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arxiv: 2605.25786 · v1 · pith:SHWCRNOYnew · submitted 2026-05-25 · 💻 cs.LG · cs.AI

NPSolver: Neural Poisson Solver with Iterative Physics Supervision

Bocheng Zeng , Rui Zhang , Runze Mao , Mengtao Yan , Xuan Bai , Yang Liu , Zhi X. Chen , Hao Sun This is my paper

Pith reviewed 2026-06-29 22:13 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords neural poisson solverphysics-informed learningpreconditioned conjugate gradientiterative supervisionirregular domainsmixed boundary conditionsneural operator
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The pith

A neural network solves Poisson equations on irregular domains without labeled solutions by supervising itself with a few PCG iterations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents NPSolver as a label-free method for solving Poisson equations that avoids both large training datasets and unstable raw residual losses. Instead of demanding fully converged numerical answers, the network refines its own output using a fixed small number of preconditioned conjugate gradient steps, which the authors show supplies a well-conditioned error signal. Theoretical analysis establishes that this iterative supervision works as a stable proxy and that a stop-gradient operation is required to keep optimization from diverging. A new Boundary-Aware Transolver architecture is added to handle mixed boundary conditions by separating interior and boundary tokens. The resulting model runs fast at inference time on complex 2D and 3D geometries and supports gradient-based control tasks.

Core claim

NPSolver is trained without solution labels via iterative physics supervision: a small number of preconditioned conjugate gradient steps refine the network's own predictions to produce a stable, well-scaled training signal. Theoretical analysis confirms that this supervision acts as a well-conditioned error proxy and that a stop-gradient design is essential for optimization stability. The Boundary-Aware Transolver architecture explicitly separates interior and boundary tokenization to capture boundary-driven features under mixed conditions.

What carries the argument

Iterative physics supervision that applies a small fixed number of preconditioned conjugate gradient steps to refine the model's predictions, together with the Boundary-Aware Transolver that tokenizes interior and boundary regions separately.

If this is right

  • The method outperforms both physics-informed and data-driven baselines on 2D and 3D irregular geometries.
  • A downstream thermal control task shows the model supports efficient gradient-based boundary control.
  • The stop-gradient design prevents optimization instability during training.
  • No labeled solution data or fully converged numerical solves are required at training time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same supervision pattern could be applied to other linear PDEs whose classical solvers are iterative but expensive to run to convergence.
  • The interior-boundary token separation may improve accuracy for other boundary-value problems where boundaries dominate the solution.
  • Hybrid use is possible in which the trained network supplies an initial guess that a classical solver then refines in very few steps.

Load-bearing premise

A small fixed number of PCG iterations supplies a sufficiently accurate and stable error proxy for training without ever needing fully converged numerical solutions or labeled data.

What would settle it

Training the network with the same small number of PCG steps yields predictions whose error does not decrease when the supervision is replaced by fully converged PCG solutions on the same geometries.

Figures

Figures reproduced from arXiv: 2605.25786 by Bocheng Zeng, Hao Sun, Mengtao Yan, Rui Zhang, Runze Mao, Xuan Bai, Yang Liu, Zhi X. Chen.

Figure 1
Figure 1. Figure 1: Cell-centered FVM mesh: cell, face, and patch. To obtain a discrete numer￾ical solution, we discretize the domain and the governing equation using a cell-centered finite volume method (FVM). As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of NPSolver. (a) Iterative Physics Supervision: The network predicts 𝒖ˆ for a sampled instance (Ω, 𝐵, 𝑓 ), which is refined by 𝐾 PCG steps to generate a self-supervision target 𝒖˜ (with stop-gradient). (b) BA-Transolver Architecture: Interior and boundary nodes are respectively tokenized into 𝒁𝑖𝑛𝑡, 𝒁𝑏𝑑 and attended jointly to model boundary-interior interactions. self-attention (MHA) to achieve bo… view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of models’ predictions for represen [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Data efficiency and scaling analysis in the RandomBC case. (a) Mean relative [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Computational Efficiency in the RandomBC case. (a) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Representative 3D predictions of NPSolver on the cube-with-cylindrical-hole Poisson problem with mixed BCs. high-speed inference with minimal memory overhead. To quan￾tify the acceleration, we compare NPSolver against the iterative numerical solver. As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes NPSolver, a neural Poisson solver for 2D/3D irregular domains trained label-free via iterative physics supervision: a small fixed number of PCG steps refines the network's own predictions to yield a stable training signal, with stop-gradient required for optimization stability as confirmed by theoretical analysis. It introduces the Boundary-Aware Transolver (BA-Transolver) architecture that separates interior and boundary tokenization under mixed boundary conditions. Empirical evaluations on irregular geometries are claimed to outperform physics-informed and data-driven baselines, with an additional demonstration on a downstream thermal control task.

