{"id":"e204fdfd-1c84-462e-b264-33625c6e5140","arxiv_id":"2412.13993","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Adding a standard-deviation term to the mean loss in physics-informed neural networks reduces peak errors on Poisson, Burgers, elasticity, and Navier-Stokes benchmarks.","lead":"A new loss function for physics-informed neural networks adds the standard deviation of the residual errors to the usual mean-squared error term, aiming to reduce large localized errors. On four benchmark PDEs, the combined loss lowers maximum error with only a small computational overhead, although a tunable weight is required.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reported gains are measured at per-problem alpha chosen from test error with one initialization per alpha, so the headline improvements may be selection artifacts rather than robust properties of the variance loss.","rationale":"The reader and I identify the same load-bearing weakness: the experimental design selects the hyperparameter alpha from the test-error curves and reports a single trajectory per setting, so the favorable comparisons are not yet evidence for the method's general behavior. The manuscript's own conclusion acknowledges that the choice alpha=0.8 lacks theoretical justification, and the Navier-Stokes MSE baseline diverging under the chosen optimizer makes that comparison especially fragile. I do not see an internal inconsistency in the derivation of Eq. (12); the loss is well-defined and the computational overhead claim is plausible. The issue is that the central claim, that the method 'ensures a more uniform error distribution' and provides 'lower maximum error', is stated more strongly than the evidence supports. A multi-seed study with a pre-specified alpha or a validation-based selection rule would materially change the strength of the claim and is the single most informative next step. Therefore I recommend keeping the reader's CONDITIONAL verdict unchanged.","tokens_in":10821,"tokens_out":4186,"duration_ms":40006,"concrete_test":"Re-run all four benchmarks with 10 independent Xavier initializations at alpha=1.0 and alpha=0.8, keeping every other setting identical, and report per-seed L2 and L_infinity errors. If alpha=0.8 does not beat alpha=1.0 in max error for at least 8 of 10 seeds on each problem, the reported 2-30x reductions are not robust to initialization and the central claim should be weakened. A secondary check: select alpha on a validation split of collocation points before evaluating against the reference solution, to test whether the method provides a usable alpha-selection rule.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, that Eq. (12) reduces localized high-error regions and yields lower maximum error, is supported only by comparisons at the best per-problem alpha chosen from the error curves in Section 3. For each benchmark, alpha is swept from 0 to 1 using the same Xavier initialization, and the best alpha (0.8, 0.8, 0.6, 0.8) is then used for the reported figures; no rule is given for selecting alpha without a reference solution, and no independent seeds or error bars are reported. Because the same initial weights are reused across alpha values, the sweep selects the best of several single trajectories rather than sampling the optimization landscape. The 30x and 15x max-error reductions in Section 3.3, the 2x reduction in Section 3.2, and the order-of-magnitude pressure improvement in Section 3.4 are therefore maxima over alpha and a single initialization, not stable expectations. The Navier-Stokes comparison in Section 3.4 compounds this: the MSE baseline (alpha=1) diverges under the fixed Adam setting, so the reported superiority there may reflect a failed baseline rather than the variance term. The conclusion itself concedes that a theoretical justification for alpha remains an open question, but the empirical support is under-powered for the strength of the claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a modified PINN loss function, Eq. (12), that combines the mean of a per-point error with its standard deviation: L = alpha * mean(e_i) + (1-alpha) * std(e_i). The authors argue that penalizing the standard deviation of the residuals, in addition to their mean, reduces localized high-error regions and leads to a more uniform error distribution. The method is tested on four problems: 1D Poisson, unsteady Burgers, 2D linear elasticity, and 2D steady Navier-Stokes. In each case, alpha is swept between 0 and 1 and the best value is selected from error-versus-iteration plots; the authors report reductions in maximum error (about 2x for Burgers, 30x/15x for elasticity, and more than 3x plus an order of magnitude for Navier-Stokes) with negligible additional computational cost. Comparisons with Huber loss and gradient-enhanced PINNs are also presented for the elasticity problem.","tokens_in":11048,"tokens_out":4577,"duration_ms":44359,"significance":"If the reported improvements are robust, the proposed loss is an attractive, one-line modification to PINN training: it is simple to implement, has negligible computational overhead, and could help with problems exhibiting localized errors or sharp gradients. The paper also provides a collocation-density experiment suggesting improved data efficiency and a direct comparison with two existing regularization