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REVIEW 3 major objections 5 minor 5 references

Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A projection layer that pins a PINN's output to the conservation manifold reduces integral-conservation error by three to four orders of magnitude while slightly improving solution accuracy.

desk verdict The quadratic projection formula is off by a factor Δx, invalidating the headline guarantee as written, but the fix is small and the rest of the paper is solid enough to send to review. read the letter →

arxiv 2511.09048 v2 pith:CRWM3FAR submitted 2025-11-12 cs.LG

Anthony Baez , Wang Zhang , Ziwen Ma , Lam Nguyen , Subhro Das , Luca Daniel This is my paper · ORCID
classification cs.LG
keywords physics-informedneuralnetworksconservationlawsprojectionlayerhardconstraintsquadratureconstrainedoptimizationlosslandscapeconditioningPDEsolving
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that conservation of linear and quadratic integral quantities can be enforced exactly in a physics-informed neural network (PINN) by projecting the raw network output onto the constraint set at every evaluation. The projection is the closed-form solution of minimizing the squared distance to the conserved manifold, derived with Lagrange multipliers, and it is differentiable, so it can be trained end-to-end. Across five PDEs in one and two dimensions, the projected network reduces conservation error by three to four orders of magnitude compared to a soft penalty, while the PDE solution error is at worst unchanged and at best reduced. The authors also report that the projection improves the conditioning of the loss landscape, making gradient descent converge in fewer epochs. The method is proposed as a general recipe: any integral quantity with a tractable differentiable projection can be conserved this way.

What carries the argument

The central object is the projection layer. For the linear integral, the projection subtracts the current spatial mean from the output and adds the target mean c(t)/(nΔx). For the quadratic integral, it rescales the entire output vector by the square root of the ratio of the target quadratic sum to the current one. For the joint constraint, it first centers the output by subtracting its mean, rescales the centered vector to satisfy the quadratic target, and adds back the mean term set by the linear target. Each formula is the closed-form solution of a constrained least-squares problem obtained with Lagrange multipliers. The layer operates on the discretized output on a uniform quadrature gri

What would settle it

Take a PDE with a known analytic conserved quantity, train PINN-Proj on a coarse collocation grid, then compute the continuous integral (e.g., ∫u² dx) with a much finer quadrature; if the continuous error does not drop alongside the discrete conservation error, the 'guarantee' is an artifact of the grid. Alternatively, feed the projection a deliberately wrong c(t) trajectory and observe whether the solution error increases; a correct mechanism should respond to a wrong constraint.

Watch

Extended reading notes

Core claim

The paper claims that a projection layer can guarantee conservation of the linear integral, the quadratic integral, and both jointly, in the sense that the only remaining errors are those from the data and from machine precision. The projection is computed by solving a constrained non-linear optimization problem — minimize the L2 distance between the network output and the set of functions with the prescribed integral — and the solution is a simple closed form: a constant shift for the linear constraint, a uniform scaling for the quadratic constraint, and a shift-around-the-mean followed by a scaling for the combined constraint. Because the projection is applied to the discretized output on

Load-bearing premise

The guarantee is defined on the discrete quadrature grid — it enforces the summed quantity, not the continuous integral — and for non-conserved systems the target c(t) is itself interpolated from data, so if that target is wrong the projection enforces the wrong conservation law.

