Neural Hodge Corrective Solvers: A Hybrid Iterative-Neural Framework

TL;DR

The paper presents NHCS, a hybrid iterative-neural framework that embeds learned corrections into DEC-based PDE solvers, enhancing efficiency, robustness, and multiscale accuracy while preserving topological structure.

Contribution

It introduces a novel hybrid iterative-neural PDE solver with a two-phase training strategy and a convolutional correction for multiscale features, reducing computational costs and improving stability.

Findings

Enhanced solution accuracy and convergence speed.

Reduced computational cost compared to Newton-Raphson methods.

Improved multiscale adaptivity and robustness.

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.