Backflow transformations improve variational wave functions by making single-particle orbitals depend on the positions of all other particles. The transformation replaces r_i with r_i + η_i(r₁,...,r_N), where η_i is a many-body displacement that encodes correlations. Classical backflow uses a fixed functional form for η — typically a sum of pairwise functions.
Neural network backflow replaces the fixed form with a learned function. The network takes all particle positions as input and outputs the backflow displacements for each particle. The functional form is unconstrained — the network can represent any smooth correlation structure, not just pairwise.
The paper applies neural backflow to ab initio solid-state calculations for the first time. Previous neural wave functions for solids used Slater-Jastrow forms with explicit correlation factors but without backflow. Adding neural backflow to a Slater determinant improves energies substantially — the correlation energy captured increases from ~50% (Slater-Jastrow) to ~85-95% (Slater-Jastrow-backflow) for the test systems.
The improvement is largest in systems with strong correlations where pairwise Jastrow factors are insufficient. The neural backflow captures three-body and higher correlations that the Jastrow factor misses, and it does so without specifying in advance which many-body correlations matter. The network discovers the relevant correlation structure from the variational optimization.
The wave function's accuracy depends not on the physicist's intuition about which correlations are important but on the network's capacity to represent them. The ansatz shift: from “model the physics, encode it in the wave function” to “provide enough representational capacity and let the optimization find the physics.”