{"id":"aa92c243-7947-4704-a055-3cb9016b8a2e","arxiv_id":"2501.14604","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Explicit backward-time steps, time-reversed, produce training pairs that satisfy implicit forward schemes and improve FNO test error, but the main experiments do not match dataset size between augmented and baseline runs.","lead":"This paper proposes generating extra training data for neural PDE solvers by running the equation backward in time with explicit numerical steps, then reversing the time order of each pair. The hope is that this cheaply creates pairs that obey stable implicit schemes and reduces the data burden of neural operators, but the experiments compare against runs with half the data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central empirical claim is confounded: augmented runs train on twice as many pairs as baselines, so the reported gains may reflect data quantity, not inverse evolution.","rationale":"The reader rejected the paper by REJECT, and this stress-test supports that verdict without changing it. The single most load-bearing concern is the uncontrolled comparison: every performance table compares N original pairs against N original plus N generated pairs. Thus the test error changes two things at once: the augmentation content and the number of training examples. Since adding extra examples can reduce test error even when the added examples are not from the intended distribution, the reported improvements in Tables 3 and 4 do not demonstrate that inverse evolution is the active ingredient. This concern is more decisive than questioning Eq. (7), because even if the initialization distribution were perfect, the current experiments would still fail to isolate the central mechanism. Conversely, if a matched-data-size baseline shows a clear and consistent gain for the generated pairs, the paper's core mechanism would have genuine support. The mathematical identities in Section 2 are not in dispute; the issue is that satisfying a discrete implicit scheme is insufficient to guarantee that a pair approximates a true PDE solution, so the augmented data could act merely as structured noise. The proposed concrete test directly separates the augmentation content from the data-size effect and should be run before the empirical claim is accepted.","tokens_in":11978,"tokens_out":4700,"duration_ms":44420,"concrete_test":"For each row of Tables 3 and 4, train the same FNO/UNet with 2N original pairs (e.g., by sampling additional trajectories with the same PDEBench or pseudo-spectral solver, or by reusing existing training pairs with reshuffled initial conditions) and compare test L2 error against the current N+N augmented runs. Also include a control of N original plus N random smooth pairs. If the 2N original control matches or outperforms N+N augmented, the inverse-evolution mechanism is not supported; if augmented data clearly wins across all equations, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.3 states: 'we use original datasets with 1000, 5000, and 10000 pairs, and the inverse evolution generates an equal number of additional pairs, resulting in augmented datasets of 2000, 10000, and 20000 pairs.' Tables 3 and 4 therefore compare N original examples against 2N examples (N original + N generated). Any improvement in the 'yes' columns is compatible with the trivial hypothesis that doubling the number of training pairs helps, independent of the inverse-evolution construction. This is the load-bearing issue for the paper's central claim that inverse-evolution data augmentation improves neural PDE solvers: the experiments never isolate the augmentation mechanism. The Section 2 identity is correct but only shows consistency of a discretization; it does not show that arbitrary initializations from Eq. (7) lie near the solution manifold or that generated outputs approximate true forward solutions. The accuracy table (Table 1) does not fully rescue the claim because the FNO experiments do not report generated-data accuracy for the exact configurations used, and the empirical comparison remains confounded. A matched-data-size control would settle whether the augmentation mechanism itself is responsible for the improvement.