Interpretable Surrogate Constraints for Physics-Informed Networks with Partially Observed Dynamics and Parameters
DOI:
https://doi.org/10.66238/gmv69t57Keywords:
Physics-Informed Neural Networks, Surrogate Modeling, Interpretable Machine Learning, Partially Observed DynamicsAbstract
The integration of deep learning with fundamental physical laws has led to the emergence of physics-informed neural networks, which have demonstrated exceptional capabilities in solving forward and inverse problems in computational mechanics and fluid dynamics. However, classical physics-informed architectures operate under the stringent assumption that the governing partial differential equations are fully specified, and that boundary or initial conditions are sufficiently resolved. In contemporary scientific computing, researchers frequently encounter scenarios characterized by partially observed dynamics, where governing equations contain unresolved terms, unknown parameter fields, or hidden physical mechanisms. This paper proposes a novel framework centered on interpretable surrogate constraints to address these challenges. By embedding symbolic and polynomial surrogate models directly into the loss landscape of physics-informed networks, the proposed methodology enforces analytical tractability and physical interpretability without sacrificing the universal approximation properties of deep neural networks. The surrogate constraints act as dynamic regularizers that identify and reconstruct the missing phenomenological behaviors from sparse observational data. Comprehensive theoretical analysis is provided alongside extensive numerical experiments involving complex spatiotemporal systems. The results indicate that the integration of interpretable surrogate formulations significantly enhances predictive accuracy, robustly recovers unknown system parameters, and provides a transparent mathematical description of the hidden dynamics.
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