A Vector-Valued Co-Chain/Chain Complex Associated to the Elasticity Complex

What the math gives to ML

The paper replaces the differential operators of the elasticity complex by algebraic maps on vector-valued chains and cochains over a simplicial mesh while preserving the complex structure. Its most transferable asset is the exact commuting relation between continuous operators and discrete interpolation, namely that discretization followed by an elasticity operator agrees with applying the operator before discretization. This suggests a mesh neural architecture with separate node, edge, face, and cell feature channels whose fixed message-passing maps satisfy complex identities exactly, reducing discretization artifacts and improving transfer across mesh resolutions.