Seven PINN papers by Karniadakis and co-authors as discourse graphs

The edge rule, translated for computational work

The key above is written for laboratory and clinical papers, where a negative counts only with a passing positive control and a stated sensitivity. Methods papers in scientific machine learning have neither, so the same rule is applied with its nearest equivalents. The comparator is a baseline that was tuned or reproduced, not a number copied from another paper. The sensitivity statement is variance over seeds or runs, a confidence interval, a significance test, or a convergence study that would have shown a difference had there been one. The missing positive control is a baseline that was never reproduced. A worse or failed result is drawn as opposing the claim only when those hold; otherwise it is drawn as merely consistent with it. A baseline that fails outright, by running out of memory or diverging, is drawn as informing the claim: it says what the baseline cannot do, not that the claim holds. The brief the extraction followed is kept with the JSON for each graph.

How the seven were chosen

All of his articles on PINNs that are open access under CC BY on the version of record, found through OpenAlex (author record, gold or hybrid open access, licence on the publisher’s own copy) and checked one by one against Crossref’s licence field or the publisher’s page. Seven qualify; the well-known 2019 PINN paper and most of the 2019 to 2021 papers do not, being bronze or closed. Ordered by forward citations (works citing the paper, OpenAlex, 18 October 2026). Four full texts are the published versions; three are the arXiv versions, because Elsevier does not serve the article to a machine, and the extraction notes under those graphs say so. None of these papers is published by ACM: his ACM record is nine supercomputing and graphics items from 1996 to 2015 and one review in ACM Computing Surveys, none with an experiment of its own.

The seven

PINN identification of internal voids and inclusions in solids (Science Advances, 2022)

Science Advances, 2022

Read left to right. The question (QUE) is addressed by 4 experiments (EXP), each producing results (RES), and each result bears on the hypothesis (HYP). 4 support it, 0 oppose it, 8 inform it. Issues (ISS) hang below: what the paper leaves open.

HYP, in the authors’ words (Discussion): “We have shown that the present framework is able to accurately estimate the unknown geometric and material parameters with a relative error O(10−2) when proper displacement data are supplied to ensure identifiability.”

• supports positive finding
• opposes a worse or failed result against a tuned baseline, with variance, an interval or a convergence study to show it is real
• informs bears on the question without deciding the hypothesis
• thin grey a relation between two experiments

• ISS Static boundary measurements carry limited information about internal features (Saint-Venant decay); dynamic loading such as ultrasound is proposed to supply time-dependent data. (next-step · Discussion)
• ISS R and μi are poorly identifiable together from boundary-only data; the fix shown (five internal points) presumes access to interior measurements that the setup originally excluded. (open-question · Results, Interpreting the convergence histories)
• ISS Training to a plateau takes ~11 hours CPU for 1M Adam iterations; Adam+L-BFGS, parallel PINNs and convergence-acceleration techniques are named as routes to cut this. (next-step · Discussion)
• ISS Continuum mechanics limits the method to defects far larger than the microstructure; parametric studies also bound reliable void sizes to no smaller than O(10^-1) of the matrix. (open-question · Discussion; Results summary of sections S7 to S10)