Editorial extensions

If this is right

  • On all five PDEs tested, PINN-Proj reduces conservation error for the linear and quadratic integrals by three to four orders of magnitude compared with the soft-constrained baseline.
  • PINN-Proj achieves the lowest PDE solution error on 11 of 15 configurations, with margins under one order of magnitude.
  • Projected models converge in the fewest epochs across all PDEs, and the Hessian spectrum is more tightly clustered, implying a better-conditioned loss landscape.
  • The method extends to 2D domains and to non-conserved systems by interpolating the time-dependent target c(t), so it applies beyond exactly conserved systems.
  • Because the projection is differentiable and applied at inference time, it can be inserted as a layer in any PINN-like architecture without changing the training objective.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the same projection recipe could be applied to other integral functionals whenever the constrained least-squares problem has a closed form; this would turn the method into a general library for hard conservation in neural PDE solvers.
  • The reported Hessian improvement suggests projection acts as a conditioning regularizer; a controlled study on stiffer PDEs could test whether the benefit grows precisely where soft constraints struggle most.
  • Because the projection is a differentiable map applied at inference time, it could serve as a post-processing step to restore conservation in an already-trained PINN without retraining.
  • A stress test on a coarse grid with a high-resolution continuous integral would reveal whether the discrete guarantee translates to the continuous conservation law, which is the quantity physics actually cares about.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes PINN-Proj, a hard-constraint layer for physics-informed neural networks that projects the network output onto a manifold where selected integral quantities (linear integral, quadratic integral, or both) are conserved. The projection is derived by solving constrained least-squares problems with Lagrange multipliers. The method is tested on 1D/2D advection, wave, Korteweg-de Vries, and reaction-diffusion equations, comparing solution error and conservation error against a vanilla PINN and a soft-constraint variant (PINN-SC). The authors report that PINN-Proj reduces conservation error by three to four orders of magnitude and marginally improves solution error, and they analyze Hessian spectral densities to argue that the projection improves the conditioning of the loss landscape.

Significance. If the claims hold, the paper offers a practically useful way to enforce hard conservation constraints in PINNs using differentiable closed-form projections, which is an active area of research. The method is simple, does not require iterative projection steps, and is demonstrated on several PDEs. The paper also provides a useful Hessian-spectrum analysis that connects the projection to improved conditioning. However, the central algebraic derivation for the quadratic projection contains an error that undermines the main guarantee, and the 'guarantee' is stated only for a discrete quadrature, not for the continuous integral.

major comments (3)
  1. [Sec. 3.7, Eq. (15); Sec. A.11, Eqs. (40)-(42)] The quadratic projection formula is algebraically incorrect. Solving the KKT system in Eq. (36)-(39) gives y_i = u_i/(1+λΔx) and, after substituting into Δx yᵀy = c(t), one obtains 1/(1+λΔx) = sqrt( c(t) / (Δx Σ_i u_i²) ). The paper instead writes 1/(1+λΔx) = sqrt( c(t) / Σ_i u_i² ), omitting the factor Δx in the denominator. Consequently, Eq. (15) and Eq. (41) do not satisfy the stated constraint: substituting the printed formula yields Δx Σ_i y_i² = Δx c(t), not c(t). With the experimental Δx = 1/128, this is a factor-of-128 discrepancy. This is a load-bearing error for the claim of guaranteed quadratic conservation. Please correct the formula, update the continuous extension in Eq. (42), and verify whether the implementation and the reported Errorc_Q values in Table 3 correspond to the corrected or the printed formula.
  2. [Secs. 3.4-3.8, 5; Abstract] The abstract and introduction state that the method 'guarantees the conservation of the linear and quadratic integrals.' As written, however, the projection enforces the discrete quadrature conditions Δx Σ_i y_i = c(t) and Δx Σ_i y_i² = c(t), not the continuous integrals ∫u dx and ∫u²dx. The discrete-to-continuous gap is a discretization error that is acknowledged only indirectly in the discussion. The guarantee should be phrased as a discrete conservation guarantee, with the continuous guarantee holding only in the limit Δx→0. This is a conceptual point that affects the interpretation of all experimental claims.
  3. [Sec. 3.9, Eq. (22); Sec. 5] For non-conserved systems (e.g., reaction-diffusion), the target c(t) at collocation times is estimated by linear interpolation from FDM-derived values using Eq. (22). The projection then enforces this estimated value, so the reported conservation error for non-conserved systems includes the interpolation error, which the paper itself notes in Sec. 5. This means the 'guarantee' is conditional on the quality of the c(t) approximation. This limitation should appear in the main text, not only in the discussion, and the effect of the interpolation error should be quantified or at least bounded.
minor comments (5)
  1. [Sec. 3.6, Eq. (10)] The notation 1_n^T uθ(x,t) Δx is confusing: it suggests a scalar Δx multiplying a scalar dot product. Please define uθ(·) as a vector over spatial indices and write the sum explicitly or use a clearer vector notation.
  2. [Sec. 3.8, Eq. (17)] The combined projection formula in Eq. (17) is visually unreadable due to the nested summations and missing parentheses. Adding brackets around the summation terms would improve clarity.
  3. [Sec. 4, Tables 2-3] The bolding in Tables 2 and 3 is described as 'lowest error for that configuration,' but it is not clear whether bold comparisons are across models within the same PDE and conserved quantity, or across conserved quantities within a model. Please specify.
  4. [Sec. 5, Figure 2] The Hessian spectral density plots are informative, but the axes are not labeled and it is unclear whether the x-axis is linear or logarithmic. Please add axis labels and specify the kernel bandwidth used for smoothing.
  5. [Sec. 5, 'PINN Training and Convergence'] The claim that the projection 'improved convergence' is based on Hessian eigenvalue clustering. This is suggestive but not a direct proof of faster convergence. Consider supplementing with convergence curves or training loss trajectories.