strategies. The main weakness is that the central claim is empirical and the supporting evidence is currently fragile: the reported gains are measured at per-problem alpha values selected from test-error curves, with a single initialization per configuration and no error bars. The Navier-Stokes comparison is further weakened by the divergence of the MSE baseline. The idea is plausible and potentially useful, but the strength of the claims in the abstract and conclusion is not yet matched by the experimental support.","major_comments":[{"comment":"The headline reductions in maximum error are maxima over alpha: for each problem the best alpha value (0.8, 0.8, 0.6, 0.8) is selected from L2-error-versus-iteration curves computed against the analytical or reference solution, and all subsequent comparisons use this selected value. Because the same Xavier initialization is reused for every alpha, the sweep selects the best of several single trajectories rather than comparing expected performance. No rule is given for choosing alpha without a reference solution, so Eq. (12) contains a free hyperparameter that is fitted to the test error. Multi-seed runs at a fixed alpha, error bars, and a validation-based alpha-selection protocol are needed before the abstract's claim that the loss 'ensures a more uniform error distribution' can be considered supported.","section":"Section 3 (alpha sweep and figures 1, 2, 5, 9)"},{"comment":"The MSE baseline (alpha=1) diverges from the reference solution under the fixed Adam setup, and the subsequent pressure and velocity comparisons are made against this failed baseline. The reported order-of-magnitude pressure improvement and the inability of MSE to capture the outlet velocity profile may therefore reflect an unstable or under-converged baseline rather than a genuine benefit of the variance term. The comparison should be repeated with a standard PINN baseline that is known to converge, for example by using more iterations, a learning-rate schedule, or a more powerful optimizer, so that the variance-based loss is measured against a successful standard PINN.","section":"Section 3.4 (Navier-Stokes example, figure 9)"},{"comment":"The efficiency claim that the proposed loss achieves a given accuracy with substantially fewer collocation points rests on only two alpha values and, as far as reported, a single training run per configuration. No repeated-seed statistics or error bars are shown, and the caption repeats 'On the left' for both panels. This is not sufficient support for the conclusion in Section 4.4 that the modified loss reduces reliance on dense spatial sampling; multiple seeds and a clearer comparison of error at matched collocation counts are required.","section":"Section 4.3 (collocation-point density experiment, figure 13)"}],"minor_comments":[{"comment":"There are several typos and grammatical errors, including 'the our new loss' in Section 3.1, 'Burger's' for 'Burgers'' in several places, and 'krieging' for 'kriging' in the Introduction. A proofreading pass is needed.","section":"Throughout"},{"comment":"The comparison with gradient-enhanced PINNs and Huber loss is qualitative: no hyperparameters for gPINNs are given, and the claimed 'marginal improvement' is not quantified. Reporting the final L2 errors for each method would make the comparison more informative.","section":"Section 4.2"},{"comment":"The text states that the new loss is 'always significantly better' than MSE based on Figure 5, but no statistical significance tests are reported and only one run per alpha is shown. The word 'significantly' should either be removed or backed by multi-seed statistics.","section":"Section 3.3"},{"comment":"The standard deviation is defined with a denominator of N rather than N-1. This is a legitimate choice for a loss term, but it should be stated explicitly since it affects the gradient scale.","section":"Eq. (12)"},{"comment":"No statement on code or data availability is included. Given the empirical nature of the claims, releasing the training code and seeds would substantially help reproducibility.","section":"Data and code availability"}],"recommendation":"major_revision","confidential_remarks":"The main risk is that the empirical claims are selected from per-problem alpha sweeps using the same single initialization, which can inflate apparent gains. If the authors add multi-seed runs, report error bars, and fix the Navier-Stokes baseline issue, the contribution could be publishable as a simple and potentially useful heuristic. I would not accept the current version without those changes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper's one-line loss change is a legitimate, reproducible idea, and the qualitative results are encouraging. The alpha-tuning and single-run issues are real and should be fixed before this is cited as strong evidence.\n\nThe proposal is exactly what it looks like: add the population standard deviation of per-sample residuals to the mean, with weight alpha, as in Eq. (12). That exact combination is not in the cited prior work; VI-PINNs learns a variance output and Markowitz is portfolio selection, so the one-line loss is new in the PINN context. It is trivially implementable, and the reported compute overhead is a few percent. The experiments on four benchmarks show lower max error and cleaner error maps with the variance term. I believe those qualitative observations.