FlowBack-Adjoint: Energy-Guided Conditional Flow-Matching for Protein Side-Chain Generation

Coarse-grained (CG) molecular models of proteins expand the time and length scales accessible to molecular dynamics simulations, but many scientific applications require recovering accurate all-atom (AA) detail. Recent work introduced FlowBack, a deep generative model that reconstructs AA ensembles from protein backbone traces using a flow-matching architecture, achieving state-of-the-art structural fidelity. However, because FlowBack is trained only on structural data, it can occasionally generate physically unrealistic or high-energy configurations. We present FlowBack-Adjoint, a lightweight physics-aware enhancement that upgrades a pre-trained FlowBack model through a one-time post-training pass. Using adjoint matching, the method steers the generative vector field toward lower-energy regions of configuration space while preserving structural diversity. In benchmark tests against FlowBack, FlowBack-Adjoint lowers single-point energies by a median of $\sim$78 kcal/mol.residue, reduces errors in bond lengths by $>$92\%, eliminates $>$98\% of molecular clashes, and maintains excellent diversity of the AA configurational ensemble. Our results demonstrate that FlowBack-Adjoint offers a practical route to integrating physical realism into protein backmapping and deep generative protein models.