{"id":"06320e4c-442d-41b1-ac95-d04fe096adbc","arxiv_id":"2502.02599","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"On one- and two-dimensional Poisson equations, finite differences beat physics-informed neural networks in accuracy, and PINNs can reconstruct source terms and coefficients in a forward-inverse setting.","lead":"This paper compares two ways of solving differential equations: the traditional finite difference method and physics-informed neural networks. It finds on simple Poisson test problems that the finite difference method is much more accurate, while the neural networks can also estimate unknown source terms and coefficients from data.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central FDM-vs-PINN error comparison is not yet substantiated: PINN errors appear to be computed on training collocation points, with no held-out test set or repeated seeds, and sections 3 and 4.1 conflict on whether 256 or 512 points were used.","rationale":"Reading in good faith, the paper's central claim is plausible: on simple Poisson problems, a well-resolved FDM should indeed outperform a default PINN. The reader's CONDITIONAL verdict correctly reflects that the reported numbers are not robust. My stress-test sharpens the same concern. The manuscript explicitly says the LHS samples were used for both training and testing, and Section 3 (256 samples) conflicts with Section 4.1 (512 collocation points per epoch). These are internal inconsistencies, not just missing details. The lack of seeds and repeats means the single PINN errors could shift substantially. I also note the 1D FDM error of 7.26e-8 is much smaller than the ~4e-6 expected from a standard second-order central-difference scheme on 512 cells for a smooth non-polynomial solution, so the exact test problem or error computation needs checking. These concerns do not overturn the likely directional conclusion, so the verdict remains CONDITIONAL rather than REJECT or ACCEPT. The proposed concrete test would settle whether the quantitative headline is robust.","tokens_in":11071,"tokens_out":8933,"duration_ms":89249,"concrete_test":"Run the provided GitHub code and instrument the evaluation: compute the PINN relative L2 error on a held-out uniform grid of 10,000 points (not the LHS collocation set), for 5 different random seeds; record the exact solution, grid size, and collocation count. Also independently assemble the standard tridiagonal FDM system for the 1D problem on 512 cells and compute the relative L2 error. If the held-out PINN error differs from 5.63e-6/6.01e-3 by more than a factor of 2, or if the FDM error is not 7.26e-8 for a smooth non-polynomial exact solution, the reported comparison is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a quantitative comparison of relative L2 errors (FDM 7.26e-8/2.21e-4 vs PINN 5.63e-6/6.01e-3). For this comparison to be meaningful, the PINN errors must be computed on points not used for training, and the runs must be representative. The manuscript states in Section 3 that 'the obtained samples were used in training the PINN as well as testing the prediction capabilities of the model'; if the same LHS points are used for both, the quoted PINN errors are in-sample training errors, not independent accuracy estimates. Section 4.1 then says 512 collocation points are sampled each epoch, while Section 3 says 256 samples were drawn, so the reported numbers cannot be assigned to a definite protocol. No random seed, learning rate, Adam/L-BFGS iteration split, or number of repeated trials is given, so there is no evidence that the single PINN runs are representative. A standard second-order FDM on 512 cells for a smooth non-polynomial solution would have relative L2 error near h^2 ~ 4e-6, not 7.26e-8, so the 1D FDM number also needs scrutiny. The directional conclusion may survive retesting, but the specific quantitative claim in the abstract is not established by the reported experiments.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11316,"tokens_out":4866,"duration_ms":49124,"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":[{"comment":"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":"Section 3 and Section 4.1"},{"comment":"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":"Section 4.1 and Section 4.2"},{"comment":"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":"Section 4.1"},{"comment":"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.","section":"Section 4.3"}],"minor_comments":[{"comment":"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.","section":"Section 1 and Section 4.1"},{"comment":"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.","section":"Figure 1 caption and surrounding text"},{"comment":"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":"Sections 2.1, 4.1, 4.2"},{"comment":"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":"Section 3"},{"comment":"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.","section":"Section 4.3"},{"comment":"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.","section":"Title and Section 4"}],"recommendation":"major_revision","confidential_remarks":"The paper is a fairly preliminary benchmark study. The main reason for major revision, rather than rejection, is that the central comparison could be made valid with a moderate amount of additional experimentation: a held-out test set, repeated seeded runs with full hyperparameter reporting, and quantitative inverse-problem errors. The manuscript also contains several internal inconsistencies and missing equations that would need careful correction. The GitHub repository is a plus, but the claims as written are not yet supported by the reported evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nWhere it falls short, in roughly increasing order of concern:\n\n1. 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.\n\n2. 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.\n\n3. 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.\n\n4. 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.\n\nNone of these break the directional conclusion, which I think holds. But the quantitative claims in the abstract are not established by the reported experiments.\n\nWho 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.","headline":"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.","tokens_in":11914,"tokens_out":3473,"would_cite":false,"duration_ms":32810,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N06","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["numerical analysis","forward-inverse problems","deep learning","physics-informed neural networks","finite difference method","Poisson equation","relative L2 error"],"falsifier":"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.","tokens_in":10808,"feed_emoji":"🧮","tokens_out":12299,"duration_ms":115936,"temperature":0.7,"pith_summary":"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.","feed_headline":"FDM beats PINNs on Poisson accuracy by 10x to 100x","feed_subtitle":"FDM is far more accurate; PINNs still recover unknown sources in one pass.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the standard PINN formulation the paper implements: a network trained on PDE residual plus boundary-condition loss.","marker":"[4]"},{"why":"Provides the prior PINN-versus-classical-solver comparison on Poisson, Allen-Cahn, and Schrödinger equations that motivates this study.","marker":"[7]"},{"why":"Supports the mesh-based numerical solution of the Poisson equation used in the FDM baseline.","marker":"[10]"},{"why":"Provides the finite-difference derivative approximations that define the FDM solver.","marker":"[11]"},{"why":"Defines the $L^2$ norm and relative error used as the shared accuracy metric.","marker":"[14]"},{"why":"Supports the forward-inverse PINN construction for simultaneous solution and parameter recovery.","marker":"[15]"}],"fun_headline_variants":["FDM beats PINNs 10–100x on Poisson forward accuracy","PINNs solve inverse Poisson while FDM leads in accuracy","Poisson test: FDM far more precise, PINN auto-infers sources"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["FDM beats PINNs 10–100x on Poisson forward accuracy","PINNs solve inverse Poisson while FDM leads in accuracy","Poisson test: FDM far more precise, PINN auto-infers sources"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000623,"raw_usage":{"total_tokens":2855,"prompt_tokens":886,"completion_tokens":1969,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":1908}},"tokens_in":502,"tokens_out":1969,"duration_ms":17065,"temperature":1.0,"reasoning_tokens":1908,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:57:31.546604+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}