{"id":"d098155b-aee7-4fe6-83dd-b0168ff537d1","arxiv_id":"1908.03190","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"NeuPDE learns ODE/PDE models from data by parameterizing the differential equation's right-hand side with a neural network over monomial and derivative features.","lead":"This paper proposes NeuPDE, a neural network method that models time-series and spatio-temporal data by learning the right-hand side of an ODE or PDE. It combines monomial feature dictionaries with small neural networks, and reports accurate fits on Lorenz, Burgers, and fluid flow, plus lower parameter counts on MNIST image classification.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Burgers experiment (§3.1) does not support governing-PDE discovery: all training ICs are sine-in-x, so y-derivative terms are never identified; the sine-in-y test IC gives train/test MSE 0.005/3.6, a gap consistent with trajectory memorization rather than operator learning.","rationale":"The reader's weakest_assumption is correct and is the most load-bearing point. The paper's own numbers in §3.1 are enough to see the mechanism: a model trained only on x-oriented initial data cannot learn coefficients for y-derivative terms, so the sine-in-y test probes an unconstrained part of the network. The large train/test gap is not a minor fitting issue; it is exactly what one expects if the learned object is a manifold-specific interpolant rather than the Burgers operator. I therefore agree with the reader's assessment. I do not see a separate, more severe objection: the ODE examples are demonstrations of trajectory fitting rather than symbolic recovery, the image-classification comparison is a narrow parameter-count claim, and the paper does not overclaim theory. The lack of error bars and absence of code make the single reported test number fragile, but those are supporting reasons for a conditional verdict, not independent fatal flaws. Keeping the verdict CONDITIONAL is appropriate: the proposed mixed-orientation retraining can directly test whether the generalization failure is due to data coverage and, if successful, would substantiate the central claim.","tokens_in":13455,"tokens_out":9713,"duration_ms":115047,"concrete_test":"Retrain the Section 3.1 Burgers model with a training set containing both sine-in-x and sine-in-y initial conditions (e.g., three of each, with the same amplitude and noise), and evaluate on held-out initial conditions of each orientation; use the same architecture, optimizer, and 2000 iterations, and report three independent seeds with error bars. If the test MSE on both orientations drops to the same order as the training MSE, the original failure is a data-coverage/identifiability problem rather than a fundamental architecture flaw. If the gap persists, the model has not learned the governing operator and the central PDE-discovery claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing assumption is that the learned right-hand side is the true differential operator, so it should predict any initial data from the same PDE. Section 3.1 is the paper's only PDE-discovery experiment and it undercuts this assumption. All five training trajectories use a sine-in-x initial condition. For scalar 2D Burgers, such initial data has u_y ≡ 0 along the entire trajectory (up to the small added noise), so the y-derivative channels (D_y, D_yy) in the dictionary are effectively unexcited on the training batches; the gradient of the loss (3.3) provides essentially no information about those channels, and the corresponding weights remain at their initialized values. The test initial condition, sine-in-y, lies in the complementary subspace and activates exactly those unconstrained terms. The reported numbers, training MSE 0.005 vs test MSE 3.6, are therefore not evidence that a governing equation was extracted: they are the expected signature of a fit to the x-dependent data manifold. This does not prove the architecture is useless, but it does mean the central claim that NeuPDE identifies an accurate governing system for PDEs is unsupported by the only PDE experiment. The MNIST/Fashion results are parameter-count comparisons and do not repair this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes NeuPDE, a framework that models time-dependent data by learning the right-hand side of an ordinary or partial differential equation. The unknown forcing term is parameterized as a shallow MLP applied to a monomial dictionary of the state variables (and, for PDEs, finite-difference derivative approximations), and the model is trained by fitting trajectories produced by a numerical integrator to observed data, using a regularized loss and a discretize-then-optimize backpropagation approach. The paper demonstrates the method on the Lorenz system, a noisy non-autonomous spiral ODE, a low-rank von Kármán vortex sheet, a 2D Burgers equation, and image classification on MNIST and Fashion-MNIST. The central claims are that NeuPDE can extract accurate governing ODE/PDE models, improve reduced-order modeling, and reduce the parameter cost relative to standard deep networks.","tokens_in":13735,"tokens_out":4944,"duration_ms":50509,"significance":"If established, the hybrid dictionary-MLP parameterization is a worthwhile contribution: it connects sparse dictionary-based equation discovery with neural network flexibility, and the finite-difference kernel constraint for image classification offers a parameter-efficient alternative to standard convolutional layers. The paper is clearly written and the formulation is easy to follow. However, the central PDE-discovery claim rests on a single Burgers experiment whose training and test initial conditions are not aligned, and the baseline comparisons are sparse. The image-classification results, while empirically plausible, are parameter-count comparisons rather