Extraction notes: Version: the text is the PubMed Central full text of the published article (Sci. Adv. 8, eabk0644, 16 Feb 2022, received 26 Jun 2021, accepted 22 Dec 2021, CC BY 4.0), with XML tags stripped; it includes main text, Table 1, figure captions, Methods and references but NOT the Supplementary Materials (sections S1-S11, figs. S1-S9, table S1). Inferences on my part: (1) the mapping of material models to cases (case 0 = linear elasticity, case 1 = hyperelasticity, case 3 = deformation plasticity) is read from 'cases 0, 1, and 3 as the baseline cases' plus Fig. 3A-C ordering and 'For case 3 where plasticity is involved'; the text never states case 0's model directly, nor explicitly the models of cases 2, 4 and 5 (Methods says most examples are incompressible Neo-Hookean, and case 5 uses μi as shear modulus, so hyperelastic is implied). (2) Loading type (uniaxial/biaxial) per case is only in Fig. 2, which is not in the text. (3) The 'reference value' for case 2 is given as −0.0392, 0.3474, 0.0392, −0.2474, which does not look like round design values; I copied it as printed. (4) All Table 1 errors are from single training runs; the paper reports no seeds, variance or confidence intervals, so none of the findings meet the brief's fairness condition for 'opposes', and I used 'informs' for the sub-par results (B in cases 0/1, μi and R in case 5). (5) There is no competing inverse method run as a baseline; the only comparison is against Abaqus FEM ground truth that also generated the data, and claims that the method avoids remeshing and problem-specific update algorithms are qualitative. (6) The 'around 30 min' Adam+L-BFGS figure and the robustness bounds (void size, moduli ratio, measurement placement) are summaries of Supplementary sections S7-S11 and table S1; the underlying numbers are not in the text. (7) The paper itself does not state which parameters in case 0 converged worse; the explicit statement about worse convergence names (A, B, Γ) in case 1 and (R, μi) in case 5 only. (8) The 'quote' for f-case4-two-voids is the setup sentence, since Table 1 gives numbers without a prose sentence for case 4. (9) Noted self-inconsistency: the Fig. 8 caption says 'Perturbation states on line 1 satisfy ΔR*Δμi* > 0. Perturbation states on line 1 satisfy ΔR*Δμi* < 0' (second 'line 1' should read 'line 2'); the main text makes the intended reading clear. Also the Results text refers to 'top left panels in Fig. 2, A to C' where the architectures are in Fig. 3. (10) Hyphen/minus/superscript characters in copied numbers (e.g. 'O(10 −2 )') follow the stripped text; I normalised them to 10^-2 in labels but kept the raw form inside quotes.

SDGD scales PINNs to 100,000-dimensional PDEs (Neural Networks, 2024)

Neural Networks, 2024

Read left to right. The question (QUE) is addressed by 5 experiments (EXP), each producing results (RES), and each result bears on the hypothesis (HYP). 4 support it, 0 oppose it, 7 inform it. Issues (ISS) hang below: what the paper leaves open.

HYP, in the authors’ words (Abstract): “Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in less than one hour for 1,000 dimensions and in 12 hours for 100,000 dimensions on a single GPU using SDGD with PINNs.”

• supports positive finding
• opposes a worse or failed result against a tuned baseline, with variance, an interval or a convergence study to show it is real
• informs bears on the question without deciding the hypothesis
• thin grey a relation between two experiments

• ISS SDGD needs a relatively large dimension batch size |I| in extremely high dimensions; too small a batch raises gradient variance and slows convergence (|I| >= 100 suggested). (open-question · Section 6 Conclusions; Section 4.4)
• ISS Outside the boundary+residual form (Schrodinger eigenvalue with an integral denominator) sampling integration points biases the gradient; an unbiased estimator is still open. (open-question · Section 5.4.3; Section 6 Conclusions)
• ISS Algorithm 3 (one sample for both passes) is biased yet is the variant behind the 100,000-D results; the convergence theory covers only the unbiased Algorithms 1 and 2. (open-question · Section 3.5; Section 4)
• ISS Applying SDGD to physics-informed neural operators (DeepONet, FNO) and to general high-dimensional regression or classification is proposed but not tested. (untried · Section 6 Conclusions)

Extraction notes: Version: the text is the arXiv preprint 2307.12306v6 dated 17 May 2024 (header on page 1), not the typeset Neural Networks article; the DOI 10.1016/j.neunet.2024.106369 was supplied with the task and does not appear in the text. Appendix A holds only proofs; the introduction promises 'more examples' in the appendices but none appear in this text, and the References start at the end of A.4. All experiments state 5 independent seeds, yet Tables 1-5 report single numbers with no standard deviation, confidence interval or range, so no finding carries variance. Self-inconsistency: Section 5.3 prose says Wang et al. [60] needed '1 day' for 1e-2 error at 250 D, while Table 5 lists 38 hours; the 3.5 hours in the prose matches the 210 minutes in the table. Two references are each called RS-PINN: Section 4.5.1 names Hu et al. [28] as RS-PINN but says He et al. [23] in Table 5 'is the RS-PINN'; the only numerical RS-PINN comparison is therefore the He et al. column of Table 5, and the rest of Section 4.5.1 (bias for nonlinear PDEs, over-smoothing, no derivatives above second order) is argument, not experiment. The HTE comparison in Section 4.5.2 concedes that under a fair JAX comparison 'they perform similarly', citing Table 1 of the HTE paper, which is not reproduced here. My readings, not the authors' words: 'N.A.' for vanilla PINN at 10,000 and 100,000 D in Table 1 is out-of-memory per the caption; 'N.A.' for (|I|,|B|)=(1,10000) in Table 2 and for Algorithm 1 at 10,000 and 100,000 D in Table 3 I read as runs not completed, since the listed times (1518.22 h, >24 h, >120 h) are given without an error; the paper does not say so explicitly. For Table 4 the paper says 'we adopt other methods' settings' but does not state whether the FBSNN, DeepBSDE and DeepSplitting numbers were rerun by the authors or copied from the original papers, so the supports valence there rests on an unverified baseline provenance; several Sine-Gordon margins are within the same order of magnitude. The HJB errors of about 5E-2 at 10,000 and 100,000 D are reported by the authors; the observation that they are far above the 1,000-D errors is my juxtaposition of their numbers. The Schrodinger result is given only as convergence curves (Figure 2) and the phrase 'less than 1e-6'; no table. The theory (Theorems 4.1-4.3: unbiasedness, variance reduction, convergence under bounded variance) is not encoded as an experiment. The question-level outcomeClass 'positive' is my summary of the paper's own verdict.

