Development and optimization of physics-informed neural networks for solving partial differential equations

Aleksei Kavokin; Batyr Sharimbayev; Shirali Kadyrov

REVIEW 4 major objections 6 minor 4 references

Development and optimization of physics-informed neural networks for solving partial differential equations

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper reports that on 1D and 2D Poisson equations, finite-difference solvers are far more accurate than a tanh-based PINN, while the same PINN can also reconstruct unknown source terms and coefficients.

desk verdict A modest, reproducible FDM-vs-PINN benchmark on 1D/2D Poisson whose directional claim is likely right, but whose quantitative error numbers are not substantiated by the reported protocol. read the letter →

arxiv 2502.02599 v1 pith:WSN3NJ3N submitted 2025-01-22 math.GM

Batyr Sharimbayev , Shirali Kadyrov , Aleksei Kavokin This is my paper · ORCID

classification math.GM MSC 65N0668T07

keywords numericalanalysisforward-inverseproblemsdeeplearningphysics-informedneuralnetworksfinitedifferencemethodPoissonequationrelativeL2error

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 tries to establish where classical numerical solvers and physics-informed neural networks each belong on simple Poisson problems. It reports that the finite difference method, which approximates derivatives on a grid, reaches $L^2$ relative errors of $7.26 \times 10^{-8}$ in 1D and $2.21 \times 10^{-4}$ in 2D, while the PINN reaches $5.63 \times 10^{-6}$ and $6.01 \times 10^{-3}$ on the same problems. It also claims that a single PINN can handle forward-inverse problems, reconstructing an unknown source term or varying coefficient together with the solution after 40,000 training epochs. The comparison matters because it gives users a concrete accuracy picture before choosing between a mesh-dependent solver and a meshless network.

What carries the argument

The comparison is carried by a common accuracy yardstick: the $L^2$ relative error between a computed approximation and the known exact solution. The FDM side uses the standard central-difference approximation of the second derivative on a grid, converting the Poisson equation into a system of algebraic equations solved by iteration. The PINN side is a fully connected feed-forward network with tanh activations and hidden width [20,20,20,1], trained by minimizing a combined loss made of the mean squared PDE residual at Latin Hypercube collocation points and the boundary-condition mismatch, first with Adam and then with L-BFGS. For forward-inverse problems, the unknown source term or varying coefficient is treated as an additional output of the same network, so one loss minimization recovers both the solution and the hidden function.

What would settle it

Run the same 2D Poisson problem with a tuned PINN (for example, a tanh network with [20,20,20,1] units trained on 1,000-4,096 Latin Hypercube collocation points, with Adam followed by L-BFGS and several random seeds), and compare every seed's best $L^2$ relative error with the reported FDM error of $2.21\times10^{-4}$; if any PINN seed falls below that value, the paper's conclusion that FDM is far more accurate on this problem would not survive that configuration.

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Extended reading notes

Core claim

The paper's central claim is that on 1D and 2D Poisson equations, FDM is substantially more accurate than a vanilla PINN measured by $L^2$ relative error: $7.26\times10^{-8}$ versus $5.63\times10^{-6}$ in 1D and $2.21\times10^{-4}$ versus $6.01\times10^{-3}$ in 2D. A second claim is that PINNs, despite this forward-problem deficit, can solve forward-inverse problems in one framework: after 40,000 epochs the network predicts the solution together with the source term (final loss $1.87\times10^{-2}$) or together with the varying coefficient (final loss $3.1563\times10^{-2}$), and these predictions agree with FDM results at observed and unobserved locations. The paper therefore positions FDM as the accuracy-first method for smooth Poisson-type forward problems and PINN as a flexible option when unknown fields must be inferred alongside the solution.

Load-bearing premise

The comparison is only representative if the single, untuned PINN runs reported here stand in fairly for what PINNs can do; the paper gives no training schedule, learning rate, random seed, or repeated trials, and uses different collocation-point counts in different sections, so a better-tuned PINN could close much of the accuracy gap.

