ReLU$^k$ Neural de Rham Complexes

What the math gives to ML

The paper provides a finite-dimensional neural analogue of the de Rham complex: fixed-neuron ReLU ridge functions are assigned to differential-form degrees so that the exterior derivative is represented exactly inside the network space. The key transferable asset is structural rather than a new scalar activation: differentiation lowers the ReLU power and becomes a fixed wedge operation, yielding neuron-wise Koszul complexes, exactness, and therefore no spurious closed-but-non-exact modes under a linear-independence condition. A practical use is a multi-form neural architecture for PDEs and geometric learning in which potentials, vector fields, curls, and divergences are connected by hard algebraic maps instead of soft penalties.