Bayesian PINNs on real-world COVID-19 case data with a damped harmonic oscillator (CMAME, 2022)

Computer Methods in Applied Mechanics and Engineering, 2022

Read left to right. The question (QUE) is addressed by 5 experiments (EXP), each producing results (RES), and each result bears on the hypothesis (HYP). 3 support it, 1 oppose it, 8 inform it. Issues (ISS) hang below: what the paper leaves open.

HYP, in the authors’ words (Abstract): “Our Physics-Informed Neural Networks can seamlessly integrate data and physics, robustly solve forward and inverse problems, and perform well for both interpolation and extrapolation, even for a small amount of noisy and incomplete data.”

• supports positive finding
• opposes a worse or failed result against a tuned baseline, with variance, an interval or a convergence study to show it is real
• informs bears on the question without deciding the hypothesis
• thin grey a relation between two experiments

• ISS PINN results are biased towards data or physics by the weighting coefficient epsilon; choosing the epsilon that minimises the sum of all four losses is left to user judgement. (open-question · Section 3.4, Section 5)
• ISS The effect of pre-conditioning and scaling techniques on SA-PINN was not explicitly shown; the method was observed to fail to converge when scaled inappropriately. (untried · Section 5 (Self Adaptive PINN paragraph))
• ISS SA-PINN training involves competing multi-term losses in a highly non-convex problem; training may not be robust or stable and does not always reach a global minimum. (open-question · Section 5 (Self Adaptive PINN paragraph))
• ISS All six methods were shown on one simple model problem only (seasonal endemic disease as a damped oscillator); generalisation to other nonlinear systems is anticipated, not shown. (next-step · Abstract, Section 5 (final paragraph))

