NPSolver: Neural Poisson Solver with Iterative Physics Supervision

2026-05-25

AI Summary

This work addresses the challenges of solving Poisson’s equation on complex irregular domains, where traditional iterative solvers are computationally expensive and existing neural operators require abundant labeled data and suffer from training instability. To overcome these limitations, the authors propose a label-free neural Poisson solver that leverages preconditioned conjugate gradient (PCG) iterations to generate well-scaled physics-informed supervision signals. A stop-gradient mechanism is introduced to ensure optimization stability, and a boundary-aware Transolver architecture is designed to explicitly distinguish interior and exterior tokens, enabling effective handling of mixed boundary conditions. Experiments on 2D and 3D irregular geometries demonstrate that the proposed method significantly outperforms both physics-informed and data-driven baselines, while exhibiting efficient and reliable gradient-based boundary control in downstream thermal regulation tasks.

Abstract

Efficiently solving Poisson equations on complex, irregular domains remains a fundamental challenge in scientific computing, as classical iterative solvers often suffer from prohibitive runtime due to ill-conditioned systems. While neural operators offer a fast alternative, they typically rely on large-scale labeled datasets or struggle with unstable training dynamics when using physics-informed residual losses. We propose \textsc{NPSolver}, a neural Poisson solver trained without solution labels via iterative physics supervision. Instead of relying on fully converged numerical solutions or raw PDE residuals, \textsc{NPSolver} utilizes a small number of preconditioned conjugate gradient (PCG) steps to refine its own predictions, providing a more stable and well-scaled training signal. Theoretical analysis confirms that this iterative supervision serves as a well-conditioned error proxy and that a stop-gradient design is essential for optimization stability. To better capture boundary-driven features under mixed boundary conditions, we further introduce the Boundary-Aware Transolver (\textsc{BA-Transolver}) architecture that explicitly separates interior and boundary tokenization. Extensive evaluations on 2D and 3D irregular geometries demonstrate that \textsc{NPSolver} outperforms both physics-informed and data-driven baselines. Furthermore, a downstream thermal control task highlights the model's capability for conducting efficient and reliable gradient-based boundary control. We will release our codes and data at https://github.com/intell-sci-comput/NPSolver.

Problem

• Poisson equation
• irregular domains
• scientific computing
• neural operators
• physics-informed learning

Innovation

• Neural Poisson Solver
• Iterative Physics Supervision
• Preconditioned Conjugate Gradient
• Boundary-Aware Architecture
• Physics-Informed Learning