Circularity Check

1 steps flagged · score 3.0 of 10

Conservation guarantee is enforced by construction; projection derivation and PDE-error comparisons are independent.

  1. self definitional [§1; §3.6 Eq. (11); §3.7 Eq. (14); §4 Table 3]
    "we must impose a hard constraint that ensures that the PINN always obeys the laws of conservation regardless of the nature of the data. ... To force the neural network output to respect the linear integral constraint, we consider the following optimization problem min_y ||uθ−y||^2 s.t. Δx 1_n^T y = c(t) ... PINN-Proj achieved the lowest Errorc L and Errorc Q on all datasets by three to four orders of magnitude compared to the next best model."

    The projection y is defined as the minimizer of ||ūθ−y||^2 subject to the very conservation constraint being 'guaranteed': Δx1^T y = c(t) for the linear integral and Δx y^T y = c(t) for the quadratic integral. The projected output is therefore constructed to satisfy that constraint, and Table 3's Errorc measures deviation from the same constraint. The 3–4 order-of-magnitude improvement over PINN-SC is thus the expected difference between hard-enforcing the target c(t) and merely penalizing it, not an independent empirical discovery. The Lagrange-multiplier derivation itself is not fitted to data, so this is a partial, definitional circularity concentrated in the conservation claim and its reported Errorc validation.

full rationale

The core derivation of the projection is self-contained: it solves a well-posed constrained optimization problem with Lagrange multipliers, and the closed-form projections are not fitted to the test data. The PDE-accuracy comparison (Erroru) and the Hessian spectral-density analysis are genuinely independent of the conservation constraint and provide external evidence for the method. The paper contains no load-bearing self-citations and does not import a uniqueness theorem from the authors' prior work; the 'unique global minimum' statement is a standard geometric fact. The only definitional element is the conservation-guarantee result: the projection is constructed to satisfy ΔxΣ y_i = c(t) or ΔxΣ y_i^2 = c(t), so the reported Errorc values are a self-consistency check. Separately, and outside circularity, the printed quadratic projection in Eqs. (15)/(41) omits the Δx factor: substituting y_i = u_i sqrt(c(t)/Σ u_i^2) into Δx y^T y = c(t) gives Δx·c(t), not c(t). That algebraic inconsistency affects the stated quadratic guarantee and should be corrected, but it is a correctness issue rather than a circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central method introduces no new physical entities and fits no constants to achieve conservation; the main free parameter is the soft-constraint baseline weight. The heavier assumptions are that discrete quadrature stands in for continuous integration and that numerical solver data provides exact conservation targets.

free parameters (2)
  • soft-constraint weight λ = 10
    Selected by a small sweep over λ in Appendix A.4; affects only the PINN-SC baseline, not the central PINN-Proj method.
  • Hessian smoothing variance = 4e-5
    Gaussian kernel variance for spectral density plots; arbitrary visualization choice, not central to the method.
assumptions (5)
  • domain assumption FDM-generated PDE solutions are accurate enough to serve as ground truth and to provide target values c(t).
    All training data and conservation targets come from numerical solvers; any solver error transfers into the projection target.
  • domain assumption Uniform-grid quadrature, Δx Σ u_i ≈ ∫ u dx and Δx Σ u_i² ≈ ∫ u² dx, faithfully represents the continuous integral.
    The projection enforces only the discrete sum, not the continuous integral; this enters at Eq. (13) and Section 3.6-3.8.
  • ad hoc to paper For non-conserved systems, c(t) at collocation times can be linearly interpolated from neighboring defined values.
    Introduced in Section 3.9 and Appendix A.9; if c(t) is not smooth on the time grid, the projection enforces an approximate target.
  • standard math The constrained optimization problems have unique global minima, so the Lagrange multiplier solution is the correct projection.
    Used in Sections 3.6-3.8 and Appendix A.10-A.12; geometrically plausible but not rigorously discussed for degenerate cases.
  • domain assumption Backpropagating through the projection layer is valid and does not destabilize training.
    The method assumes the non-linear projection is differentiable enough for gradient-based optimization; no proof or stability analysis is provided.

