Fig. 1. (a) An illustration of equivariant message-passing. In order to address the atom-atom interactions, the scalars xi were updated invariant while the tensors $\overrightarrow{x_{i}}$ were updated equivariant. mij and $\overrightarrow{m_{i j}}$ were separately the scalar and directional outputs by the message passing over edge (i,j). (b) the eNequIP architecture, a residual network was added following the NequIP, with applied electric field interact with node features in an invariant scheme. $\overrightarrow{e_{n}}$ donates the applied electric field, Zn the atomic number, and $\overrightarrow{r_{i j}}$ the edge vector. (c) The message-passing blocks in eNequIP. The applied electric field $\overrightarrow{e_{i}}$ was treated similarly with edge vector $\overrightarrow{r_{i j}}$ to maintain equivariant. (d) The architecture of GCMC simulations. Oxygen atom addition, oxygen atom removal, and Cu/O atom displacement are all possible MC actions. When the structure is modified, the Eapplied is updated with PZC-NN to calculate the UPZC. The structure optimization and energy calculation are carried out in eNequIP.

Fig. 2. Evaluation of eNequIP model performance on the Cu-O system. (a) Parity plot of predicted vs. DFT-calculated energies. The model demonstrates high accuracy across a wide energy range (-3000 to 0 eV), with an MAE of 4.2 meV per atom. The inset shows the average atomic energy. (b) Parity plot of predicted vs. DFT-calculated atomic forces. The employed sampling strategy ensured adequate representation of configurations with high-force atoms, resulting in a force MAE of 0.052 eV/?.

Fig. 3. (a) Illustration of the evolution process of CuO (111) surface at -0.8 V vs. RHE w/o eNO3RR; (b-d) The KDE distribution of GCN of surface atoms on operando reduced CuO (111) surfaces. OGCN represents the generalized coordination number of Cu with O atoms, while CuGCN represents the generalized coordination number of Cu with Cu atoms. (b) The OGCN of equilibrium surface Cu over different potentials, with eNO3RR. (c) The CuGCN of equilibrium surface Cu over different potentials, with eNO3RR. (d) Comparison of CuGCN distribution at -0.4 V vs. RHE, with and without eNO3RR.

Fig. 4. Representative equilibrium structures of reduced CuO(111) (a), CuO(100) (b), Cu2O(111) (c), and Cu2O(100) (d) surfaces obtained from GCMC simulations at -0.6 V vs. RHE. Snapshots are taken at the end of the simulations after convergence is achieved. The oxygen number evolution history in GCMC simulation: Cu2O(111) (e) and Cu2O(100) (f). Different color represents the different electrode potential. Blue, orange, green, red and pink represents potential at 0.0, -0.2, -0.4, -0.6 and -0.8 V vs. RHE, respectively. The KDE distribution of CuGCN among CuO(111) (g), CuO(100) (h), Cu2O(111) (i), and Cu2O(100) (j). The potential is set as -0.6 V vs. RHE.

Fig. 5. The microkinetic modeling results. Theoretical results of the selectivity on Cu(100) (a), Cu(111) (b), defect-Cu(100) (c), and defect-Cu(111) (d). (e,f) are the potential-dependent Faradaic efficiency compared with experimental results. (e) is the reduced CuO(111) surface, corresponds to the defect-Cu(100) sites, and is compared with the experimental results of the reduced CuO nanosheets [66] (measured in 1 mol/L KOH + 0.2 mol/L KNO3, alkaline condition). (f) The reduced Cu2O(100) surface, corresponds to a combination of defect-Cu(100) sites and perfect Cu(111) sites. And is compared with the experimental results of the reduced Cu2O nanocubes [67] (measured in 0.1 mol/L Na2SO4, pH-neutral condition).