Neural Hodge Corrective Solvers: A Hybrid Iterative-Neural Framework

TL;DR

The paper addresses the challenge of efficiently solving PDEs while preserving geometric and topological structure. It introduces the Neural Hodge Corrective Solver (NHCS), a hybrid method that learns data-driven DEC operators and a CNN-based multiscale correction, embedded within a Jacobi–Richardson solver to avoid costly Jacobian evaluations. The approach uses a two-phase training regime: Phase 1 learns the DEC operators by residual minimization, and Phase 2 tunes them inside the iterative solver, yielding a stable update rule for the learned operator with $ ilde{\Delta_k} = \Delta_k + \mathrm{CNN}(\Delta_k)$. Experiments on Poisson, Helmholtz, and linear elasticity demonstrate faster convergence, improved accuracy, and robust multiscale resolution compared to prior DL-assisted methods, while preserving structure-preserving properties and reducing computational overhead. Overall, NHCS offers a scalable, interpretable, and efficient framework for hybrid neural–numerical PDE solvers in multiscale and geometry-rich settings.

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.

Figures (13)

• Figure 1: Commutative diagram showing relationships between classical and data-driven DEC operators.
• Figure 2: NHCS training pipeline for Phase 1 and Phase 2. Phase 1 minimizes $\| f_{\text{est}} - f_{\text{data}} \|$ to update DDEC operators, while Phase 2 optimizes the hybrid Jacobi + Neural Hodge correction solver.
• Figure 3: Poisson equation training time plot. Training time (in seconds) for 1 epoch for the PDE constrained optimization algorithm in TRASK2022110969 with varying grid size, represented by the blue dashed line. The component wise neural networks have one hidden linear layer, with a width of $10$, and ReLU activation functions between successive layers. A dataset with $512$ samples, and batch size $1$ is used. The red line represents $t=11063s$, the time taken for Phase 1 + Phase 2 of training for the $64\times64$ model.
• Figure 4: Phase 1 Training performance on the 2D Poisson Equation (a, b), 2D Helmholtz equation (c, d), and the 2D Linear Elasticity Problem (e,f). (a, c) Relative MSE of the forcing term vs. epochs. (b, d) Relative MSE of the forcing term vs. training time (in seconds). For the Helmholtz problem, we train for wave numbers $\in \{4, 8, 16, 20, 24, 28\}$. (e) RMSE of the forcing term vs. epochs. (f) RMSE of forcing term vs. training time (in seconds).
• Figure 5: (Phase 2 Training performance on the 2D Poisson Equation (a, b), 2D Helmholtz equation (c, d), and the 2D Linear Elasticity Problem (e,f). (a, c, e) Relative MSE of the solution vs. epochs. (b, d, f) Relative MSE of the solution vs. training time (in seconds).

Theorems & Definitions (1)

• Remark 2.1