Classical, Machine Learning, and Hybrid Solvers for Partial Differential Equations: An Evidence-Weighted Review and Decision Framework

1.1 The practitioner’s dilemma

• The introduction identifies a credibility problem: 60 of 76 ML fluid-PDE papers claiming superiority compared with a weak numerical baseline, while reporting biases further inflated apparent advantage.A controlled study also found PINNs did not beat FEM on combined solution time and accuracy, although trained models could be cheaper at evaluation.Source: §1.1 · p. 2
• The review frames solver choice around PDE regime, geometry, reliability, query volume, and trusted-data availability rather than asking only whether ML can approximate solutions.Its motivating comparison includes one-time certified elliptic solves versus many-query parametric surrogates, and supervised operator learners versus residual-trained approximators that need no solution corpus.Source: §1.1 · p. 2

1.2 What existing reviews cover and what they leave open

• Existing reviews are organized mainly by architecture, loss, and application, and rarely compare classical solvers and ROMs under matched costs or grade individual claims.The review positions its contribution as cross-family evidence grading and replication-based auditing.Source: §1.2 · pp. 2–3

1.3 Scope, research questions, and contribution

• The survey covers forward deterministic and parametric PDE solving through a structured narrative review curated by domain expertise, not a protocol-driven systematic review.Inverse problems appear only where they clarify data scarcity, supervision, or solver choice; PDE discovery, pure control, and weather or climate modeling are outside scope.Source: §1.3 · p. 3
• The review’s contributions combine a cross-family decision framework, an amortization law tested on Darcy flow, an evidence ladder and claim rubric, and audits of benchmarks and three case studies.It asks six research questions spanning solver choice, PDE characteristics, training data and amortization, claim credibility, reporting standards, and research priorities.Source: §1.3 · pp. 3–5
• A surrogate trained on N_tr solutions costing at least one classical solve each must be queried more than N_tr times before it saves cost.In the measured Darcy study, break-even rises from about 3 × 10^4 queries at 64^2 to 9 × 10^5 at 512^2 with 103 training samples; with fewer samples at 512^2, FNO never breaks even.Source: §1.3 · pp. 5–6
• In all three case studies, speedup, generalization, efficiency, or structure receive the weakest grades or negative verdicts, while some in-distribution accuracy is supported for each method.The review also finds that reported accuracy may not survive reproduction: GON mean squared errors were 1.8 to 40 times larger than reported on every non-trivial case, while the re-implemented FNO Darcy study reached 0.6–0.7 times the reported error.Source: §1.3 · p. 6
• PDEBench, CFDBench, and APEBench results vary by task: FNO beats U-Net on every PDEBench task, but on only 43% of CFDBench and 20% of APEBench tasks.Across eighteen PINNacle tasks, eleven PINN variants have eight different winners, and the task-dependent part of log error is five to seven times their average difference.Source: §1.3 · p. 6

2 Decision regimes and taxonomy

• Five primary regimes are distinguished: one-off high-fidelity solves, many-query amortized inference, high-dimensional or mesh-free solves, complex geometry, and long-time rollout of nonlinear or chaotic dynamics.Data-informed inverse modeling is included as a bounded secondary regime, where data-scarce methods combine sparse observations with PDE constraints.Source: §2 · p. 6
• Regime suitability also depends on query volume, data availability, regularity, geometry, reliability requirements, and offline budget, so a solver family can be appropriate in one setting but not another.The decision landscape plots query volume against data availability and offline solver budget, with certification pressure as a side constraint.Source: §2 · p. 6, Figure 1 · p. 7
• Solution regularity separates failure pressures: conservative finite-volume methods build in shock handling, while smooth network bias, residual imbalance, and boundary treatment make sharp layers difficult for neural approximators.The taxonomy treats regularity as a cross-cutting axis alongside the decision regimes.Source: §2 · p. 6
• Classical solvers use the mathematical model directly, function learners use residual or variational signals, operator learners usually need paired solution data, and autoregressive emulators need trajectories.Whether paired data exist or can be generated by a trusted solver at counted cost helps determine feasibility before accuracy comparisons.Source: §2 · p. 7
• The taxonomy flags risky recommendations, including operator learning without a feasible corpus budget, certified use without error control or a mature classical baseline, and long-horizon surrogates tested only on short rollouts.It also cautions against treating residual-trained solvers as cost-free or hybrids as automatically robust.Source: §2 · p. 7