\n\nThe soft spots are in the evidence, not the idea. Alpha is swept from 0 to 1 on each problem and the best value is read off the test-error curves; all headline numbers are at that chosen alpha. With one initialization shared across the sweep, you are comparing single trajectories, not sampling the optimization landscape. No seeds, no error bars. For Navier-Stokes, the MSE baseline diverges under the fixed Adam setting, so the comparison there is against a failed run, not a healthy baseline. The positive results on Poisson and Burgers are smaller and more believable, but still single runs. The paper admits the choice of alpha is empirical and gives no rule for picking it without a reference solution, which limits the drop-in story.\n\nThat said, the paper is not sloppy in its claims: it connects to Markowitz, compares against Huber and gPINNs, and reports computational cost. The writing is clear and the method is simple enough to reproduce. The problem is under-powered validation, not incoherence.\n\nMy take: this deserves a serious referee, but it needs revision before acceptance. Ask for multiple seeds, error bars, a pre-specified alpha rule or a sensitivity analysis showing the gain holds across a range of alpha, and a Navier-Stokes baseline that does not diverge. If the max-error reduction survives that, it is a useful contribution. If not, it is a cautionary tale about tuning on the test set.\n\nI would bring it to reading group as a case study in loss-function evaluation. I would not cite it yet.","headline":"A simple, plausible one-line loss for PINNs, but the evidence is under-powered by test-tuned alpha and single runs; worth refereeing, not yet citable.","tokens_in":11588,"tokens_out":2676,"would_cite":false,"duration_ms":24749,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","65N35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A PINN loss that combines the mean and the standard deviation of pointwise errors produces more uniform error fields and lower maximum errors than minimizing the mean alone.","keywords":["physics-informed neural networks","loss function","standard deviation","variance regularization","error distribution","maximum error","collocation points","partial differential equations"],"falsifier":"Train the four benchmarks, or a single smooth Poisson case, with the proposed loss and the mean-squared loss across twenty random seeds each, fixing $\\alpha=0.8$, and compare maximum pointwise error on the same collocation set; if the variance-based loss does not reach a lower or equal maximum error in a clear majority of seeds, the central claim fails.","tokens_in":10591,"feed_emoji":"📉","tokens_out":7554,"duration_ms":59820,"temperature":0.7,"pith_summary":"Physics-informed neural networks (PINNs) solve partial differential equations by minimizing a loss that averages pointwise errors over collocation and boundary points, but a mean-based loss tolerates large localized errors wherever the solution has steep gradients. This paper claims that adding the standard deviation of those pointwise errors to the loss, so that both the average error and its spread are minimized, forces the network to allocate effort to the worst regions rather than the bulk. On four test problems, from a smooth 1D Poisson equation to a steep Burgers shock and a 2D Navier-Stokes flow, the proposed loss is reported to reduce the maximum absolute error by a factor of about 2 to 30, while adding only a few percent to training time. The paper also reports that the same accuracy can be reached with substantially fewer collocation points, which would matter for expensive high-dimensional problems. The simple form of the loss, one line changed in existing code, is the practical payoff.","feed_headline":"Variance term in physics-informed neural networks cuts max error 30x","feed_subtitle":"Regularizing error spread, not just its mean, tames shocks and boundary layers in PDE-solving neural nets.","key_machinery":"The load-bearing object is the variance-augmented loss of Eq. (12): $L=\\alpha\\,\\mathrm{mean}(e_i)+(1-\\alpha)\\,\\mathrm{std}(e_i)$, with $e_i$ the pointwise squared error of the PDE residual, initial condition, or boundary condition. The mean keeps the overall fit tight; the standard-deviation term smooths the error landscape by making the optimizer reduce the largest deviations, which is what transfers effort to high-gradient regions. $\\alpha$, chosen per problem by sweeping from 0 to 1, sets how much weight the spread term receives; best values were 0.8, 0.8, 0.6 and 0.8 across the four tests.","core_discovery":"The central claim is that replacing the usual mean-error loss $L=\\frac{1}{N}\\sum_i e_i$ with $L=\\alpha \\frac{1}{N}\\sum_i e_i + (1-\\alpha)\\sqrt{\\frac{\\sum_i(e_i-\\bar e)^2}{N}}$, where $e_i$ is the squared pointwise residual or boundary mismatch and $\\alpha$ balances the two terms, yields a more uniform error distribution and a lower maximum error than the mean alone. The standard deviation term penalizes the dispersion of errors, so the optimizer cannot hide a few large residuals inside a small average. In the experiments this shows up as an $L^\\infty$ error reduced by roughly 2x on Burgers, 15–30x on the 2D elasticity displacement, and more than 3x on velocity plus an order of