NPSolver: Neural Poisson Solver with Iterative Physics Supervision
2026-05-25 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors address the problem of solving Poisson equations, which are important but hard to solve on complex shapes because traditional methods can be slow or unstable. They introduce NPSolver, a new neural network approach that trains itself using a few steps of a classic solver, making training more stable without needing full solutions as labels. They also design a special model called BA-Transolver that better understands boundary conditions by treating boundary and interior parts separately. Tests show their method works better than existing neural or physics-based methods, and it can also help with controlling temperature settings efficiently.
Poisson equationneural operatorsphysics-informed learningpreconditioned conjugate gradientboundary conditionsiterative supervisiontransolver architecturegradient-based controlill-conditioned systemsscientific computing
Authors
Bocheng Zeng, Rui Zhang, Runze Mao, Mengtao Yan, Xuan Bai, Yang Liu, Zhi X. Chen, Hao Sun
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.