Extraction notes: VERSION: the text is the arXiv preprint 2205.08304v2 (cs.LG, 24 May 2022), not the CMAME version of record; the DOI 10.1016/j.cma.2022.115346 was supplied by the caller and does not appear in the text. SCOPE: there is no separate synthetic damped-harmonic-oscillator benchmark; the oscillator is the physics model applied to the 2021 worldwide COVID-19 case data throughout, so the network-size grid search (Figure 5), the epsilon study (Figures 6-7) and all other experiments run on the COVID-19 data. SELF-INCONSISTENCIES IN THE TEXT: (a) the training window is 225 days everywhere except Figure 7 and its caption, which use 255 descriptive days and sum the predictive loss from day 256 to 365 while dividing by 140 (256-365 is 110 days); (b) Section 5 says the BNN has 1053 x 2 = 2306 unknowns while Table 1 and the NN paragraph give 1153 parameters (1153 x 2 = 2306, so 1053 looks like a typo); (c) Section 4.2 gives zero-mean normal priors for BNN weights, but the Discussion says log-normal; (d) the B-PINN's predictive verdict differs between the Figure 14 caption ('fail to predict the behavior outside the training window'), the Discussion ('reasonable predictive potential' and 'perform best'), Table 1 ('moderate predictive potential', 'requires large training data set'), the Figure 13 caption ('performs reasonably well for small data sets but only improves marginally') and the end of Section 4.4 ('more expensive ... but also less robust'); (e) equation (19) lists m among the learned physics parameters although m is fixed to 1 elsewhere; (f) Section 4.1 cites equation (33) before it is defined and Section 4.3 cites Bayes theorem (29) where (32) is meant; (g) Figure 6 varies epsilon from 0 to 1 while Figure 7 varies it from 1e-6 to 1e-2. GARBLED TEXT: the epsilon list in Section 3.4 reads '10-inf, 10-5, 10-4, 10-3, 10-2, 10-0', read here as 0, 1e-5, 1e-4, 1e-3, 1e-2, 1. EVIDENCE QUALITY (my reading): the NN-family comparisons (Figures 3, 5, 6, 13, 14, 15) are single-run qualitative figure readings with no seeds, no variance and no numeric error values in the text; the loss values in Figures 7 and 13 are plotted but never tabulated, so no numbers could be copied for them. The only variance reported is the posterior SDs of BI and B-PINN (Figures 11-12), which is why the B-PINN-vs-BI finding is the only one given valence 'opposes'; that SD is parameter uncertainty from one HMC configuration, not variance over repeated runs, so a reviewer may prefer 'consistent-with'. The percentage differences (18%, 21%, 6%) and SD ratios (3, 23, 10) are the authors' own statements, not my computations. CLASSIFICATION CHOICES THAT ARE MINE: 'negative-not-achievable' for the BNN failing to extrapolate and for the epsilon = 1 trivial solution (the brief's examples are crashes, these are runs that produce useless outputs); 'negative-no-effect' for the one-layer and 16/32-node grid-search results; question-level and B-PINN outcomeClass 'partial'. The hypothesis quote is 31 words. The 'supports' valence on the SA-PINN epsilon finding rests on the authors' statement that the learned 0.007 +/- 0.003 is of the same order as the hand-estimated 1e-3. Issues iss-2 and iss-3 both come from the SA-PINN paragraph of the Discussion; iss-2 is the authors' explicit 'we have not explicitly shown' statement.

APINN soft domain decomposition versus PINN and XPINN on five PDEs (Engineering Applications of Artificial Intelligence, 2023)

Engineering Applications of Artificial Intelligence, 2023

Read left to right. The question (QUE) is addressed by 5 experiments (EXP), each producing results (RES), and each result bears on the hypothesis (HYP). 2 support it, 5 oppose it, 4 inform it. Issues (ISS) hang below: what the paper leaves open.

HYP, in the authors’ words (Abstract): “Specifically, we present examples where XPINN performs similarly to or worse than PINN, so that APINN can significantly improve both. We also show cases where XPINN is already better than PINN, so APINN can still slightly improve XPINN.”

• supports positive finding
• opposes a worse or failed result against a tuned baseline, with variance, an interval or a convergence study to show it is real
• informs bears on the question without deciding the hypothesis
• thin grey a relation between two experiments

• ISS Initialisation decides APINN performance: a trained gate keeps its initial type (MPINN-type cannot become XPINN-type), so how to design the optimal decomposition stays open. (open-question · Section 6.5.4 and Abstract)
• ISS APINN shares more data points and parameters than XPINN and can therefore be more expensive in parallel settings; no wall-clock or memory cost is measured. (open-question · Section 7)
• ISS Four sub-nets do worse than two for both XPINN and APINN on Boussinesq-Burger, attributed to the complexity-versus-samples tradeoff; how many subdomains to use is unresolved. (open-question · Section 6.5.3)
• ISS Errors of APINN, XPINNv2 and PINN concentrate near the boundary on Helmholtz, attributed to gradient pathology, which the method does not address. (untried · Section 6.2.3)

