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REVIEW 2 major objections 1 minor 8 references

Reformulating zeroth-order terms into aligned constraints mitigates gradient conflicts in PINNs.

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2026-06-30 12:36 UTC pith:VJS6WXCH

load-bearing objection CAML reformulates boundary terms in PINNs for gradient alignment but the abstract shows no derivation that the new loss keeps the same solutions as the original BVP. the 2 major comments →

arxiv 2605.25001 v1 pith:VJS6WXCH submitted 2026-05-24 cs.LG

Mitigating Gradient Pathology in PINNs through Aligned Constraint

Yichen Luo , Peiyu Zhu , Dongxiao Hu , Jia Wang , Tailin Wu , Dapeng Lan , Yu Liu , Zhibo Pang This is my paper
classification cs.LG
keywords Physics-Informed Neural NetworksPINNsGradient PathologyConstraint AlignmentLoss LandscapePDE SolvingOptimization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to show that gradient pathology in Physics-Informed Neural Networks arises when gradients from PDE residuals and boundary constraints point in opposite directions, trapping the model in poor solutions. By converting all such zeroth-order terms into aligned constraints through a new loss called Constraint-Aligned loss with Manifold Lifting, and adding a delay factor during optimization, the conflicts are reduced. A sympathetic reader would care because this could make PINNs reliable for solving complicated partial differential equations where standard training fails due to ill-conditioned landscapes.

Core claim

The central claim is that gradient pathology in PINNs is caused by opposing gradients between PDE residuals and boundary constraints. Reformulating all zeroth-order terms into aligned constraints via Constraint-Aligned loss with Manifold Lifting effectively mitigates these conflicts, and introducing a delay factor helps the optimizer avoid high-curvature regions, resulting in enhanced numerical stability and efficiency for complex problems.

What carries the argument

Constraint-Aligned loss with Manifold Lifting (CAML), which reformulates zeroth-order terms as aligned constraints to eliminate gradient opposition.

Load-bearing premise

Gradient pathology stems mainly from opposing directions of PDE residual gradients and boundary constraint gradients, and reformulating the terms will align them without creating new optimization difficulties or reducing accuracy.

What would settle it

Running the CAML method on a known complex PDE where gradient directions remain opposing after reformulation, or where accuracy drops compared to baseline, would disprove the central claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Training of PINNs avoids local minima caused by gradient opposition between residuals and boundaries.
  • Methods like adaptive weighting or geometry-limited hard constraints become unnecessary for many cases.
  • Numerical stability and training efficiency improve significantly in highly complex PINN problems.
  • The optimizer can skip high-curvature areas using the introduced delay factor.

Where Pith is reading between the lines

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

  • This alignment strategy could apply to other neural network tasks where multiple loss terms conflict.
  • Manifold lifting might offer a general way to handle constraint satisfaction in optimization without explicit enforcement.
  • Further tests on time-dependent or high-dimensional PDEs would show if the benefits hold beyond the reported experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that gradient pathology in PINNs is caused by opposing gradients from PDE residuals and boundary constraints. It proposes Constraint-Aligned loss with Manifold Lifting (CAML), which reformulates all zeroth-order terms into aligned constraints to mitigate conflicts, plus a delay factor to avoid high-curvature regions. The authors state that experiments show significantly improved numerical stability and efficiency on complex PINN problems, with code released openly.