Editorial extensions

If this is right

For smooth Poisson problems in 1D and 2D, the accuracy-first choice is FDM: the reported relative $L^2$ errors are $7.26\times10^{-8}$ versus $5.63\times10^{-6}$ in 1D and $2.21\times10^{-4}$ versus $6.01\times10^{-3}$ in 2D.
A vanilla PINN, though less accurate on these forward problems, can be trained once to solve a forward-inverse problem, jointly producing the solution and an unknown source term or varying coefficient.
After 40,000 epochs the forward-inverse PINN reaches training losses of $1.87\times10^{-2}$ (source-term reconstruction) and $3.1563\times10^{-2}$ (coefficient reconstruction), with predictions matching FDM at observed and unobserved points.
The paper's evidence is confined to second-order Poisson-type equations; the authors state that higher-order equations, where they expect PINNs may do better, are left for future research.

Reading between the lines

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

Beyond the paper: the quoted accuracy gap should be read as specific to the reported setup, not as a general statement about PINNs; a tuned PINN with residual weighting, more collocation points, or different seeds could plausibly close much of the gap.
Beyond the paper: a natural next experiment is the 2D forward-inverse Poisson problem, since the paper's inverse demonstration is one-dimensional and it is not clear that the same reconstruction quality carries over.
Beyond the paper: the authors' hint that PINNs may outperform classical methods on higher-order equations can be tested directly by running the same $L^2$ relative error comparison on, for example, the biharmonic or Allen-Cahn equations.
Beyond the paper: the success of the forward-inverse reconstruction suggests a hybrid strategy in which FDM supplies cheap supervision data for PINN training, combining the grid solver's accuracy with the network's ability to infer hidden parameters.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper is a straightforward benchmark: FDM vs vanilla PINN on 1D and 2D Poisson equations, plus a PINN demo for a forward-inverse ODE. The punchline—FDM wins handily on small smooth forward problems, PINNs are flexible enough for inverse tasks—is probably correct and already known from the literature. The paper does not introduce a new method, theory, or dataset; its value is as a clean, reproducible confirmation with code on GitHub.

What it does well: the setup is clear, the comparison is against analytical solutions, and the forward-inverse experiments are a nice illustration of the framework. The reference list includes the key comparison paper [7], which already makes much the same point.

Where it falls short, in roughly increasing order of concern:

1. Reporting. Section 3 says 256 LHS samples for training/testing; Section 4.1 says 512 collocation points per epoch. The numbers in the abstract cannot be assigned to a definite protocol. No seeds, learning rates, or repeated runs are given, so the single quoted errors are not demonstrably representative.

2. Testing. The text says the same samples were used for training and testing. If that's literal, the PINN errors are in-sample optimization residuals, not generalization errors. That alone undermines the headline accuracy comparison.

3. Conceptual slips. Section 4.1 calls an FDM step a 'weak form'—only FEM does that. The introduction says 'finite element method' where the paper actually uses finite differences. Figure 1 caption similarly says FEM in the body text. These are fixable but should be caught in revision.

4. The 1D FDM error (7.26e-8) is surprisingly low for a second-order method on a 512-cell grid unless the exact solution is a low-degree polynomial, in which case the central difference is exact and the error is roundoff. The equations are missing from the text, so I can't check. The authors need to state the exact solution explicitly and describe the error computation.

None of these break the directional conclusion, which I think holds. But the quantitative claims in the abstract are not established by the reported experiments.

Who is this for? A practitioner wanting a quick, simple example of PINN vs FDM on Poisson, or someone teaching a scientific ML course. It deserves a serious referee—the topic is timely and the code is a plus—but it needs a heavy revision rather than acceptance as is. I'd recommend send to peer review with a request for major revisions: fix the errors, add seeds/repeats/error bars, and clarify the test set.