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a data augmentation strategy for training neural operators on evolution equations. The idea is to initialize a backward-time integration from randomized combinations of existing solution snapshots, take one explicit (or Taylor-based high-order) step of the inverse equation, and then reverse the pair so that it formally satisfies an implicit scheme of the forward equation. The authors validate the accuracy of generated pairs on heat, Burgers, Allen-Cahn, and Navier-Stokes equations and report that adding such pairs improves FNO and UNet test errors (Tables 3 and 4).","tokens_in":12139,"tokens_out":7563,"duration_ms":59767,"significance":"If the method worked as claimed, it would be a useful and cheap way to generate training pairs for neural PDE solvers without solving implicit systems. The first-order equivalence is simple and correct, and the computational-time comparison in Table 2 is attractive. However, the experimental evidence does not isolate the proposed mechanism: every augmented run contains twice as many training pairs as the corresponding baseline run. The high-order equivalence is also stated without proof. These gaps are central to the paper's claims, so the contribution is not yet established.","major_comments":[{"comment":"The reported gains are confounded with dataset size. The text states that the augmented datasets contain the original pairs plus an equal number of generated pairs, so the 'yes' rows train on 2000, 10000, and 20000 pairs while the 'no' rows train on 1000, 5000, and 10000 pairs. Any improvement in the augmented columns is therefore compatible with the trivial hypothesis that doubling the number of training examples helps, independent of the inverse-evolution construction. To support the claim that inverse evolution itself improves performance, the authors should add a matched-data-size control: either train baselines on the same total number of original pairs, or train augmented models on a random subsample of the full augmented set of the same size as the baseline. Without such a control, the central empirical claim of the paper is not established.","section":"Section 3.3 and 3.4, Tables 3 and 4"},{"comment":"The statement that data pairs from the Taylor-based high-order inverse schemes 'can be proved' to satisfy the corresponding high-order implicit schemes is asserted without proof, and no reference is given. This claim is load-bearing because the main FNO experiments use the third-order scheme. The authors should either provide the explicit high-order implicit relations (for the second- and third-order cases) or state the precise approximation property that the reversed pairs satisfy. As written, the theoretical justification for the high-order augmentation is missing.","section":"Section 2, High-order Schemes (after Eq. (13))"},{"comment":"The initialization via U*_i = sum_j lambda_j U_{R_j(i)} + C is a random linear combination of shuffled solution snapshots plus a constant, but the paper gives no evidence that such initializations lie close to the solution manifold of the evolution equation. The assertion that nonlinearity makes the generated pairs distinct from the original data does not establish that the pairs are physically relevant or that one inverse-evolution step maps them to states on the solution manifold. A concrete check is needed, such as measuring the distribution shift between generated inputs and test-time inputs, or reporting the residual of the generated pairs under a high-accuracy reference solver. Without this, the generated data may be out-of-distribution, and the experimental gains may be due to a broadened input distribution rather than to the inverse-evolution mechanism.","section":"Section 2, Eq. (7)"},{"comment":"The accuracy of the generated data is not reported for the exact settings used in the neural-operator experiments. Table 1 covers Allen-Cahn with epsilon=0.05 and Navier-Stokes with nu=0.001, but the FNO experiments also use Allen-Cahn epsilon=0.01 and Navier-Stokes nu=0.0001; Burgers accuracy is reported only for nu=0.1, while the FNO experiments use nu=0.001 as well. Since the reliability of the augmented data is the premise of the method, the authors should report the generated-data accuracy for those configurations, or clearly state which settings in Table 1 correspond to the data actually added in Tables 3 and 4.","section":"Section 3.1 vs. Section 3.3"}],"minor_comments":[{"comment":"The definition of (V^n, V^{n+1}) := (U^{n+1}, U^n) is notationally confusing because the superscripts imply a forward time step while the pair is time-reversed; please define the time levels explicitly (for example, V^n = U^{n+1} and V^{n+1} = U^n) to avoid apparent inconsistency.","section":"Section 2, Eq. (5)"},{"comment":"The 'Magnitude' row in Table 1 is ambiguous: it is unclear whether the scaling factor applies to all entries in the column section or only to the first row, and the caption does not define the em dash placeholder. Please clarify by using scientific notation in each cell and defining '—' in the caption.","section":"Table 1"},{"comment":"Hyperparameters for Eq. (7) are