Data driven non-equilibrium moist phase exchanges for atmospheric convection within a discontinuous Galerkin model of the compressible Euler equations

What the math gives to ML

The paper provides a concrete metriplectic-style construction for inserting learned irreversible source terms into a conservative Hamiltonian PDE. Its transferable asset is that a neural constitutive law can predict phase-transfer rates while a skew-symmetric operator preserves the discrete energy identity and a thermodynamic correction determines entropy consistently rather than asking the network to learn all conservation laws from data. This suggests a general neural residual or neural ODE block in which learned forces are projected into a structure-preserving operator, with conservation guaranteed algebraically at every step. The most promising tests are learned physical simulators and latent dynamics models, evaluated by long-horizon energy drift, entropy behavior, and prediction accuracy.