Referee Report

4 major / 6 minor

Summary. The paper reports a comparison between the Finite Difference Method (FDM) and Physics-Informed Neural Networks (PINNs) on one- and two-dimensional Poisson equations. It reports relative L2 errors of 7.26e-8 and 2.21e-4 for FDM versus 5.63e-6 and 6.01e-3 for PINNs, concluding that FDM is far more accurate for these forward problems. The paper further demonstrates PINN-based reconstruction of a source term and a varying coefficient in a forward-inverse setting. The manuscript also provides a GitHub link to the code used for the experiments.

Significance. If the quantitative comparison were substantiated, the paper would provide a useful, if modest, benchmark for the well-known accuracy gap between FDM and PINNs on small elliptic problems. The availability of code on GitHub is a concrete reproducibility asset, and the forward-inverse examples illustrate a real strength of the PINN framework. However, the central quantitative claims are currently supported only by single unrepeated runs with no held-out test set, no hyperparameter reporting, and an internal inconsistency in the number of collocation points. The directional conclusion may well survive a more careful experiment, but the specific error numbers in the abstract and Section 4 are not established by the reported methodology.

major comments (4)

[Section 3 and Section 4.1] The PINN errors appear to be in-sample training errors rather than independent generalization errors. Section 3 states that 'the obtained samples were used in training the PINN as well as testing the prediction capabilities of the model', meaning the same LHS points are used for training and evaluation. Section 4.1 then says that 512 collocation points are sampled each epoch, while Section 3 says 256 samples were drawn. If the reported relative L2 errors of 5.63e-6 and 6.01e-3 are computed on training collocation points, they are not a fair measure of accuracy against the FDM solution. The authors must evaluate the PINN on a held-out set of points not used during training, and report the number of training versus test points consistently.
[Section 4.1 and Section 4.2] The quantitative comparison rests on single runs with no random seed, no learning rate, no specification of the Adam/L-BFGS iteration split, and no repeated trials. A PINN trained with a different initialization or optimization schedule can easily change the reported errors by orders of magnitude, so the central claim that FDM has errors of 7.26e-8 and 2.21e-4 versus PINN errors of 5.63e-6 and 6.01e-3 is not yet robust. The authors should report the full training protocol and provide mean and standard deviation over at least several independent runs for the PINN errors.
[Section 4.1] The 1D FDM relative error of 7.26e-8 with 512 cells is not justified by the information given. For a standard second-order central-difference discretization and a generic smooth non-polynomial solution, one would expect an error on the order of h^2, i.e. roughly 4e-6 at h = 1/512, not 7e-8. The exact solution and source term are not displayed in the text, so it is impossible to check whether the solution is a low-degree polynomial, for which the central scheme can be exact up to roundoff, or whether a different discretization/solver was used. The authors must state the exact PDE, the analytical solution, the FDM stencil, the linear solver, and the stopping criterion.
[Section 4.3] The forward-inverse results are not quantitatively validated. The paper reports only the final loss values of 1.87e-2 and 3.1563e-2 after 40,000 epochs, but a loss value is not an accuracy metric for the reconstructed source term Q(x) or the varying coefficient a(x). The claims that the predictions are in 'excellent agreement' with FDM and that the model 'accurately reconstructs' the coefficient are supported only by visual inspection of figures. The authors should report measured errors (e.g., relative L2 errors) for Q(x), a(x), and U(x) against the true or FDM reference, together with the number and placement of the observed data points used in the inverse setting.

minor comments (6)