incompletely specified: the choice of lambda_j and C is given only for Burgers nu=0.001 (C in [-0.1, 0.1]), while the experiments for Allen-Cahn and Navier-Stokes do not state the values or ranges used, which prevents reproduction.","section":"Section 3.3"},{"comment":"There are several typographical errors, including 'generated reliable data' in Section 2 and 'H 2 norm' in Section 2, which should read 'H^2 norm'; additionally, the caption and Eq. (14) do not specify how the rescaling coefficient a is chosen for the experiments.","section":"Section 2 and Figure 2"}],"recommendation":"major_revision","confidential_remarks":"The first-order equivalence and the computational cost comparison are sound, but the causal claim is not supported by the current experimental protocol. I would encourage a revision that adds matched-data-size baselines and a proof of the high-order equivalence; both are within scope. The novelty relative to the authors' own prior inverse-evolution work [20] should also be clarified explicitly, since the manuscript cites it for the concept but does not state precisely which components are new."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the inverse-evolution swap is a genuinely neat trick. The statement that one explicit backward step followed by time reversal yields a pair satisfying implicit Euler is elementary and correct. Using that as a data generator for neural operators, with random-combination initialization and Taylor-based high-order variants, is new relative to the cited augmentation literature (Lie point symmetry, general covariance). Self-citation to [20] is appropriate; the augmentation pipeline is distinct from the regularizer there.\n\nWhat the paper does well: the method is cheap (Table 2), the generated data are checked against true solutions in Table 1, and the authors are honest about limitations\\u2014instability for chaotic or sharply interfaced problems and the restriction of preprocessed data to a specific space. The FNO gains at 1000 pairs are sometimes large (Allen-Cahn from 4.1e-2 to 5.2e-3), which is not nothing.\n\nThe soft spots are real, and one is load-bearing. Section 3.3 states that augmented runs use 2N training pairs when baselines use N. Tables 3 and 4 therefore compare one dataset size against twice that size. Any benefit is compatible with the trivial hypothesis that more data helps; the inverse-evolution mechanism is never isolated. That is not a minor issue\\u2014it is the central empirical claim. The paper needs a matched-size control (for example, N original + N additional original, or N original + N pairs generated by a non-physics random process) before the efficacy conclusion is supported. Second, the claim that high-order Taylor pairs \\u201ccan be proved to satisfy the corresponding high-order implicit schemes\\u201d is asserted without proof. It may be true, but it is not shown, and Table 1 shows the high-order schemes degrade badly at larger time steps. Third, Equation (7) random combinations are asserted to stay near the solution manifold without any quantitative check; Figure 3 shows some generated inputs look plausible, but there is no distribution-shift measure. Minor: no code, seeds, or error bars, so the magnitude of the gains is hard to assess.\n\nOn balance, the paper is worth engaging. The mathematical core is sound at first order, the method is cheap, and the confound is fixable with one additional experiment. I would send it to review with a request for matched-size comparisons and code, but I would not accept the efficacy claim as it stands.","headline":"Clever augmentation idea with a correct first-order identity, but the headline gain is confounded because augmented runs double the training set.","tokens_in":12695,"tokens_out":1792,"would_cite":false,"duration_ms":20840,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","35Q35","65N99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that one explicit backward-time step, after time reversal, yields data pairs satisfying implicit forward schemes, and shows that adding these pairs improves neural operators on Burgers, Allen-Cahn, and Navier-Stokes…","keywords":["neural operators","data augmentation","inverse evolution","partial differential equations","high-order inverse evolution schemes","Fourier Neural Operator","UNet","evolution equations"],"falsifier":"Take a fixed set of solution snapshots, build initializations with progressively larger mixing constants in Eq. (7), run one third-order inverse step, and compare the time-reversed pair against a high-accuracy forward reference over the same interval; if the relative $L^2$ error grows with the mixing constants or exceeds the error of the