Significance. If the central claims hold, the work provides a concrete mechanism for stable, label-free training of neural PDE solvers that avoids both large labeled datasets and raw residual losses, while the boundary-aware architecture and downstream control example indicate applicability to practical scientific computing problems. The code release supports independent verification.

major comments (2)
  1. [Abstract, paragraph on iterative physics supervision] Abstract, paragraph on iterative physics supervision: the central assumption that a small fixed number of PCG iterations supplies a sufficiently accurate and stable error proxy without ever requiring fully converged numerical solutions is load-bearing for the label-free claim; the manuscript should report quantitative ablation on iteration count versus solution accuracy and training stability to substantiate that this proxy remains well-conditioned across the tested geometries.
  2. [Theoretical analysis] Theoretical analysis (referenced in abstract): while the stop-gradient design is stated to be essential for stability, the manuscript should explicitly connect the derived error-proxy conditioning bounds to the observed training dynamics (e.g., via a theorem or corollary that predicts the instability without stop-gradient) so that the necessity claim can be directly verified against the reported experiments.
minor comments (2)
  1. [Abstract] The abstract states that evaluations 'demonstrate that NPSolver outperforms both physics-informed and data-driven baselines' but does not preview the magnitude of improvement or the precise metrics; the results section should include a summary table with mean errors and standard deviations across all compared methods.
  2. Ensure that the released code repository contains the exact PCG preconditioner implementation, iteration counts, and random seeds used for the 2D/3D experiments so that the iterative supervision procedure can be reproduced exactly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation of minor revision. We address each major comment below and will update the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract, paragraph on iterative physics supervision] Abstract, paragraph on iterative physics supervision: the central assumption that a small fixed number of PCG iterations supplies a sufficiently accurate and stable error proxy without ever requiring fully converged numerical solutions is load-bearing for the label-free claim; the manuscript should report quantitative ablation on iteration count versus solution accuracy and training stability to substantiate that this proxy remains well-conditioned across the tested geometries.

    Authors: We agree that a quantitative ablation on PCG iteration count would strengthen the central claim. In the revised manuscript we will add a new figure and table reporting solution accuracy (relative L2 error) and training stability (loss curves and final convergence) for iteration counts of 1, 5, 10 and 20 on both the 2D and 3D irregular-geometry benchmarks. These results will confirm that the small fixed count used in the main experiments remains well-conditioned. revision: yes

  2. Referee: [Theoretical analysis] Theoretical analysis (referenced in abstract): while the stop-gradient design is stated to be essential for stability, the manuscript should explicitly connect the derived error-proxy conditioning bounds to the observed training dynamics (e.g., via a theorem or corollary that predicts the instability without stop-gradient) so that the necessity claim can be directly verified against the reported experiments.

    Authors: We appreciate the suggestion to make the link between theory and experiment more explicit. We will add a short corollary in the theoretical analysis section that derives the divergence of the gradient signal in the absence of stop-gradient, directly referencing the conditioning bounds already obtained and showing consistency with the instability observed in our existing stop-gradient ablation experiments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external PCG

full rationale

The core training signal is generated by a fixed number of steps from the classical preconditioned conjugate gradient algorithm, an external numerical routine whose behavior is independent of the neural network parameters. The claimed theoretical analysis of the error proxy and stop-gradient necessity is presented as a separate derivation that does not reduce to a redefinition of the network output or to any fitted quantity internal to the model. No self-citation chain, ansatz smuggling, or renaming of known results is required for the central claim; the argument therefore remains non-circular and externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or newly postulated entities; the central claim rests on the unstated premise that PCG steps remain a reliable proxy across the tested geometries and boundary conditions.