Significance. If the reformulation is proven equivalent to the original BVP and the stability gains are demonstrated with quantitative metrics and baselines, the approach could offer a general technique for resolving ill-conditioning in PINN optimization across geometries. The open-sourced code is a strength for reproducibility.

major comments (2)
  1. [Method] The central claim requires that the aligned-constraint reformulation preserves the original BVP minimizer. No derivation establishing mathematical equivalence between the transformed loss and the original PDE+BC problem is provided (see the method description and any equations defining the manifold lifting or alignment enforcement).
  2. [Abstract / Experiments] The abstract asserts that 'experiments demonstrate that our CAML significantly enhances numerical stability and efficiency', yet supplies no quantitative metrics, baselines, problem specifications, error tables, or convergence plots. This absence prevents evaluation of whether the claimed gains are load-bearing for the stability conclusion.
minor comments (1)
  1. [Method] Notation for the delay factor and its integration into the optimizer is introduced without a clear algorithmic description or pseudocode.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the two major comments point-by-point below.

read point-by-point responses

  1. Referee: [Method] The central claim requires that the aligned-constraint reformulation preserves the original BVP minimizer. No derivation establishing mathematical equivalence between the transformed loss and the original PDE+BC problem is provided (see the method description and any equations defining the manifold lifting or alignment enforcement).

    Authors: We agree that the manuscript lacks an explicit derivation establishing equivalence. In the revised version we will insert a dedicated subsection deriving that the manifold lifting and alignment operations preserve the original BVP minimizer, showing that the transformed loss shares the same critical points under the stated assumptions on the constraint functions. revision: yes

  2. Referee: [Abstract / Experiments] The abstract asserts that 'experiments demonstrate that our CAML significantly enhances numerical stability and efficiency', yet supplies no quantitative metrics, baselines, problem specifications, error tables, or convergence plots. This absence prevents evaluation of whether the claimed gains are load-bearing for the stability conclusion.

    Authors: The current abstract is deliberately concise. We will expand it to report the key quantitative metrics (error norms, iteration counts, stability indicators) and baseline comparisons already present in Section 4, while ensuring the experimental section contains the requested tables and plots with problem specifications. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation chain self-contained with no reductions to inputs

full rationale

The abstract describes an analysis of gradient pathology in PINNs leading to the CAML method via reformulation of zeroth-order terms into aligned constraints, plus a delay factor. No equations, loss functions, or derivations are presented that could reduce a claimed prediction or result to a fitted parameter or self-citation by construction. No self-citations, uniqueness theorems, or ansatzes are invoked in the provided text. The central claim rests on empirical experiments rather than any load-bearing mathematical equivalence that collapses to the inputs. This matches the default expectation for non-circular papers.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are stated; the method description does not introduce new physical quantities or unproven lemmas.

pith-pipeline@v0.9.1-grok · 5721 in / 1075 out tokens · 32752 ms · 2026-06-30T12:36:13.318991+00:00 · methodology

0 comments
read the original abstract

While Physics-Informed Neural Networks (PINNs) are powerful for solving Partial Differential Equations (PDEs), their training is often paralyzed by gradient pathology. The gradients from the PDE residuals and boundary constraints oppose each other, trapping the model in local minima. Current solutions, such as adaptive weighting or hard constraints, either fail to fundamentally resolve this ill-conditioning or are limited to simple geometries. In this study, we systematically analyze the possible causes of this gradient pathology from the perspectives of loss landscapes and optimization dynamics. Based on the obtained conclusion, we propose Constraint-Aligned loss with Manifold Lifting (CAML). By reformulating all zeroth-order terms into aligned constraints, our method effectively mitigates gradient conflicts. In addition, we introduce a delay factor to help the optimizer skip the high-curvature area. Experiments demonstrate that our CAML significantly enhances numerical stability and efficiency in highly complex PINN problems. Our code is open-sourced on https://github.com/YichenLuo-0/CAML.

Figures

Figures reproduced from arXiv: 2605.25001 by Dapeng Lan, Dongxiao Hu, Jia Wang, Peiyu Zhu, Tailin Wu, Yichen Luo, Yu Liu, Zhibo Pang.

