Modeling of an ODE-constrained optimization problem describing tumor dynamics, and numerical approximation via sequential physics-informed neural networks

What the math gives to ML

The transferable contribution is a time-domain domain-decomposition strategy for neural differential-equation solvers: replace one long-horizon network by locally trained networks whose initial values are inherited exactly from the preceding interval. This reduces the burden of representing long trajectories and prevents accumulated interface drift, while retaining a shared physics residual objective. The most promising neural-network use is a hard-constrained sequential architecture for long-horizon neural ODEs, PINNs, differentiable simulators, and adjoint-based training. The paper's numerical evidence suggests that exact interface handling is more robust than merely penalizing continuity, although it costs additional forward passes and parameters.