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Cite this review

Pith. "Pith review of Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks." pith.science (2026). https://pith.science/paper/CRWM3FAR

@misc{pith2026251109048,
  author       = {Pith},
  title        = {Pith review of: Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CRWM3FAR}},
  note         = {Machine review of arXiv:2511.09048}
}
read the original abstract

We propose a novel projection method that guarantees the conservation of integral quantities in Physics-Informed Neural Networks (PINNs). While the soft constraint that PINNs use to enforce the structure of partial differential equations (PDEs) enables necessary flexibility during training, it also permits the discovered solution to violate physical laws. To address this, we introduce a projection method that guarantees the conservation of the linear and quadratic integrals, both separately and jointly. We derived the projection formulae by solving constrained non-linear optimization problems and found that our PINN modified with the projection, which we call PINN-Proj, reduced the error in the conservation of these quantities by three to four orders of magnitude compared to the soft constraint and marginally reduced the PDE solution error. We also found evidence that the projection improved convergence through improving the conditioning of the loss landscape. Our method holds promise as a general framework to guarantee the conservation of any integral quantity in a PINN if a tractable solution exists.

Figures

Figures reproduced from arXiv: 2511.09048 by the authors.

Figure 1
Figure 1. c(t) values over time for the conserved system PDEs with one conserved quantity [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Spectral densities of Hessian for the conserved system PDEs with one conserved quantity [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

5 extracted references · 3 linked inside Pith

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    Challenges in training pinns: A loss landscape perspective.arXiv preprint arXiv:2402.01868,

    Pratik Rathore, Weimu Lei, Zachary Frangella, Lu Lu, and Madeleine Udell. Challenges in training pinns: A loss landscape perspective.arXiv preprint arXiv:2402.01868,

    arXiv

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    Learning differentiable solvers for systems with hard constraints.arXiv preprint arXiv:2207.08675,

    Geoffrey N´egiar, Michael W Mahoney, and Aditi S Krishnapriyan. Learning differentiable solvers for systems with hard constraints.arXiv preprint arXiv:2207.08675,

    arXiv

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    Harnessing the power of neural operators with automatically encoded conservation laws.arXiv preprint arXiv:2312.11176,

    Ning Liu, Yiming Fan, Xianyi Zeng, Milan Kl¨ower, Lu Zhang, and Yue Yu. Harnessing the power of neural operators with automatically encoded conservation laws.arXiv preprint arXiv:2312.11176,

    arXiv

  4. [2024]

    Lagrangian neural networks

    Miles Cranmer, Sam Greydanus, Stephan Hoyer, Peter Battaglia, David Spergel, and Shirley Ho. Lagrangian neural networks. InICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations,

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  5. [4090]

    Each 2D PDE dataset had parameters ∆x= 1/16,X= [0,2],∆t= 0.01,T= [0,0.99],(n x, ny)= (32, 32), andn t = 100, so each had 25,600 total state valuesu

    A.2 PDE SETUP Each 1D PDE dataset had parameters∆x= 1/128,X= [0,2],∆t= 0.01,T= [0,0.99],n x = 256, andn t = 100, so each had 25,600 total state valuesu. Each 2D PDE dataset had parameters ∆x= 1/16,X= [0,2],∆t= 0.01,T= [0,0.99],(n x, ny)= (32, 32), andn t = 100, so each had 25,600 total state valuesu. We included for only the 1D conserved systems to isolat...

    2020

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