Hybrid Modeling
Hybrid modeling combines mechanistic, first-principles equations with data-driven components, typically by representing unknown terms (rate laws, flows, closures) inside a known differential or differential-algebraic model with a neural network or other regression model trained on observed data. Neural ODEs, universal differential equations and neural DAEs are common forms; the mechanistic structure supplies physical constraints and extrapolation ability, while the learned part captures relationships that are hard to derive.
Training neural DAEs with the simultaneous approach (2026)
Lueg, Alves, Schicksnus, Kitchin, Laird and Biegler train neural DAEs by fully discretizing the DAE with orthogonal collocation and solving one NLP in which states, unknown terms and network weights are all variables (Lueg et al. 2026). A smooth initialization problem, gray-box network constraints evaluated with JAX and L-BFGS Hessians make small to medium problems tractable, and a bi-level decomposition (Adam on the weights, parallel per-trajectory NLPs differentiated implicitly) scales to 128 trajectories and eight hidden layers. Because constraints are enforced exactly, the method handles an index-2 tank network, a Lyapunov-descent path constraint that improves generalization of a predator-prey model, and constraint-respecting inference for a fed-batch bioreactor where plain integration violated a nonnegativity bound. Summary: lueg2026.