NPSolver: Neural Poisson Solver with Iterative Physics Supervision

• URL: http://arxiv.org/abs/2605.25786v1
• Date: Mon, 25 May 2026 12:33:52 GMT
• Title: NPSolver: Neural Poisson Solver with Iterative Physics Supervision
• Abstract summary: textscNPr is a neural Poisson solver trained without solution labels.<n>textscNPr utilizes a small number of preconditioned conjugate gradient steps to refine its own predictions.<n>textscBA-Transolver architecture explicitly separates interior and boundary tokenization.
• Score: 13.109423311684134
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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