FunctionalODE: Reduced-Order Neural Dynamics for Extreme-Deformation Lagrangian Systems

Lagrangian methods discretize a continuum into particles whose positions and properties are tracked over time. Learned graph-based simulators model these dynamics through repeated message passing over particle interaction graphs, so every particle is updated at every step, and the connectivity becomes unstable under extreme deformation and fragmentation. We propose FunctionalODE, a reduced-order alternative that views the system state as a vector-valued function over the fixed reference configuration. FunctionalODE learns an orthonormal basis for the low-dimensional subspace occupied by the states, encodes each state by projection onto this basis, and evolves the resulting compact coefficients with a NeuralODE, avoiding particle-level message passing; the full state can be evaluated at arbitrary material points. On SPH simulations of hypervelocity plate impact, FunctionalODE reaches $98.5\%$ $R^2$ on the fragmenting projectile, compared with $91.8\%$ for the best baseline, and completes a rollout in under a second, about $100\times$ faster than graph-based simulators and the numerical solver. FunctionalODE could serve as a cheap surrogate for expensive hypervelocity-impact simulations and enable the integration of simulation into prediction for physical impact experiments.