3.1 Classical solvers as regime-specific baselines

• Classical methods need no training data and remain defaults for one-off high-fidelity solves, unavailable paired data, certification needs, or regimes with mature solver stacks.Comparisons require a named, regime-appropriate baseline and relevant solver configuration.Source: §3.1 · pp. 8–9

3.2 ML only solvers

• The review compares ML-only solvers by training signal, data needs, optimization reliability, generalization, and amortized cost, alongside their evidence requirements and failure modes.The family includes function learners, operator learners, and autoregressive emulators.Source: §3.2 · p. 9, Table 6 · p. 11

3.2.1 Function learning solvers.

• Function learners need no paired corpus, but small training residuals do not guarantee solution accuracy, and credible studies need independent reference error and classical-cost comparisons.They are most credible when the PDE is known and dimension or geometry makes meshing difficult.Source: §3.2.1 · pp. 9–10

3.2.2 Operator learning and interpretability guided variants.

• Most supervised operator learners require paired solution data, so their claims should report corpus cost and query volume; linear reconstruction also inherits reduced-basis limits on transport-dominated or shock-forming families.Resolution transfer is strong evidence only when tested under input-distribution shift; interpolation within a covered corpus can look like broad generalization.Source: §3.2.1 · p. 11, §3.2.2 · pp. 11–12

3.2.3 Autoregressive emulators.

• Autoregressive emulators can accumulate one-step errors into long-time drift, especially in chaotic regimes, so credible studies report rollout horizons, stability failures, conserved quantities, and attractor statistics.Refinement-based rollout and energy-conserving Hamiltonian networks are described as partial remedies.Source: §3.2.3 · p. 12

3.3 Hybrid solvers

• Hybrid methods retain classical equations or iterations while learning components, but can inherit risks such as instability, shifted conservation, or training-distribution dependence.The review classifies hybrids by where learning enters the pipeline and cautions that combining physics with learning alone does not justify a recommendation.Source: §3.3 · p. 12
• HINTS combines a DeepONet step with classical relaxation, exploiting networks’ low-frequency fitting and relaxation’s high-frequency damping; its multigrid variant outperformed default HYPRE AMG in wall-clock time on large high-wavenumber Helmholtz problems.The reported problems had more than 106 unknowns.Source: §3.3 · p. 12
• Learned discretization and correction methods have the strongest hybrid evidence, while learned iterative solvers have the weakest; classical-dominant hybrids are the most defensible starting point when monitoring and fallback remain.Physics-informed operators have moderate evidence; the review distinguishes evidence maturity from design.Source: §3.3 · pp. 12–13

4.1 Benchmark suites and coverage

• Benchmark suites show task-dependent rankings and stronger in-distribution than shifted generalization, while none combines strong classical and ROM baselines with out-of-distribution stress tests.The suites cover distinct settings, including broad time-dependent PDEs, PINNs, flow shifts, autoregressive rollout, and measurement-paired tasks.Source: §4.1 · pp. 13–14

4.2 Recommended core PDE problems

• A cross-family benchmark should span varied PDE regimes, pair each with suitable classical and relevant reduced-basis baselines, and report regime-appropriate errors, uncertainty, and phase-separated costs.Examples include near-boundary error for boundary-sensitive problems, conservation and shock diagnostics for conservation laws, and rollout error and invariants for time-dependent systems.Source: §4.2 · pp. 14–15

4.3 Evidence weighted benchmarking protocol

• The benchmarking protocol fixes the PDE regime, reference source, data splits, and baselines before tuning, then grades accuracy, cost, distribution shift, variation, and reproducibility together.It distinguishes many-query settings, where reduced-basis baselines are included, from one-off certified solves.Source: §4.3 · p. 15

4.4 Worked cost comparison: Darcy flow parametric benchmark

• For Darcy flow, estimated FNO or DeepONet offline cost is 10^3–10^4 classical solves, and operator advantage over direct FEM appears only after roughly 10^3–10^4 queries in that estimate.The measured reduced-basis approximation needs 125–335 modes to match FNO accuracy, while full-system reassembly caps its per-query speedup at about 20×.Source: §4.4 · pp. 15–16

4.5 An amortization law for learned surrogates

• For solver-trained surrogates, break-even depends on data-generation and training costs as well as inference speed; training-set size imposes a query-volume floor when data are generated by a solver.Hidden tuning and reference-resolution costs can raise the threshold, while sunk or shared data reduce the cost borne by a deployment.Source: §4.5 · pp. 16–17