Extraction notes: Version: the text is the arXiv preprint 2211.08939v3 (header 'arXiv:2211.08939v3 [cs.LG] 30 Sep 2023', title 'Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology'), not the Engineering Applications of Artificial Intelligence typeset version; the DOI was supplied by the caller and does not appear in the text. Self-inconsistencies in the paper: (a) Section 6.1.3 says 'APINN-X-F with a fixed gate function performs slightly worse than PINN and APINN', but Table 1 gives APINN-X-F 1.293E-3 versus PINN 1.620E-3, i.e. better than PINN; I used the table. (b) Section 6.4 claims 'APINN still performs slightly better than XPINN', but in Table 4 only APINN-M (1.299E-3) has a lower mean than XPINNv2 (1.378E-3); APINN-X (1.492E-3) is worse. This is recorded in the wave finding. (c) Section 6.1.4 says the two APINN-M seeds in Figure 7 'differ a lot' in performance, but the caption gives 6.936E-4 and 8.284E-4, a smaller gap than between the APINN-X seeds' trained gates would suggest; recorded as the authors' statement. (d) Section 6.2.4 says 'XPINN outperforms PINN' on Helmholtz, which holds for XPINNv2 only. (e) Section 6.1.4 says 'APINN-M is worse than its XPINN counterpart in the Burgers problem'; I read 'XPINN counterpart' as APINN-X (the XPINN-initialised APINN), since APINN-M beats both actual XPINNs by two orders of magnitude. Garbled text: Section 6.4 reads 'its relative L2 error is 27However'; the 27% figure was recovered from Section 6.4.3 ('reduces by 27% the relative L2 of PINN'). Missing in the text: the Helmholtz constants a1, a2, k are never given; XPINNv1 is not reported for the wave equation; the quantity after ± is never defined (I assume a standard deviation over the 10 runs, inferred); no significance tests, timings or memory numbers are reported. Model selection differs across problems and is my reading of a concern, not the authors': Burgers and Helmholtz report the best (lowest) test L2 over the whole optimisation, wave and Boussinesq-Burger report the error at the epoch with the lower training loss among the last 10% of epochs, Klein-Gordon does not say. The Klein-Gordon finding that APINN-M-F is about 10x worse than PINN (3.197E-2 vs 3.565E-3) is my reading of Table 3; the authors do not comment on it. Judgements that mean differences on Helmholtz, Klein-Gordon and wave are 'within the reported spread' are mine from comparing means to the ± values; the authors call them improvements. Valence choice: the three findings where XPINN does worse than PINN are classed negative-not-replicated with valence 'opposes' following the brief's rule for an earlier method doing worse than a simpler one; the edge reads against XPINN's claimed advantage (Jagtap and Karniadakis 2020), and these results are in fact consistent with this paper's own framing. The fairness condition holds in all cases: PINN and XPINN baselines are trained by the authors with stated settings, and every number comes with 10-run variance; note however that in the wave experiment XPINN was trained at learning rate 1e-4 while PINN used 8E-4, and in Boussinesq-Burger APINN4 sub-nets use width 18 versus 20 elsewhere. The MPINN method that APINN-M mimics is cited only as an anonymous under-review ICLR 2022 submission. Experiment-level outcomeClass values and the question-level 'partial' are my summary of the tables, not the authors' labels.

When XPINNs beat, match or lose to PINNs: a generalization tradeoff (SIAM J. Sci. Comput., 2022)

SIAM Journal on Scientific Computing, 2022

Read left to right. The question (QUE) is addressed by 5 experiments (EXP), each producing results (RES), and each result bears on the hypothesis (HYP). 1 support it, 2 oppose it, 8 inform it. Issues (ISS) hang below: what the paper leaves open.

HYP, in the authors’ words (Section 1 (Introduction)): “XPINNs facilitate parallel computing, accelerate convergence, and improve generalization empirically.”

• supports positive finding
• opposes a worse or failed result against a tuned baseline, with variance, an interval or a convergence study to show it is real
• informs bears on the question without deciding the hypothesis
• thin grey a relation between two experiments

• ISS Adaptive domain decomposition: compute the bound during training and restructure subdomains (give data-poor subnets more points) when overfitting dominates; proposed, not tested. (next-step · Section 6 (Conclusion))
• ISS XPINN loses on Poisson because boundary and interface losses trade off under fixed weights; the authors call for XPINNs with adaptive weight adjustment, which is not tried here. (untried · Section 5.4.2)
• ISS The theory assumes a linear second-order operator with K-Lipschitz bounded coefficients and a differentiable activation (no ReLU); extension beyond this class is not addressed. (open-question · Section 3, Assumption 3.1; Section 2.3)
• ISS Long-time integration, where only XPINNs are reported to remain accurate, is offered as a partial explanation of the theory but is not tested in the paper. (untried · Section 6 (Conclusion))

