Energy-Decrease Training for Learned Conjugate Directions

Learned iterative solvers are often trained without solution labels by minimizing the residual left after a correction. For a symmetric positive-definite system this under-rewards removing low-eigenvalue error: squared residual weights each error eigenmode by ${\lambda }^2$ while energy weights it by $\lambda$, so such error can be large in the solution yet small in the residual. We show that, in a learned conjugate-direction solver, changing only the training objective substantially changes how much of this error is removed. Energy-DCDM (E-DCDM) maximizes the exact energy decrease of each step, divided by the error energy visible on a small coarse space (32 sine modes); it needs no solution labels and leaves the deployed solver unchanged. When both losses train on the same solver states, E-DCDM lowers relative solution error after eight operator products by at least $3.3\times$ on every paired seed on public Darcy and screened-Poisson problems, both under a shared optimizer recipe ($6.6$ to $9.8\times$) and when each loss's learning rate and clipping are tuned separately ($3.3$ to $5.4\times$ on screened Poisson). Numerically verified Darcy eigenvectors indicate that energy training removes more low-eigenvalue error. The gain needs a suitable state weighting: in 2D, energy decrease weighted by $1/\Vert r{\Vert }_2^2$ is worse than the residual loss, and a generic coarse-free normalizer does not reproduce the 2D gain. E-DCDM can have higher residual error, the method and main comparison were designed after earlier methods were scored on the same evaluation data, and on 47 held-out 3D operators only a coarse-free variant was tested ($1.8\times$ lower one-step solution error). The study isolates the training objective rather than proposing a competitive solver: with stencil access, algebraic multigrid reaches far higher accuracy on Darcy within seven V-cycles.