4.6 Measured break-even on parametric Darcy flow

• For errors below 1%, FNO break-even requires at least 3.1 × 10^4 queries with the coarsest-grid comparator, or 4.0 × 10^4 with interpolated costs.No model below 1% error breaks even sooner; finer resolutions can raise the threshold or make FNO queries slower than the matched classical solve.Source: §4.6 · p. 18, Table 8 · p. 18
• At 64² with 10^3 training solutions, a network ten times faster would cut break-even from 3.1 × 10^4 to 3.8 × 10^3 queries when effective training cost is unchanged.With fixed training cost, the predicted reduction is 7%; across resolutions, measured break-even rises to 1.5 × 10^5 at 256² and 9 × 10^5 at 512².Source: §4.6 · p. 18, Table 8 · p. 18

4.7 Operational claim strength rubric

• The rubric grades evidence quality separately from claim verdicts, assigns grades per claim, and requires at least Moderate quality for a positive recommendation.It is an expert-consensus instrument, so its grades are illustrative guidance rather than calibrated probabilities.Source: §4.7 · pp. 18–20

5.1 Full replication: Green’s function guided operator network

• The GON replication reproduced five original cases with three seeds and 95% confidence intervals, but could not rerun unreleased neural baselines.It added an independent finite-element reference, cost anchor, shift tests, data-efficiency sweep, and structural audit.Source: §5.1 · p. 21
• GON’s reproduced errors exceed original values by about 40× on 3D Poisson, 1.8× on Plate, 2.8× on Pipe, and 2.3× on Stokes.Reported neural baselines were not retrained under matched splits, seeds, and hardware, capping accuracy grades at Moderate.Source: §5.1 · p. 21
• One finite element solve takes 4.35 ms, while GON costs 6.86 h to train and 28.06 ms per end-to-end query; it never breaks even end to end.Counting only the 2.82 ms operator forward pass, GON breaks even after about 1.6 × 10^7 queries; the case has only 47 free degrees of freedom.Source: §5.1 · pp. 21–22
• GON’s 3D Poisson robustness is specific to a linear boundary-harmonic map with an inert source, and a fourfold boundary-amplitude increase raises error about 16 times.Errors also fall about 100 times on 2D Poisson and 1000 times on Plate from the smallest to full training corpus; Plate fails with two training instances.Source: §5.1 · p. 22
• A controlled Green’s-function test finds accurate predictions but an invalid learned kernel: symmetry error is about 0.18, and the kernel differs from analytic and discrete references by about 47% and 30%.The test uses a case where the interior integral supplies the whole prediction, so an accurate model must encode the Green’s function.Source: §5.1 · p. 23
• GON’s accuracy claims are supported at Moderate quality only for diffusion and Stokes, while its speedup and interpretability claims are not supported on Strong evidence.The review therefore treats GON as a case-dependent operator variant, not a default solver for these PDE families.Source: Table 11 · p. 22, §5.1 · p. 23

5.2 Evidence extraction audits of FNO and PINN

• The FNO audit reports about 440× speedup for Navier–Stokes inference, but excludes about 12 h of data generation and training; its MCMC application is about 1.5× faster end to end.Its baselines are mostly neural, entries are single-run, and test instances share the training input distribution; PINN reporting also lacks classical baselines and uses single-run estimates.Source: §5.2 · p. 23, Table 12 · p. 24
• Across the three case studies, accuracy or feasibility evidence is generally stronger than evidence for speedup, generalization, efficiency, and robustness.The decision guidance makes FNO on many-query Darcy conditional on corpus cost and a reduced-basis comparison, and PINN-style learning conditional on reference and boundary error, multiple seeds, and a classical baseline.Source: §5.2 · pp. 23–24

6 Decision matrix

• High-confidence recommendations require strong evidence across at least two PDE families or mature classical theory plus benchmark evidence; Red Flag and Not supported claims never become positive recommendations.FNO for many-query Darcy is Moderate confidence with incomplete shift evidence and ROM comparison, while learned iterative solvers for general linear systems are Low confidence.Source: §6 · p. 24
• Classical solvers remain the High-confidence default for one-off, high-fidelity solves requiring error control; learned alternatives should beat a tuned classical workflow.ROMs or operator learners become attractive when many queries amortize offline costs; measured Darcy FNO errors below 1% require at least 3×10^4 queries.Source: §6 · pp. 24–25
• Operator learners require feasible paired data, and ROMs should be considered first when intrusive access, low-dimensional parametric structure, and error estimation are available.Recommendations should state the query volume that justifies the offline stage and be recomputed for each regime.Source: §6 · p. 25
• PINN-style learning is a Moderate- or Low-confidence option for data-scarce, inverse, high-dimensional, or mesh-difficult settings, conditional on reference error, boundary error, multiple seeds, and a classical comparison.Autoregressive emulators are recommended only for repeated dynamic prediction with long-horizon rollout diagnostics.Source: §6 · p. 25
• Figure 2 orders selection by paired-data availability, one-off versus many-query use, and then certification, geometry, dimensionality, smoothness, shock, and long-horizon reliability filters.Hybrids are classified by learned component and retained structure; evidence is more credible when classical structure, residual monitoring, or fallback remains.Source: §6 · p. 25