Extraction notes: Version: the text is the published SIAM J. Sci. Comput. Vol. 44 No. 5, pp. A3158-A3182 (published electronically September 27, 2022), CC BY, with a download stamp of 10/18/26. Framing choice (mine, at the coordinator's direction): the hypothesis is the prior claim the five experiments decide, namely that XPINNs generalize better than PINNs (Section 1: 'XPINNs facilitate parallel computing, accelerate convergence, and improve generalization empirically'; the abstract poses it as 'understanding how and when XPINNs outperform PINNs'). The paper's own framing is different: it proposes a tradeoff theory (domain decomposition yields simpler per-subdomain targets but less data per subdomain) with prior and posterior generalization bounds (Theorems 3.1, 3.5, 3.6) and three analytical examples (sections 4.1-4.3: Barron-norm comparisons 27 vs 9/sqrt(2); 15.625 vs about 16.70; threshold q about 0.683), and under that framing every one of the five PDE results, including the losses on heat and Poisson, counts as confirmation because the posterior bound ordering matched the error ordering in every reported case. This graph is drawn against the prior claim instead, so the heat and Poisson results are recorded as negative-not-replicated / opposes, and the theory and complexity-percentage findings as informs. The fairness condition for 'opposes' holds on heat and Poisson: the PINN baseline was trained by the authors under the same procedure (same epochs, learning rate, model structure, weight decay), 5 runs with seeds 0-4, mean±std printed, and the printed std intervals do not overlap. Note also that the bound is a posterior quantity computed from the same trained networks whose test error is compared, so the bound-vs-error agreement is a consistency check, not a prediction made before training. The brief caps experiments at five, so the theoretical results and analytical examples are not listed as experiments. Self-inconsistencies in the paper: (1) Section 5.3.2 prose says 'XPINN (2.052e-1) performs better than PINN (1.617e-1)', but Table 3 assigns 2.052e-1 to PINN and 1.617e-1 to XPINN; I used the table and the bound ordering (XPINN 66.59%). (2) The Conclusion lists KdV, heat, advection and Euler as cases where XPINN outperforms PINN, and heat and 'wave equation' as cases where PINN wins, and advection as the balanced case; this contradicts section 5 (KdV similar, heat PINN wins, advection XPINN wins) and there is no wave equation experiment in the paper. The experiment entries follow section 5. (3) Section 4.1 writes the Omega_2 Barron norm as '||u*||_W2(Omega_1) = 1' and section 4.2 says 'subnets on Omega_1 and Omega_3'; both look like typos. Variance caveats (mine): on advection the two means lie within one printed standard deviation of each other (PINN 2.052e-1±1.001e-1 vs XPINN 1.617e-1±3.582e-2), on Euler XPINN-TB vs PINN (3.5722e-2±4.290e-3 vs 3.4604e-2±7.385e-3) likewise overlap, and on KdV the difference (6.899e-1±8.015e-3 vs 6.955e-1±9.905e-3) is within one std; the authors do not report a significance test, so these three are recorded as partial / informs rather than as wins or losses, even though the authors describe advection as an XPINN win and Euler-TB as a failure. On KdV both models have relative L2 error near 0.69, i.e. neither solves the problem well at 5000 epochs; the authors do not comment on this. On advection XPINN's training loss (4.239e-3) is about two orders larger than PINN's (1.387e-5) yet its test error is lower; the authors do not discuss this. Missing details: the Euler experiment does not state the number of runs or seeds (the other four use seeds 0-4), the number of boundary points, or the epoch count; Table 6 labels the complexity column 'Norms' where Tables 1-3 and 5 say 'Complexity'. Boundary-loss versions of the comparisons and some proofs are in a supplement (supp.pdf) that was not provided. The bounds in the tables are reported only as percentages of the PINN bound; absolute values are not given. Issue iss-3 is the paper's stated assumption (it says the restriction is 'required for our estimation'); the observation that KdV (third order, nonlinear) and Euler (nonlinear first-order system) fall outside that assumption is my reading, not the authors' statement.

AI-Aristotle: X-TFC and PINNs plus symbolic regression for gray-box identification in systems biology (PLoS Computational Biology, 2024)

PLoS Computational Biology, 2024

Read left to right. The question (QUE) is addressed by 5 experiments (EXP), each producing results (RES), and each result bears on the hypothesis (HYP). 5 support it, 0 oppose it, 7 inform it. Issues (ISS) hang below: what the paper leaves open.