forward solver used to create the original data, the generated pairs are not reliable implicit-scheme solutions.","tokens_in":11709,"feed_emoji":"🧮","tokens_out":8468,"duration_ms":77765,"temperature":0.7,"pith_summary":"This paper argues that a backward-time trick can produce abundant training data for neural operators that solve evolution equations, at a fraction of the cost of running forward solvers. The trick is to take a later-time state, advance it one explicit step of the inverse equation $u_t=-F(u)$, and then reverse the pair; the reversed pair satisfies the implicit Backward Euler scheme of the forward equation. The paper proves the same for second- and third-order Taylor-based inverse schemes, adds a mixing-and-shifting initialization step to create varied starting states, and tests the generated pairs by training Fourier Neural Operators and UNets on Burgers, Allen-Cahn, and Navier-Stokes equations. The reported result is lower test error and better robustness to noisy inputs, with the largest gains when the original dataset is small.","feed_headline":"One reverse step yields reliable data for PDE neural nets","feed_subtitle":"Flipping one explicit backward step gives pairs that obey implicit schemes and cut FNO error on three equations.","key_machinery":"The carrying mechanism is the reversal identity: take one explicit backward step $U^{n+1}=U^n-\\Delta t\\,F(U^n)$ and then swap the pair so $(V^n,V^{n+1})=(U^{n+1},U^n)$; substitution gives $(V^{n+1}-V^n)/\\Delta t = F(V^{n+1})$, which is exactly the implicit Backward Euler scheme of the forward equation. This identity turns an unstable explicit backward computation into a legitimate implicit-scheme training pair. The paper's high-order variants repeat the same argument with Taylor-expansion inverse steps, yielding data that satisfy higher-order implicit schemes, while the randomized mixing initialization Eq. (7) and the rescaling preprocessing in Eq. (14) keep the generated inputs within a useful neighborhood of the solution space.","core_discovery":"The central claim is that inverse evolution is a reliable and cheap data source for learning evolution operators. If $(U^n, U^{n+1})$ is computed by one explicit Forward Euler step of the inverse equation, then the time-reversed pair $(V^n,V^{n+1})=(U^{n+1},U^n)$ satisfies the implicit Backward Euler formula $(V^{n+1}-V^n)/\\Delta t = F(V^{n+1})$ of the original forward equation. The paper extends this identity to Taylor-expansion inverse schemes up to third order, claiming the generated pairs satisfy the corresponding high-order implicit schemes, so that explicit computation with relatively large time steps yields data normally associated with stable implicit solvers. The initialization combines random permutations of existing solution snapshots with constants, Eq. (7), and a normalization/rescaling preprocessing step handles sharp interfaces. Experiments on three evolution equations show that adding these pairs to the training set improves FNO accuracy in all tested settings and improves UNet in most, with particularly clear gains at 1000-pair datasets.","pith_inferences":["Testable extension: because the reversal identity is algebraic, the same augmentation could be applied to any architecture that learns time-stepping maps, including Transformers or graph-based operators; the measured gain would likely grow with the model's ability to represent high-frequency features, paralleling the FNO-versus-UNet gap reported here.","The initialization in Eq. (7) is essentially a linear mixing of solution snapshots, so the central unresolved risk is distribution shift; a direct test would be to measure the distance between generated inputs and the true solution manifold as the mixing constants $\\lambda_j$ and $C$ vary, and to compare against training on randomly perturbed snapshots of equal size.","The paper's robustness result for Allen-Cahn suggests the high-frequency content in augmented inputs acts as a regularizer; a natural way to check this would be to compare the spectral bias or the norm of the learned operator's Jacobian with and without augmented data.","Using pseudo-spectral spatial discretization for the inverse step, which the paper already finds less sensitive to $C$ for Navier-Stokes, could be combined with adaptive time stepping to push the method into regimes where explicit high-order inverse steps currently diverge."],"forward_implications":["Every pair produced by one explicit inverse step of the form (4) is, after reversing the time order, a Backward Euler pair of the forward equation, so the augmented data inherit the stability and consistency of an implicit scheme without solving nonlinear equations.","The high-order