7 Open problems and research priorities

• Operator-learning evidence supports interpolation within benchmark distributions more strongly than deployment shifts, so future benchmarks should distinguish interpolation, moderate extrapolation, and deployment shift with fixed splits.The decision matrix’s regime entries are traceable to supporting theory, studies, graded claims, measurements, and remaining evidence gaps.Source: §7 · p. 25, Table 13 · p. 26
• Open evaluation barriers include uncertainty calibration against classical stochastic baselines, geometry generalization, shock preservation, long-horizon reliability, hybrid convergence guarantees, and high-dimensional reference protocols.Relevant diagnostics include boundary error, conservation, shock location, rollout statistics, and dimension-scaling comparisons.Source: §7 · p. 27
• Multi-fidelity operators, foundation models, and generative solvers need the same data, cost, and shift accounting as established families.Reported promise remains qualified by narrow problem coverage, degradation on structurally distant PDEs, or thin evidence for fair cost accounting and out-of-distribution reliability.Source: §7 · p. 27
• Future studies should state decision regime, data regime, query volume, baseline class, cost model, and failure conditions before making superiority claims.This would make comparisons more useful for practitioners choosing among classical, ML-only, and hybrid solvers under real constraints.Source: §7 · p. 27
• The review proposes testing whether tighter tolerances favor operator learning exactly when training-error decay exponent α exceeds classical cost exponent p/d.The Darcy study reports p≈1.4; α is 0.7–0.9 over 100–2000 training samples and falls to 0.4–0.6 above 1000, motivating a matched-accuracy 3D test.Source: §7 · pp. 27–28
• The proposed next advance is cross-family evaluation infrastructure with documented references, strong classical and ROM baselines, fixed shifts, standardized cost accounting, reproducibility checks, physical diagnostics, and failure reporting.The review also calls for testing proxy-versus-decision-quantity rankings, structural validity, and the trade-off between hybrid speedup and retained structure.Source: §7 · p. 28

8 Conclusion

• Classical methods remain the strongest default for one-off high-fidelity, conservation-critical, and certified workflows, while learned and hybrid methods address specific data-scarce, inverse, high-dimensional, and many-query bottlenecks.The choice depends on PDE regime, available data, offline budget, query volume, geometry, regularity, time horizon, and certification needs.Source: §8 · p. 28
• The central practical output is a decision matrix and flowchart that begin with data availability, separate one-off from many-query use, and raise evidence thresholds for certification, conservation, or long-horizon reliability.Classical solvers use the model directly, operator learners need paired data whose generation cost must be counted, and hybrids are judged by where learning enters.Source: §8 · p. 28, Figure 2 · p. 29
• No FNO model below 1% error in the measured Darcy study broke even before about 3 × 10^4 queries, and finer resolution raised break-even at least thirtyfold.The review also finds that reduced-basis comparison depends on parametric structure, since matching FNO on the thresholded Darcy benchmark requires 125–335 modes.Source: §8 · p. 28
• Across the case studies, accuracy is supported at Moderate quality for at least some cases of each method, while speedup, generalization, efficiency, and structural claims receive the weakest grades or negative verdicts.The review identifies establishing rubric inter-rater reliability on a prespecified corpus as a next step.Source: §8 · pp. 28–29
• Credible superiority claims require fair classical and, where applicable, reduced-basis baselines, transparent reference generation, separated costs, multiple seeds, deployment-matched shift tests, and PDE-specific diagnostics.The review recommends that developers compute break-even for the claimed use case, while practitioners check whether their query volume exceeds it and treat unsupported claims as a reason to stay with classical workflows.Source: §8 · pp. 28–30
• The review prioritizes better cross-family evaluation infrastructure so solver families can be compared as decision alternatives rather than isolated research communities.Table 14 condenses the evidence ladder, amortization law, and rubric into ten questions for reviewing superiority claims.Source: §8 · p. 30, Table 14 · p. 30

