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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.25786 |
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| _version_ | 1866911716103684096 |
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| author | Zeng, Bocheng Zhang, Rui Mao, Runze Yan, Mengtao Bai, Xuan Liu, Yang Chen, Zhi X. Sun, Hao |
| author_facet | Zeng, Bocheng Zhang, Rui Mao, Runze Yan, Mengtao Bai, Xuan Liu, Yang Chen, Zhi X. Sun, Hao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25786 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | NPSolver: Neural Poisson Solver with Iterative Physics Supervision Zeng, Bocheng Zhang, Rui Mao, Runze Yan, Mengtao Bai, Xuan Liu, Yang Chen, Zhi X. Sun, Hao Machine Learning Artificial Intelligence 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. |
| title | NPSolver: Neural Poisson Solver with Iterative Physics Supervision |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2605.25786 |