这一局限性。这类网络结合了有理函数的逼近能力与表示算术运算的灵活性，并推广了符号回归中用于物理定律学习的ParFam和EQL类架构。我们进一步建立了这些符号网络的正则性结果。我们的主要贡献是一个重建结果：如果存在一个可在符号网络架构内表达的可行物理定律，那么在无噪声且完备测量的极限条件下，符号网络可恢复该PDE模型内可由该架构表示的物理定律。此外，恢复的定律对应一个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。在额外的可辨识性条件下，可恢复唯一真实的物理定律。由于在函数空间中进行了公式化，这些重建与正则性结果是在离散化之前的连续层面推导得出的。使用ParFam架构的实证结果与理论发现一致，并表明在实践中恢复可解释的物理定律是可行的。