• The listed authors are Yingchao Huang, Xin Wang, Shanshan Yao, Fanhua Zeng, and Wei Peng.Source: Supplementary Material for: Classical,… · p. 36
• The supplement contains full protocols and case-study records summarized in the main article.Source: Supplementary Material for: Classical,… · p. 36
• Section, table, and figure numbers with an S prefix refer to the supplement; unprefixed numbers refer to the main article.Source: Supplementary Material for: Classical,… · p. 36

S1 Reproducibility checklist

• The reproducibility checklist requires solver, reference, data-generation, cost, seed, and failure-case details before results inform evidence grading.It makes solution-corpus sources and costs explicit for supervised and residual-trained methods.Source: S1 Reproducibility checklist · pp. 36–37, Table S1 · p. 37

S2 Evidence extraction schema

• The evidence schema records regime, split, reference solver, baseline strength, costs, and out-of-distribution tests to support conflict-aware comparisons.It requires solution-corpus accounting for supervised learning and optimization and validation evidence for residual-trained methods.Source: S2 Evidence extraction schema · p. 37, Table S2 · p. 38

S3 Operational claim strength rubric

• Evidence quality and verdict are separate, and each claim receives its own grade; a positive recommendation requires a Supported verdict and at least Moderate quality.A well-measured negative finding can have Strong quality with a Not supported verdict.Source: S3 Operational claim strength rubric · p. 38, Table S3 · p. 39

S4 GON replication protocol

• The replication separates faithful reproduction from decision stress testing and covers five original cases, but cannot retune neural baselines because their implementations were unavailable.The study includes classical references, cost analysis, out-of-distribution tests, data-efficiency sweeps, and interpretability checks.Source: S4 GON replication protocol · p. 39
• On non-trivial cases, reproduced GON errors exceed original reports; it does not beat reported FNO on 3D Poisson but remains below reported baselines on Plate, Pipe, and Stokes.The baselines were not retrained under matched splits, seeds, and hardware, so the comparison mixes reproduced and reported values.Source: S4 GON replication protocol · p. 40
• GON is unaffected by input noise, mild extrapolation, sixfold interior-mesh refinement, and boundary-mean shifts, but fourfold boundary-amplitude growth raises error about 16 times.The observed robustness is attributed to a linear boundary-harmonic map with an inert source, not general out-of-distribution generalization.Source: S4 GON replication protocol · p. 42
• GON’s forward pass breaks even with finite elements only after about 16 million queries; end-to-end queries are slower than a finite-element solve in this 47-free-degree-of-freedom regime.Its learned kernel also fails structural tests despite accurate predictions, including about 0.18 symmetry error and about 47% and 30% differences from analytic and discrete Green’s functions.Source: S4 GON replication protocol · pp. 44–45
• GON’s speedup and Green’s function interpretability claims are not supported on Strong evidence, while its accuracy advantage over reported baselines is confined to diffusion and Stokes.The accuracy comparisons rely partly on reported rather than reproduced baselines.Source: S4 GON replication protocol · p. 47

• The extraction record identifies careful accuracy reporting and available code, but no matched-accuracy classical solver and a headline speedup that separates online inference from offline cost [27].Source: Table S12 · p. 48, S5 Evidence extraction audit: Fourier… · p. 49
• The audit finds Moderate evidence for in-distribution FNO accuracy on smooth parametric PDEs, but Weak-to-Red-Flag evidence for speedup and out-of-distribution generalization.Independent benchmark re-evaluations find regime-dependent accuracy and degraded zero-shot transfer outside training regimes.Source: S5 Evidence extraction audit: Fourier… · p. 49
• The resulting many-query Darcy recommendation is Moderate-confidence only if corpus-generation cost is counted and a reduced-basis alternative is compared or ruled out by a decay test.The single reduced-basis entry reported in the paper is insufficient.Source: S5 Evidence extraction audit: Fourier… · p. 49

• The audit grades inverse discovery and forward solution as distinct claim types and distinguishes the paper’s measured claims from stronger downstream citation practice [37].Source: S6 Evidence extraction audit:… · p. 50
• The foundational framework paper demonstrates feasibility on selected nonlinear PDEs but lacks classical baselines, cost accounting, and optimization-variability reporting for solver-replacement decisions.Source: S6 Evidence extraction audit:… · p. 50
• Raissi et al. provides Moderate evidence for residual-trained forward and inverse PDE-solving feasibility, but Weak evidence for efficiency and robustness claims.Source: S6 Evidence extraction audit:… · p. 51
• PINN-style function learning is most defensible in data-scarce and inverse regimes with residual or variational signals, while replacement recommendations require reference, boundary, seed, and classical-baseline reporting.Source: S6 Evidence extraction audit:… · p. 51

