). Then, we introduce the TENN-e3, with E(3) equivariant neural network (E3NN) considering time-reversal symmetry shown in Figure . TENN-e3, with spin as input, can be used to construct the spin potentials and Hamiltonian for the magnetic material. The input of TENN-e3 for a structure graph not only includes bond displacement vectors 𝑟 𝑖𝑗 ⃗⃗⃗⃗ between atoms indexed with 𝑖 and 𝑗 and atomic numbers 𝑧 𝑖 that included in E3NN, but also the spin vector 𝑠 𝑖 ⃗⃗⃗ of magnetic atoms with odd time-reversal index. The time-reversal and parity symmetry of some possible input scalars and vectors for TENN-e3 are listed in based on noncollinear magnetic. For the status of considering collinear magnetic, the input spin will be represented as a scalar with an odd time-reversal index and even parity. In the same way, the scalars are easy to be considered in node embedding and edge featuring including scalars and vectors listed in Table . In a word, with TENN, the invariant spin model can be manufactured considering time-reversal E(3) or E(3) or time-reversal SE(3) or SO(3) symmetry and whether to include spin-orbit coupling for both noncollinear and collinear magnetic states. For building the spin potential based on TENN, a spin model can be constructed according to the situation considered and use the output scalar with an even time-reversal index and even parity as the final energy. features of other edges are set to zero to distinguish edges that 𝑠 𝑖 ⃗⃗⃗ ⋅ 𝑠 𝑗 ⃗⃗⃗ = 0. At last but not least, the forces on the spin can be computed using auto differentiation according to its definition as the gradient of the total energy with respect to the spin vector of the atom just as same as how forces are computed. Benefited from the strict locality of the design of Allegro, spin forces can be calculated based only on the neighbor atoms of the central atoms and Spin-Allegro is easy to be parallelized across different devices, which makes it powerful to do large-scale spin-lattice dynamics. More to say, for simple Heisenberg-type spin potentials, the edge energy in Allegro can be treated as the Heisenberg coefficients and express the energy with ∑ 𝐽 𝑖𝑗 Allegro (𝑟)⟨𝑠 𝑖 ⃗⃗⃗, 𝑠 𝑗 ⃗⃗⃗ ⟩ 𝑖,𝑗