magnitude on pressure in the Navier-Stokes case, with best results usually at $\\alpha=0.8$. The paper frames the term as regularization, notes that no theoretical justification for the $\\alpha$ choice is yet known, and contrasts the method with gradient-enhanced PINNs and Huber loss, finding comparable or better accuracy at a fraction of the computational cost.","pith_inferences":["Editorial extension: applying the same mean-plus-standard-deviation construction separately to each term of a multi-task PINN loss, such as PDE residual, initial condition, and boundary, would add per-task variance penalties and might outperform the single global $\\alpha$ used here.","Editorial extension: because the variance term already pushes optimization toward high-error zones, it should reduce or delay the need for residual-based adaptive sampling; a direct test is to compare adaptive-sampling PINNs with and without the proposed loss at fixed point counts.","Editorial extension: the reported gains grow with problem sharpness—small on smooth Poisson, large on the Burgers front and elasticity corners—so the loss is likely most valuable precisely where mean-squared PINN training is known to struggle."],"forward_implications":["On the four tested problems, the variance-based loss lowers the maximum absolute error compared with mean-squared loss: roughly 2x for Burgers, 15–30x for the elasticity displacements, and an order of magnitude for the Navier-Stokes pressure.","Because the loss is a one-line change in standard frameworks, the improvement comes with negligible added compute: one to two seconds extra on problems that take 8–45 seconds, and about one extra minute on the 15-minute Navier-Stokes run.","The same target accuracy can be reached with fewer collocation points when the variance term is active, which the paper argues is decisive for high-dimensional or expensive PDE problems.","Against gradient-enhanced PINNs and Huber loss on the elasticity benchmark, the proposed loss matches or approaches the accuracy of gradient-enhanced training while avoiding the multi-fold cost increase and the extra implementation burden."],"supporting_citations":[{"why":"Introduces the PINN framework whose standard mean-squared loss is the baseline the paper modifies.","marker":"[7]"},{"why":"Supplies the 2D linear elasticity problem and the five-network architecture used in Section 3.3.","marker":"[10]"},{"why":"Defines gradient-enhanced PINNs, the higher-cost comparison method for the elasticity test.","marker":"[32]"},{"why":"Proposes variance-predicted PINNs, the alternative variance-based approach the paper distinguishes from its own.","marker":"[33]"},{"why":"Defines the Huber loss, the outlier-resistant regression baseline compared on the elasticity problem.","marker":"[30]"},{"why":"Provides the analytic Burgers solution used to measure the $L^2$ error reduction.","marker":"[36]"}],"fun_headline_variants":["Variance term in loss flattens error spikes in PINNs","PINN loss that penalizes error variance cuts worst errors 30x","Add error spread to PINN loss for more uniform accuracy","Variance-based loss regularization reduces peak errors in PINNs","Penalize error variance in PINNs to shrink max error"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reported gains assume that the per-problem hyperparameter $\\alpha$, selected by sweeping its value against a known reference solution, is representative enough that a user can pick it without such a reference; all headline numbers are single runs at the chosen $\\alpha$, and no choice rule is given.","fun_headline_variants_meta":{"raw":{"variants":["Variance term in loss flattens error spikes in PINNs","PINN loss that penalizes error variance cuts worst errors 30x","Add error spread to PINN loss for more uniform accuracy","Variance-based loss regularization reduces peak errors in PINNs","Penalize error variance in PINNs to shrink max error"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000583,"raw_usage":{"total_tokens":2757,"prompt_tokens":972,"completion_tokens":1785,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":1698}},"tokens_in":588,"tokens_out":1785,"duration_ms":12444,"temperature":1.0,"reasoning_tokens":1698,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:34:44.775079+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the four benchmarks, or a single smooth Poisson case, with the proposed loss and the mean-squared loss across twenty random seeds each, fixing $\\alpha=0.8$, and compare maximum pointwise error on the same collocation set; if the variance-based loss does not reach a lower or equal maximum error in a clear majority of seeds, the central claim fails.","supporting_citations":[{"cited_title":"Haghighat, M","cited_arxiv_id":null,"evidence_quote":"Supplies the 2D linear elasticity problem and the five-network architecture used in Section 3.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proposes variance-predicted PINNs, the alternative variance-based approach the paper distinguishes from its own."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Huber loss, the outlier-resistant regression baseline compared on the elasticity problem."},{"cited_title":"Basdevant, M","cited_arxiv_id":null,"evidence_quote":"Provides the analytic Burgers solution used to measure the $L^2$ error reduction."}],"review_version":1}