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    hot spots

    ∈ [0,1). Lemma C.1 (Norm eqivalence under symmetric precon- ditioning).Let 𝒚=𝑴 1/2𝒖 and 𝒚★ =𝑴 1/2𝒖★. Then 𝒚★ solves 𝑪𝒚=𝑴 −1/2𝒃, and for any𝒖, ∥𝒖−𝒖 ★∥𝑨 =∥𝒚−𝒚 ★∥𝑪 . Proof. We have 𝑨𝒖=𝒃⇐ ⇒𝑴 −1/2𝑨𝑴 −1/2𝒚=𝑴 −1/2𝒃, i.e., 𝑪𝒚=𝑴 −1/2𝒃. Moreover, ∥𝒖−𝒖 ★∥2 𝑨 =(𝒖−𝒖 ★)⊤𝑨(𝒖−𝒖 ★) =(𝒚−𝒚 ★)⊤𝑴 −1/2𝑨𝑴 −1/2 (𝒚−𝒚 ★) =∥𝒚−𝒚 ★∥2 𝑪 . □ Lemma C.2 (Kantorovich ineqality).Let 𝑪 be S...

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    Number of hot spots: 𝐾∼Unif{𝐾 min, 𝐾max}

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    from {1,

    Hot-spot centers: we sample 𝐾 indices {𝑐𝑘 }𝐾 𝑘=1 i.i.d. from {1, . . . , 𝑛}, and set 𝝁𝑘 =𝒙 𝑐𝑘 =(𝜇 𝑘,𝑥 , 𝜇𝑘,𝑦 )

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    and we use isotropic widths 𝜎𝑘,𝑥 =𝜎 𝑘,𝑦 =𝑠 𝑘

    Amplitudes and widths: 𝐴𝑘 ∼Unif(𝑎 min, 𝑎max), 𝑠 𝑘 ∼Unif(𝑠 min, 𝑠max), Figure A.4: Relationship between the estimated condition number and the relative 𝐿2 error gap between residual super- vision and iterative supervision on C4. and we use isotropic widths 𝜎𝑘,𝑥 =𝜎 𝑘,𝑦 =𝑠 𝑘

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    Unnormalized field: for each point𝒙 𝑖 =(𝑥 𝑖, 𝑦𝑖 ), 𝑓0 (𝒙𝑖 )= 𝐾∑︁ 𝑘=1 𝐴𝑘 exp − 1 2 𝑥𝑖 −𝜇 𝑘,𝑥 𝜎𝑘,𝑥 +𝜀 2 − 1 2 𝑦𝑖 −𝜇 𝑘,𝑦 𝜎𝑘,𝑦 +𝜀 2! , 𝜀=10 −8

  58. [58]

    Then the returned heat-source field is scaled: 𝑓(𝒙 𝑖 )=𝑓 0 (𝒙𝑖 ) · 1 ¯𝑓0 +𝜀

    Mean-power scaling: define the sample mean ¯𝑓0 = 1 𝑛 𝑛∑︁ 𝑖=1 𝑓0 (𝒙𝑖 ). Then the returned heat-source field is scaled: 𝑓(𝒙 𝑖 )=𝑓 0 (𝒙𝑖 ) · 1 ¯𝑓0 +𝜀 . In practice, we choose 𝐾min = 2, 𝐾max = 6, 𝑎min = 0.5, 𝑎max = 2.0, 𝑠min = 0.02𝐿, 𝑠max = 0.08𝐿, 𝐿= 2𝜋 to produce spatially sparse, inhomogeneous sources that resemble heat injection from discrete components. F...

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