S7 Justification of the landscape table marks

• “Cross family” marks whether reviews compare classical, ML-only, and hybrid solvers within one framework, cover only two families or a slice, or focus on one family.“Evidence graded” marks whether claims are weighed against fair baselines and cost accounting fully, for a subset, or not at all.Source: S7 Justification of the landscape… · p. 51, Table S16 · p. 52

S8 Rank concordance in published benchmark suites

• Method rankings were transcribed from four benchmark suites, with non-converged cells ranked last and unsuitable or unreported tasks removed.The analysis uses Kendall’s W for ranking agreement and a Friedman test for consistent ranking differences.Source: S8 Rank concordance in published… · p. 52, Table S17 · p. 52
• No PINN variant is reliably best across tasks: rankings agree only weakly, and the task-by-method interaction is five to seven times the method effect.Only LAAF’s interval excludes 1; the best variant depends on the task.Source: Table S17 · p. 52, S8 Rank concordance in published… · p. 53

S9 Evidence trace for the decision matrix

• The trace identifies evidence types and missing evidence for each confidence level, making individual matrix cells checkable and disputable.Confidence can change as evidence changes.Source: S9 Evidence trace for the decision… · pp. 53–54, Table S18 · p. 53, Table S19 · p. 54
• Every High-confidence decision-matrix entry rests on theory plus independent empirical evidence; learned-method entries are no higher than Moderate.Each learned-method entry lacks at least one equal-accuracy, cost-matched comparison or deployment-shift test.Source: S9 Evidence trace for the decision… · pp. 53–54, Table S18 · p. 53, Table S19 · p. 54

S10 Measured break-even study: protocol and full results

• The study estimates N*=(N_tr c_gen+C_train)/(c_cl−c_nn), matching classical costs to FNO error and reporting training effort of 359–1310 network evaluations per sample across resolutions.The classical comparator is at least as accurate as FNO, favouring FNO in the comparison.Source: S10 Measured break-even study:… · pp. 54–55
• All 60 FNO trainings completed, and the stated sanity checks found no failure in errors, classical convergence, reference resolution, or GPU use.The full results, logs, environment record, and analysis scripts are released.Source: S10 Measured break-even study:… · p. 55, Table S20 · p. 55
• At fixed training cost, speedups S≥10 raise break-even N* by at most 11%; the GON replication’s N*≈1.6 × 10^7 contrasts with the original FNO study’s ≈2 × 10^4.For the original FNO study, the split between data generation and training is unreported; its break-even lies below the 3 × 10^4 queries of its MCMC application.Source: Figure S3 · p. 56
• For the reduced-basis check, the published Darcy result’s 2.4–2.6% error lies between the study’s 2.1% best-approximation and 3.8–4.4% POD–Galerkin errors at k=30.The projection error is the same at every resolution because the grids solve the same instances.Source: S10 Measured break-even study:… · p. 57, Table S21 · p. 57
• POD costs 6–35 classical solves for 2000 snapshots, but the coefficient does not compress, leaving reduced-basis online costs high; matching FNO accuracy would require an uncomputed several hundred to several thousand modes by extrapolation.FNO training costs about thirty times its data-generation cost.Source: S10 Measured break-even study:… · p. 57

S11 Tables and figures moved from the main article

• Tables S22 and S23 and Figure S5 cover benchmark suites, recommended core PDE problems, and the evidence-weighted benchmarking protocol.These materials support Section 4.Source: S11 Tables and figures moved from the… · p. 57, Table S22 · p. 58, Table S23 · p. 59
• The moved materials also include a Darcy cost comparison, a GRADE mapping, and research priorities linked to decision barriers.These support Sections 4.4, 4.7, and 7, respectively.Source: S11 Tables and figures moved from the… · p. 57, Table S24 · p. 61, Table S25 · p. 61
• Figure S4 presents the taxonomy of classical, ML-only, and hybrid PDE solvers, distinguishing families reviewed in depth from methods mentioned to clarify decision regimes.The moved figure supports Section 2.Source: S11 Tables and figures moved from the… · p. 57, Figure S4 · p. 58