PhDN: A Physics-embedded Dictionary Network for Data-Driven Discovery of Physical Relationships

Discovering physical relationships from data requires flexible approximation, symbolic interpretability, and incorporation of physical priors. Black-box neural networks offer strong approximation but limited access to physical structure, while interpretable dictionary and symbolic methods remain sensitive to predefined features, operator grammars, or discrete structural choices. We propose the Physics-embedded Dictionary Network (PhDN), a compositional explicit-dictionary architecture that organizes SINDy-style local dictionaries as a multilayer differentiable DAG with reusable intermediate states. Local dictionaries combine symbolic expressions, problem-specific operators, and approximation bases, so symbolic structure can be compiled, refined, and augmented within PhDN. PhDN can exactly represent finite symbolic expressions built from its admissible operators, while increasing polynomial or neural dictionaries provide universal approximation on compact domains. When informative structure is unavailable, SR supplies a skeleton for PhDN compilation; partial physical knowledge can enter as soft structural priors. Additional dictionary bases enable local correction without renewed symbolic search. Experiments on Feynman relationships and two dynamical systems show favorable ID/OOD accuracy. Staged ablations show fixed-support gains mainly reflect parameter refinement, while added bases address structural mismatch. Informative priors improve recovery and rollout robustness under derivative-label noise.