Scientists often have a decent set of equations for how a system evolves, say part of a biological or physical process, but not all of it. One piece of the mechanism is unknown, and worse, the sensors capture only some of the moving parts, not the full internal state. Purely physical models cannot fill the gap, and a purely neural model throws away the physics you do trust.

This work keeps the known equations as they are and hands only the missing piece to a neural network. Training alternates between two jobs: holding the network fixed, a Rauch-Tung-Striebel smoother (a classic tool for reconstructing hidden states from noisy partial measurements) infers the unobserved variables; then, treating those trajectories as ground truth, it trains the network by backpropagation, repeating until things settle. Across linear, nonlinear, and stiff test systems, it recovers the unknown dynamics while keeping interpretable structure and improving long-range predictions.

That comes from the abstract, so the paper is where the benchmarks and limits are laid out.