Figure 1
Figure 1. Figure 1: Visualization of the PDE residual loss landscape in both the function space and the parameter space, projected onto a 2D subspace. The red curve indicates the loss valley. While the landscape is relatively simple in the function space, it becomes highly distorted and non-convex when mapped to the parameter space by the neural network. More details are in Appendix G. where {xi} are collocation points sample… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic illustration of the intersection between PDE residual loss landscape and boundary-condition constraint land￾scape in function space. The zero-residual solutions of the govern￾ing PDE form a manifold, while boundary conditions restricting the admissible solutions to a subset. Their intersections correspond to valid solutions, which can be (a) unique or (b) non-unique. 4.1.2. BOUNDARY CONDITION TER… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration of the two-phase optimization dynamics of PINNs under gradient conflict in parameter space. (a) Phase I: dominance of the PDE residual gradient leads to a rapid descent into the low-residual loss valley. (b) Phase II (success): boundary-condition gradients steer the optimizer along the valley toward the global optimum. (c) Phase II (failure): high-curvature regions induce oscillatory… view at source ↗
Figure 4
Figure 4. Figure 4: Conceptual illustration of the proposed optimization mechanism. (a) Aligned constraints enlarge the admissible solution range, while (b) delaying the residual loss guides the optimizer into the PDE loss valley from a favorable region, enabling smoother convergence to the global optimum. Here, in the early iteration steps, we recommend using New￾ton’s method several times to update c. For more complex PDEs,… view at source ↗
Figure 6
Figure 6. Figure 6: Variation of the c curve across iterations on all bench￾marks (MLP backbone). loss landscape. Consequently, it is not equivalent to genuine progress toward the correct solution [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Relative L2 error and cumulative parameter update distance P∥θt+1 − θt∥2 versus training iterations on Heat bench￾mark (MLP backbone). 6.2. Results and Discussions RQ1: Performance Analysis [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visualization of the loss valleys in the parameter space under 4 group of different random seeds. We also conducted experiments with more random initializations, as shown in [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of loss components and gradient statistics during PINN training, including the PDE residual loss, boundary condition loss, cosine similarity between their gradients, and the parameter optimize trajectory. 0 5 10 15 proj axis 1 1 0 1 2 3 proj axis 2 ep0 ep2000 ep10000 2.10 2.55 3.00 3.45 3.90 4.35 4.80 5.25 5.70 6.15 lo g10 ( res) 14.6300 14.6325 14.6350 14.6375 14.6400 proj axis 1 1.18 1.20 1.22 … view at source ↗
Figure 9
Figure 9. Figure 9: Visualization of parameter trajectories and the corresponding PDE residual loss landscape, from both (a) global and (b) local perspective. While the global loss curve appears nearly linear in loss valley, the local optimization path displays substantial oscillatory behavior, indicating persistent gradient conflicts within the residual-induced valley. Phase I: Rapid PDE Residual Minimization (0–2000 epochs)… view at source ↗
Figure 10
Figure 10. Figure 10: Visualization results on the Heat benchmark, showing the ground truth, prediction and error distributions during training with CAML integrated into the MLP backbone. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Visualization results on the Poisson benchmark, showing the ground truth, prediction and error distributions during training with CAML integrated into the MLP backbone. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0.002 0.004 0.006 0.008 0.010 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0… view at source ↗
Figure 12
Figure 12. Figure 12: Visualization results on the NS benchmark, showing the ground truth, prediction and error distributions during training with CAML integrated into the MLP backbone. -1.0 -0.5 0.0 0.5 1.0 x -1.0 -0.5 0.0 0.5 1.0 y 99.5 100.0 100.5 101.0 101.5 -1.0 -0.5 0.0 0.5 1.0 x -1.0 -0.5 0.0 0.5 1.0 y 99.5 100.0 100.5 101.0 101.5 -1.0 -0.5 0.0 0.5 1.0 x -1.0 -0.5 0.0 0.5 1.0 y 0.005 0.010 0.015 0.020 0.025 (a) Ground T… view at source ↗
Figure 13
Figure 13. Figure 13: Visualization results on the Helmholtz benchmark, showing the ground truth, prediction and error distributions during training with CAML integrated into the MLP backbone. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_13.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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