HYP, in the authors’ words (Summary and discussion): “The results demonstrated the capability of both X-TFC and PINNs to accurately estimate parameters even with limited data, showcasing their potential for model calibration in real-world scenarios.”

• supports positive finding
• opposes a worse or failed result against a tuned baseline, with variance, an interval or a convergence study to show it is real
• informs bears on the question without deciding the hypothesis
• thin grey a relation between two experiments

• ISS PINN accuracy plateaus with more data, attributed to optimization error; deeper networks, other optimizers or alternative techniques are proposed but not tested. (untried · Section 4.1)
• ISS X-TFC is sensitive to hyperparameters (points per subdomain, neurons, step length); an ablation study is required to reach the desired accuracy. (next-step · Summary and discussion; Section 4.1)
• ISS In X-TFC on the ultradian model, iterations beyond the first subdomain might produce higher errors; more careful hyperparameters and per-subdomain initial guesses are needed. (open-question · Section 4.2)
• ISS Extending the framework to stiff systems via operator splitting (Strang or Godunov) in PINNs is suggested but not demonstrated here. (untried · Summary and discussion)

Extraction notes: Version: the text is the PubMed Central published version (PMC10931529, PLoS Comput Biol 20(3): e1011916, received 4 Oct 2023, accepted 13 Feb 2024, published 12 Mar 2024, CC BY 4.0), with XML tags stripped. Supplementary S1 Text (Tables A-E, ablation studies for X-TFC and PINNs) is referenced but not in the file, so the claim that X-TFC errors can be 'reduced by several orders of magnitude' with ablation is unverified here. Self-inconsistencies in the paper: (1) Section 2.1 states a time range of [0, 10] hours for the pharmacokinetics model, but all results tables and Section 4.1 use [0, 50] hours. (2) Sections 4.1 and 4.2 and Table 7's caption say accuracy is the 'absolute difference' between nominal and inferred values, while the column headers in Tables 2 and 7 say 'relative error (%)'; I used the table header wording. (3) The noise study is introduced as 'four different levels of noise' but lists six (1, 2, 3, 4, 5, 10%), and the text then refers to 'all five values of noise std'. (4) The closing paragraph of Section 4.2 and the Summary say PINNs achieve greater accuracy on noisy data, but Table 6 shows X-TFC with lower relative error at every noise level except 1%; my finding pk-h-noise-10pct records the table, and this is my reading of the conflict. (5) Section 4.1 says both methods give 'great precision even for a meager dataset (10 data points)' while Table 2 shows X-TFC at 41.66% relative error on k_g; I classed that finding as partial. (6) gplearn expression lengths for the pharmacokinetics candidates are given as 7 and 19 in the text but 9 and 19 in the Fig 9 caption and Fig 10; for the ultradian g_sym they are 13 and 18 in the text but 13 and 25 in the Fig 12 caption. (7) Eq 22 writes dI_p/dt = R_m f + f_sym, which does not match Eq 5 (dI_p/dt = f_1(G) + f(t)); I did not resolve this. (8) The organisation paragraph says conclusions are in Section 4.3.2, but they are in 'Summary and discussion'. Fairness of comparisons: only the PINN pharmacokinetics parameter-discovery runs are averaged over 10 seeds; every other number appears to be a single run with no variance, so no finding is marked 'opposes'. PINNs ran on an NVIDIA RTX 3090 GPU and X-TFC on an Intel Xeon W-2255 CPU, so the timing comparison crosses hardware. Synthetic data were generated with Runge-Kutta for PINNs and with RPNNs for X-TFC, so the two methods were not fitted to byte-identical data. X-TFC noisy-data results were obtained without domain decomposition 'to avoid overfitting', unlike the clean-data runs. The 'true expressions' in Table 13 are the paper's own statement; I did not derive them from Table 1. The ratios in my outcomeSummary fields ('about 700x', 'about 1/3000', 'about 8000x') are my rough readings of the tabulated times, not the paper's statements; all finding labels and effects use the paper's numbers verbatim. The que outcomeClass 'positive' and the finding-level classes are my assignments. The PINN local-minima finding (Section 3.2.2) has no numbers; I kept it because the brief asks for stated failures. The claim that PINNs discovered up to 21 parameters from G and I_p data, and 17 from 360 G points, 'not possible for the X-TFC algorithm', is attributed to ref [30] and is not an experiment in this paper, so I did not make it a finding.