[Section 1 and Section 4.1] The Introduction states that the purpose is to assess the 'finite element method' against PINNs, but the experiments use the Finite Difference Method; Section 4.1 also contains the phrase 'using FEM and PINNs'. The method names should be made consistent throughout the manuscript.
[Figure 1 caption and surrounding text] The paragraph under Figure 1 says that a one-node, one-hidden-layer PINN 'performs poorly and fails to satisfy the boundary conditions' and then immediately says that 'all approximations are very close to the exact solution.' This is contradictory or at least ambiguous; clarify whether the one-node setup is part of the figure or a separate negative result.
[Sections 2.1, 4.1, 4.2] Several key equations, including the 1D and 2D Poisson equations, their boundary conditions, and their analytical solutions, are missing from the displayed text. Without these equations, the reported error values cannot be independently checked. The authors should ensure all equations are rendered and numbered.
[Section 3] Section 3 says 256 samples were drawn from the range plus 2 boundary points, while Section 4.1 says 512 collocation points are sampled each epoch using LHS. These numbers should be reconciled, or the text should explain that they refer to different protocols.
[Section 4.3] The problem in Section 4.3 is repeatedly called a 'thermal' problem and the solution a 'temperature distribution', but the displayed equation is a generic second-order ODE with a source term and a varying coefficient. The physical interpretation should be stated explicitly or the thermal terminology should be removed.
[Title and Section 4] The title promises 'development and optimization' of PINNs, but the paper does not describe any systematic hyperparameter optimization, architecture search, or comparison of alternative designs. Either add such an optimization study or soften the title to match the actual content.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the forward comparisons use analytical solutions and independent FDM references, and the inverse problems are standard benchmark reconstructions, not results equivalent to their own inputs.

full rationale

The paper's central quantitative claims are the relative L2 errors of FDM versus PINNs on 1D and 2D Poisson equations. These errors are computed against the exact analytical solutions stated in Sections 4.1 and 4.2, not against any quantity fitted into the methods. FDM and PINN are each implemented from their standard formulations in Sections 2.2 and 2.3, and no parameter is fitted to the error values and then reused as the benchmark. The forward-inverse experiments in Section 4.3 reconstruct a known source term and a known varying coefficient from data generated by the same differential equation; this is a standard inverse-problem benchmark, not a circular derivation. The manuscript does contain methodological weaknesses: the LHS samples are described as used for both training and testing, the number of collocation points differs between Section 3 (256) and Section 4.1 (512), no random seed or repeated trials are reported, and the in-text reference [20] for the network architecture is missing from the reference list. These are reproducibility and validation concerns, not instances where a predicted result reduces by construction to the paper's inputs. No self-citation chain is load-bearing, no uniqueness theorem is imported, and no known result is merely renamed. The derivation chain therefore shows no significant circularity, and the honest finding is a score of 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central comparison depends only on standard PDE theory and hand-chosen experimental settings; no fitted constants enter a derivation. The main hidden burden is the assumption that the reported PINN runs are representative.

free parameters (4)

PINN architecture (hidden layers 20-20-20, tanh) = 20-20-20, tanh
Chosen by hand; no architecture search or justification is given in Section 4.1.
Number of collocation points = 256 (Section 3) vs 512 (Section 4.1)
Inconsistent in the text; this setting affects the reported L2 error and training cost.
Training epochs for forward-inverse problems = 40,000
Chosen by hand; the loss is reported after a fixed epoch count with no convergence criterion.
Grid cells for FDM = 512
Chosen by hand; the FDM error scale depends directly on this resolution.

assumptions (4)

standard math The Poisson equation with Dirichlet boundary conditions on [0,1] has unique solutions, and the stated analytical solutions are correct.
Used to define exact solutions against which the numerical errors are measured; Sections 2.1, 4.1, and 4.2.
standard math The finite difference discretization converges to the exact solution as the grid is refined.
Assumed when treating FDM as a reliable reference benchmark; Section 2.3.
domain assumption A fully connected neural network with tanh activations can approximate the Poisson solution and the source term or coefficient in the inverse tasks.
PINN training relies on universal approximation and the chosen architecture being adequate; Sections 2.2 and 4.3.
domain assumption Latin hypercube sampling provides representative collocation points in the domain.
Used to generate training points for the PINN; Section 3.

