Hybrid Generative Modeling for Incomplete Physics: Deep Grey-Box Meets Optimal Transport

Physics phenomena are often described by ordinary and/or partial differential equations (ODEs/PDEs), and solved analytically or numerically. Unfortunately, many real-world systems are described only approximately with terms in the equations missing. This makes the distribution of the model differ from the true data-generating process (DGP). Using limited and unpaired data between DGP observations and the imperfect model simulations, we investigate this particular setting by completing the known physics model, combining theory-driven models and data-driven as to describe the shifted distribution involved in the DGP. We present a novel hybrid generative model approach combining Deep grey-box modeling with Optimal Transport (OT) methods to enhance incomplete physics models. Our method implements OT maps in data space while maintaining minimal source distribution distortion, demonstrating superior performance in resolving unpaired marginal problems and ensuring correct usage of physics parameters. Unlike black-box alternatives, our approach leverages physics-based inductive biases to accurately learn system dynamics while preserving interpretability through its domain knowledge foundation. Experimental results validate our method's effectiveness in both generation tasks and model transparency, offering detailed insights into learned physics dynamics.