Technical Summary: NPSolver – Neural Poisson Solver with Iterative Physics Supervision

1. Problem Statement

Solving Poisson equations on complex, irregular domains is a fundamental bottleneck in scientific computing. Classical iterative solvers, such as Preconditioned Conjugate Gradient (PCG), often suffer from prohibitive runtimes due to ill-conditioned systems arising from irregular geometries and challenging boundary conditions (BCs). While neural operators offer a fast inference alternative, they typically rely on large-scale labeled datasets generated by expensive numerical solvers. Conversely, Physics-Informed Neural Networks (PINNs) and physics-informed operator learning methods avoid labeled data but struggle with unstable training dynamics, slow convergence, and poor gradient scaling when directly minimizing PDE residuals, particularly in stiff or ill-conditioned regimes. Furthermore, many existing approaches are limited to regular grids or simple geometries, hindering their generalization to irregular domains with mixed BCs.

2. Methodology

2.1 Iterative Physics Supervision

The core innovation of NPSolver is a training paradigm that eliminates the need for fully converged numerical solution labels. Instead of minimizing raw PDE residuals or relying on ground-truth pairs, NPSolver utilizes iterative physics supervision:

• Process: For a given input (domain geometry $\Omega$, boundary conditions $B$, and forcing field $f$), the neural network predicts an initial solution $\hat{\mathbf{u}}$.
• Refinement: A small number ($K$, where $K \ll T$, and $T$ is the full convergence count) of PCG steps are applied to $\hat{\mathbf{u}}$ to produce an updated iterate $\tilde{\mathbf{u}} = F_K(\hat{\mathbf{u}})$.
• Objective: The network is trained to minimize the discrepancy $\|\hat{\mathbf{u}} - \tilde{\mathbf{u}}\|_2^2$.
• Stop-Gradient: Crucially, gradients are stopped through the PCG operator $F_K$. The PCG steps act solely as a target generator, providing a self-consistency signal without backpropagating through the iterative solver.

2.2 Theoretical Guarantees

The authors provide theoretical analysis supporting this approach:

• Fixed-Point Consistency: The fixed points of the $K$-step PCG map correspond exactly to the solutions of the linear system $A\mathbf{u} = \mathbf{b}$.
• Well-Conditioned Error Proxy: The self-consistency residual $\|\mathbf{u} - F_K(\mathbf{u})\|$ is proven to be a well-conditioned proxy for the true energy error $\|\mathbf{u} - \mathbf{u}^*\|_A$. Unlike the PDE residual $\|A\mathbf{u} - \mathbf{b}\|$, which can be poorly scaled by the spectrum of $A$ (especially in ill-conditioned cases), the iterative supervision signal remains stable.
• Optimization Stability: The stop-gradient design is shown to yield more favorable optimization dynamics compared to full-gradient updates on the residual, preventing the training from becoming overly sensitive to step sizes near the solution.

2.3 Architecture: Boundary-Aware Transolver (BA-Transolver)

To handle irregular meshes and mixed boundary conditions effectively, the authors introduce the BA-Transolver architecture:

• Dual-Stream Tokenization: Unlike standard Transolvers that apply a shared slicing operation to all nodes, BA-Transolver separately tokenizes interior nodes and boundary nodes.
• Boundary-Interior Interaction: The model learns separate slice-assignment weights for interior and boundary embeddings. These are aggregated into token sets, concatenated, and processed via multi-head self-attention.
• Mechanism: This design explicitly exposes boundary information to the global attention mechanism, allowing the model to better capture complex boundary-interior interactions and generalize across varying geometries and mixed BCs (e.g., Dirichlet and Neumann patches).

3. Key Contributions

1. Label-Free Training: NPSolver is the first neural Poisson solver trained without solution labels, utilizing iterative physics supervision to bypass the cost of generating fully converged numerical pairs.
2. Theoretical Foundation: The paper establishes that iterative supervision provides a well-conditioned error proxy and that the stop-gradient mechanism is essential for stable optimization, offering a theoretical alternative to direct residual minimization.
3. Architecture Innovation: The introduction of BA-Transolver addresses the specific challenges of mixed boundary conditions and irregular geometries by explicitly modeling boundary-interior interactions.
4. Comprehensive Evaluation: The method is validated on 2D and 3D irregular domains with varying forcing fields and boundary regimes, as well as a downstream thermal control task.

4. Experimental Results

4.1 2D Generalization

• Setup: Experiments were conducted on irregular domains (corner-removed squares) with varying forcing fields and three boundary regimes: All-Dirichlet, All-Neumann, and RandomBC (patch-wise mixed BCs).
• Performance: In the challenging RandomBC regime, NPSolver achieved a relative $L_2$ error of 8.17% without any labeled data. This outperformed the strongest supervised baseline (Transolver trained with 7,000 labeled samples), which achieved 10.17% error.
• Efficiency: NPSolver demonstrated a 15× speedup at matched accuracy compared to classical iterative numerical solvers. It also significantly outperformed other self-supervised baselines (PI-DeepONet, PINO) and per-instance optimization methods (PINN) in terms of both accuracy and inference time.

4.2 3D Scalability

• Setup: Evaluated on a 3D "cube-with-cylindrical-hole" family with mixed BCs.
• Performance: NPSolver achieved a mean relative $L_2$ error of 8.67% with an inference time of 11.8 ms. In contrast, a supervised Transolver baseline trained with 1,000 labeled samples yielded a significantly higher error of 27.22%.

4.3 Downstream Application: Thermal Control

• Task: A gradient-based thermal control task on a perforated plate, aiming to maintain temperature below a safety threshold while minimizing cooling effort.
• Results: NPSolver served as an efficient differentiable surrogate. It achieved an 83% success rate in satisfying safety constraints. The optimization process using NPSolver was over 10× faster than using the numerical solver directly, reducing peak temperature from ~33.42 to ~25.31 with minimal cooling cost.

4.4 Ablation Studies

• Supervision: Iterative physics supervision consistently outperformed direct residual supervision, confirming the benefits of the well-conditioned error proxy.
• Architecture: BA-Transolver outperformed the vanilla Transolver under data supervision, validating the importance of explicit boundary tokenization.
• Stop-Gradient: Removing the stop-gradient operator increased the error from 8.17% to 11.90%, confirming its necessity for stability.
• PCG Steps ($K$): Performance peaked at intermediate $K$ values (e.g., $K=40$), balancing the strength of the correction signal against optimization difficulty.

5. Significance and Claims

The paper claims that NPSolver offers a favorable error–cost trade-off for complex boundary-value Poisson problems. By leveraging iterative physics supervision, it achieves robust generalization across irregular geometries and mixed boundary conditions without the prohibitive cost of generating large labeled datasets. The method effectively bridges the gap between the speed of neural surrogates and the physical consistency of classical solvers, demonstrating particular utility in scenarios where classical solvers are slow due to ill-conditioning and where data-driven methods fail due to a lack of labeled data or poor generalization to irregular domains.

The authors note that while the current work focuses on the Poisson equation with SPD systems, the core idea of using truncated solver maps for self-supervision could potentially extend to other Krylov solvers and PDE families, though this remains a direction for future work.