Taylor inverse schemes (8)-(10) yield pairs that satisfy the corresponding high-order implicit schemes, which the accuracy tests show reduces the relative $L^2$ error of generated data at moderate $\\Delta t$.","The method reduces the cost of training-data generation: for 100 pairs at resolution 256, the measured time falls from about 3.2 s to 0.27 s for 1D Burgers, from 11.8 s to 0.15 s for 2D Allen-Cahn, and from 74.8 s to 0.033 s for Navier-Stokes.","FNO trained on augmented data achieves lower test error than on original data alone across Burgers, Allen-Cahn, and Navier-Stokes settings, with the biggest relative gains at 1000 training pairs; the Allen-Cahn noise experiments indicate the augmented training also improves robustness to input perturbations.","UNet also benefits in most settings, though less than FNO, consistent with the paper's suggestion that architectures with larger capacity extract more from the more complex inputs produced by inverse evolution."],"supporting_citations":[{"why":"introduces the inverse-evolution concept that this paper re-purposes as a data-generation mechanism","marker":"[20]"},{"why":"supplies the Fourier Neural Operator architecture used in the experiments","marker":"[17]"},{"why":"provides the benchmark data generation method and baseline code for the Burgers experiments","marker":"[27]"},{"why":"grounds the rescaling preprocessing step as adaptive moving-mesh r-adaptivity for reducing error","marker":"[11]"},{"why":"supplies the finite-difference and spectral spatial discretization framework used in the inverse evolution schemes","marker":"[24]"},{"why":"underlies the stability discussion that motivates avoiding small explicit forward time steps","marker":"[26]"}],"fun_headline_variants":["Reverse steps generate cheap training data for PDE nets","Flip time to create data that satisfies implicit schemes","Inverse evolution yields reliable data for neural PDE solvers","One backward step gives implicit-valid pairs for FNO training","Reverse PDE steps cut data cost, boost FNO accuracy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method's reliability rests on the assumption that the randomly mixed and shifted snapshots in Eq. (7) are close enough to genuine solutions of the evolution equation that one explicit backward step, after time reversal, still lands near a true forward trajectory.","fun_headline_variants_meta":{"raw":{"variants":["Reverse steps generate cheap training data for PDE nets","Flip time to create data that satisfies implicit schemes","Inverse evolution yields reliable data for neural PDE solvers","One backward step gives implicit-valid pairs for FNO training","Reverse PDE steps cut data cost, boost FNO accuracy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000154,"raw_usage":{"total_tokens":1227,"prompt_tokens":975,"completion_tokens":252,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":174}},"tokens_in":591,"tokens_out":252,"duration_ms":3399,"temperature":1.0,"reasoning_tokens":174,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:58:38.572943+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a fixed set of solution snapshots, build initializations with progressively larger mixing constants in Eq. (7), run one third-order inverse step, and compare the time-reversed pair against a high-accuracy forward reference over the same interval; if the relative $L^2$ error grows with the mixing constants or exceeds the error of the forward solver used to create the original data, the generated pairs are not reliable implicit-scheme solutions.","supporting_citations":[{"cited_title":"Inverse Evolution Layers: Physics-informed Regularizers for Deep Neural Networks","cited_arxiv_id":"2307.07344","evidence_quote":"introduces the inverse-evolution concept that this paper re-purposes as a data-generation mechanism"},{"cited_title":"Fourier neural operator for parametric partial differential equations","cited_arxiv_id":null,"evidence_quote":"supplies the Fourier Neural Operator architecture used in the experiments"},{"cited_title":"Pdebench: An extensive benchmark for scientific machine learning","cited_arxiv_id":null,"evidence_quote":"provides the benchmark data generation method and baseline code for the Burgers experiments"},{"cited_title":"Adaptive moving mesh methods , volume 174","cited_arxiv_id":null,"evidence_quote":"grounds the rescaling preprocessing step as adaptive moving-mesh r-adaptivity for reducing error"},{"cited_title":"Numerical solution of partial differential equations: finite difference methods","cited_arxiv_id":null,"evidence_quote":"underlies the stability discussion that motivates avoiding small explicit forward time steps"}],"review_version":1}