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Cite this review

Pith. "Pith review of Development and optimization of physics-informed neural networks for solving partial differential equations." pith.science (2026). https://pith.science/paper/WSN3NJ3N

@misc{pith2026250202599,
  author       = {Pith},
  title        = {Pith review of: Development and optimization of physics-informed neural networks for solving partial differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WSN3NJ3N}},
  note         = {Machine review of arXiv:2502.02599}
}

read the original abstract

This work compares the advantages and limitations of the Finite Difference Method with Physics-Informed Neural Networks, showing where each can best be applied for different problem scenarios. Analysis on the L2 relative error based on one-dimensional and two-dimensional Poisson equations suggests that FDM gives far more accurate results with a relative error of 7.26 x 10-8 and 2.21 x 10-4, respectively, in comparison with PINNs, with an error of 5.63 x 10-6 and 6.01 x 10-3 accordingly. Besides forward problems, PINN is realized also for forward-inverse problems which reflect its ability to predict source term after its sufficient training. Visualization of the solution underlines different methodologies adopted by FDM and PINNs, yielding useful insights into their performance and applicability.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

4 extracted references · 3 canonical work pages

[1]

meshless

Introduction PDEs would form a crucial backbone in the understanding and modeling of various real-world problems. In simple words, PDEs describe how something changes through time and space based on very well-defined mathematical rules. The central issue with PDEs is determining whether these rules suffice to guarantee a unique solution to the problem [1]...

work page
[2]

It provides significant applications to physics and engineering, modeling electrostatics, heat conduction, and fluid dynamics among many others

Mathematical Background 2.1 Poisson Equation The Poisson equation is one of the most basic PDEs, which results in a potential field created by a given source. It provides significant applications to physics and engineering, modeling electrostatics, heat conduction, and fluid dynamics among many others. References [8][9][10]. In one dimension, the Poisson ...

work page
[3]

The data to be used in these tasks were sampled from a given range by using the Latin Hypercube Sampling (LHS) technique

Methodology This work concentrates on developing and enhancing PINNs for the solutions of the 1D and 2D Poisson equations and comparing them with the FDM. The data to be used in these tasks were sampled from a given range by using the Latin Hypercube Sampling (LHS) technique. The range of interest for the problem at hand was between 0 and 1, and the LHS t...

work page
[5]

Conclusion & Future work This work compared the accuracy of FDM and PINNs for solving 1D and 2D Poisson equations. Results show that the FDM has much smaller relative errors compared with PINNs for both cases: the relative errors are 7.26 × 10 -8 versus 5.63×10 -6 for the 1D problem and 2.21× 10 -4 versus 6.01 × 10 -3 for the 2D problem. Although PINNs ar...

arXiv 2014

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

meshless

Introduction PDEs would form a crucial backbone in the understanding and modeling of various real-world problems. In simple words, PDEs describe how something changes through time and space based on very well-defined mathematical rules. The central issue with PDEs is determining whether these rules suffice to guarantee a unique solution to the problem [1]...

work page

[2] [2]

It provides significant applications to physics and engineering, modeling electrostatics, heat conduction, and fluid dynamics among many others

Mathematical Background 2.1 Poisson Equation The Poisson equation is one of the most basic PDEs, which results in a potential field created by a given source. It provides significant applications to physics and engineering, modeling electrostatics, heat conduction, and fluid dynamics among many others. References [8][9][10]. In one dimension, the Poisson ...

work page

[3] [3]

The data to be used in these tasks were sampled from a given range by using the Latin Hypercube Sampling (LHS) technique

Methodology This work concentrates on developing and enhancing PINNs for the solutions of the 1D and 2D Poisson equations and comparing them with the FDM. The data to be used in these tasks were sampled from a given range by using the Latin Hypercube Sampling (LHS) technique. The range of interest for the problem at hand was between 0 and 1, and the LHS t...

work page

[4] [5]

Conclusion & Future work This work compared the accuracy of FDM and PINNs for solving 1D and 2D Poisson equations. Results show that the FDM has much smaller relative errors compared with PINNs for both cases: the relative errors are 7.26 × 10 -8 versus 5.63×10 -6 for the 1D problem and 2.21× 10 -4 versus 6.01 × 10 -3 for the 2D problem. Although PINNs ar...

arXiv 2014