Neural Hodge Corrective Solvers: A Hybrid Iterative-Neural Framework

Abstract: We introduce the Neural Hodge Corrective Solver (NHCS), a hybrid iterative-neural framework for partial differential equations that embeds learned corrective operators within the Discrete Exterior Calculus (DEC) formulation. The method combines classical Jacobi-Richardson iterations with data-driven corrections to refine numerical solutions while preserving the underlying topological and metric structure. NHCS employs a two-phase training strategy. In the first phase, DEC operators are learned through relative residual minimization from data. In the second phase, these operators are integrated into the iterative solver, and training targets the improvement of convergence through learned corrective updates that remain effective even for inaccurate intermediate solutions. This staggered training enables stable, progressive refinement while maintaining the structure-preserving properties of DEC discretizations. To improve multiscale adaptivity, NHCS introduces a convolutional neural network-based correction term capable of capturing fine-scale solution features via localized updates informed by global context, improving scalability over mesh component-wise neural approaches. Moreover, the proposed framework substantially reduces computational cost by avoiding Newton-Raphson-based training and the associated Jacobian evaluations of parameterized operators. The resulting solver achieves improved efficiency, robustness, and accuracy without compromising numerical stability.