Learning piecewise-smooth dynamical systems

What the math gives to ML

The paper offers a constructive identification framework for piecewise-smooth vector fields: estimate affine switching hyperplanes, partition state space into polyhedral cells, and learn a separate smooth vector field in each cell. Its transferable asset is an explicitly prescribed discontinuity geometry rather than an implicit approximation by a very steep smooth network. For neural ODEs, state-space models, and world models, this suggests architectures with learned hyperplane gates and region-specific dynamics, together with Filippov convexification at switching surfaces to represent sliding motion. The sharp experimental signature is that errors and regime transitions should align with learned hyperplanes, while trajectories near a surface should follow the convexified tangential flow rather than numerically chatter.