Scalable PINNs and physics-informed graph networks (Data-Centric Engineering, 2022)

Data-Centric Engineering, 2022

Read left to right. The question (QUE) is addressed by 5 experiments (EXP), each producing results (RES), and each result bears on the hypothesis (HYP). 3 support it, 0 oppose it, 9 inform it. Issues (ISS) hang below: what the paper leaves open.

HYP, in the authors’ words (Abstract): “We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.”

• supports positive finding
• opposes a worse or failed result against a tuned baseline, with variance, an interval or a convergence study to show it is real
• informs bears on the question without deciding the hypothesis
• thin grey a relation between two experiments

• ISS Manual interface choice caused load imbalance (subdomain 7 has 800 residual points); point-cloud or K-way partitioning is proposed to pack subdomains for communication. (next-step · Section 2.4)
• ISS Efficiency of generic data-parallel and model-parallel training for scientific machine learning, dominated by high-dimensional sparse data, has not been explored. (open-question · Section 2.3)
• ISS Multi-GPU PINN implementations need more algorithmic and implementation work, including mixed-precision calculations that could significantly speed up PINNs. (untried · Section 4, Summary and Outlook)
• ISS Several parallel frameworks exist for scaling plain GNNs, but scaling physics-informed GNNs (PIGNs) has not been attempted and is named as the new direction. (untried · Section 4, Summary and Outlook)

Extraction notes: Version: published Cambridge Core HTML of Data-Centric Engineering vol. 3 (2022) e24, published online 29 June 2022, labelled 'Survey paper', CC BY 4.0; text begins with site navigation and ends with references. Because it is a survey, the hypothesis is the stated purpose from the abstract; the closest claim-like statements are 'domain decomposition approaches paired with physics-based synchronization help more than the vanilla data-parallel approaches' (Section 2.3) and 'a hybrid multi-GPU/multi-CPU paradigm based on domain decomposition seems to emerge as the most efficient solution' (Section 4). Valences of 'supports' were assigned against that scaling claim, which is my reading; 'informs' was used for the loss-function and graph-network examples that do not bear on scaling. None of the experiments appears to be a new run for this paper: Figure 5 is from Yu et al. 2021, Figures 6, 10 and 11 are adapted from Shukla et al. 2021b, Figure 12 from Trask et al. 2019, Figure 13 from Trask et al. 2022; the DistDGLv2 and Pfaff et al. (10-100x speedups) numbers are third-party results. No experiment reports seeds, repeats, variance or confidence intervals, so no finding was given valence 'opposes'; the vanilla-PINN boundary-layer failure is classed negative-not-achievable on the baseline per the brief. Table 1 (network architecture and data points per subdomain) is present only as a caption in the text; its contents are missing, so the per-subdomain point counts behind the '800 residual points' remark cannot be checked. Figure 8 weak-scaling values are not in the text. Numeric speedups (9X/10X CPUs, 7X/9X GPUs, factor 2.5, 30%) and the gPINN 40 vs >100 points were each found in only one place in the text. Apparent typos in the paper: Equation (6) has 'L_b(w,lambda_b) -' where '=' is expected, Equation (8) writes w_{k-1} for the updated weights, Equation (22) labels the curl as 'grad', and Section 2.1 writes 'lambda_n' for the initial-condition weights; none affects the extracted findings.

Licence and what this page is

All seven papers are CC BY 4.0 on the version of record, so the graphs are published here with attribution; each links to the paper it was read from and quotes it only in short. The graphs are a view of what each paper claims and found, drawn by one extraction pass against a written brief and checked by hand on the headline numbers. They are not the authors’ own summary, and the extraction notes under each graph record what was inferred and what could not be resolved. Corrections from the authors are welcome.

Method: OpenAlex author A5009658255, title and abstract search for physics-informed or PINN, gold or hybrid open access, licence cc-by on the primary location, verified per paper via Crossref (content-version vor) or the publisher page read in a browser; full texts from PubMed Central XML, Cambridge Core HTML, the SIAM PDF, or arXiv; one extraction pass per paper following a written brief, each finding with a locator and a quote; the JSON for each graph is kept with the brief.