than demonstrations of equation discovery. The manuscript would be strengthened materially by a corrected or extended PDE experiment and by quantitative evaluations of the ODE examples.","major_comments":[{"comment":"The Burgers experiment does not support the claim that NeuPDE identifies a governing PDE. All five training trajectories use a sine-in-x initial condition; for the 2D Burgers equation, such initial data has u_y and u_yy identically zero along the entire trajectory (up to the small added noise). Consequently, the dictionary channels containing y-derivatives are unexcited during training, the gradient of the loss provides essentially no information about their weights, and those weights remain unconstrained. The test initial condition, sine-in-y, activates exactly those channels, and the reported train/test MSE of 0.005/3.6 is the expected signature of a fit to the x-dependent training manifold rather than of a discovered differential operator. To support the PDE-discovery claim, the authors must either train on initial conditions that excite all derivative directions, or demonstrate generalization on multiple genuinely unseen initial conditions from the same PDE, ideally also inspecting the learned RHS against the true Burgers operator.","section":"Sec. 3.1, Eq. (3.3)"},{"comment":"The comparisons to SINDy and LASSO are shown only for the non-autonomous noisy spiral of Section 2.2, and SINDy fails on that example while LASSO fails in the z-component. No data-discovery baseline is reported for the Burgers PDE experiment, even though the abstract states a comparison to \"other data-discovery methods\" and the PDE case is the central novelty. Please add a PDE-discovery baseline (for example, PDE-FIND-style sparse regression or a dictionary-based method) to the Burgers example, and also provide quantitative comparisons for the autonomous Lorenz ODE example so that the reader can judge the performance relative to standard sparse identification.","section":"Sec. 2.3"},{"comment":"The Lorenz experiment is evaluated only qualitatively, with the statement that the learned system generates a \"high-fidelity trajectory for the first part of the time-interval.\" No quantitative error, prediction horizon, or learned coefficients are reported. For a chaotic system, short-horizon agreement can be achieved by models that do not identify the true governing equations, so this demonstration is weak evidence for the method's discovery capability. Please report the normalized prediction error as a function of lead time, the length of the accurate prediction interval, and, if possible, a comparison of the learned dynamics with the true Lorenz system in phase-space or via coefficient recovery.","section":"Sec. 2.1, Fig. 2.1"}],"minor_comments":[{"comment":"The definition of σ_ELU appears to be inverted: the standard ELU is x for x >= 0 and e^x - 1 for x < 0, whereas the text states e^x - 1 for x >= 0 and x for x < 0. Please correct this, as it affects the implementation.","section":"Sec. 2, definition of ELU"},{"comment":"There is a typo: \"we purpose a different parameterization\" should read \"we propose a different parameterization.\"","section":"Sec. 2, text"},{"comment":"The word \"semi-discete\" should be \"semi-discrete\" in the description of the PDE block.","section":"Sec. 3.2, text"},{"comment":"The relationship between the continuous-time derivative regularizer in Eq. (2.1) and the finite-difference sum in Eq. (2.2) is stated only as \"β2 rescaled by the time-step.\" Please spell out the rescaling formula so that the discretization is unambiguous.","section":"Eq. (2.1) and Eq. (2.2)"},{"comment":"The caption for Figure 3.1 says \"(a) Training data. (b) Learned surface, test data,\" while the text says the learned solution is plotted on both training and test sets. Please make the caption consistent with the text and clarify which quantity is shown at the terminal time.","section":"Fig. 3.1 caption"}],"recommendation":"major_revision","confidential_remarks":"The Burgers experiment is the only PDE-discovery experiment in the paper, and the train/test initial-condition mismatch makes the reported gap consistent with memorization rather than operator discovery. This is a load-bearing issue for the central claim, but it is fixable by retraining on a diverse set of initial conditions and/or by adding a PDE baseline. I recommend major revision rather than rejection, provided the authors can supply the new experiments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one if you are curious about the latest dictionary-plus-MLP combinations for equation discovery, but do not send it to press as is. The architecture is sensible: parameterize the RHS of an ODE/PDE as an MLP applied to monomials and finite-difference derivatives, then train through a solver. The Lorenz and spiral fits are fine as demonstrations, and the finite-difference kernel substitution in MNIST is a neat trick that saves parameters at no accuracy cost. But the paper's only PDE-discovery experiment, the Burgers example in Section 3.1, does not support the central claim. All five training initial conditions are sine-in-x, so along those trajectories u_y and u_yy are zero up to noise; the y-derivative dictionary channels are never excited, and the training loss carries no information about them. The test IC is sine-in-y, exactly the direction those unconstrained terms control. The gap between training MSE 0.005 and test MSE 3.6 is the signature of a fit to the training manifold rather than an extracted operator. The stress-test note nailed this. It does not mean the architecture is useless, but it does mean the claim that NeuPDE identifies accurate governing PDEs is unsupported by the only PDE experiment. Elsewhere, missing error bars and the selective use of SINDy/LASSO baselines only on the hard non-autonomous noisy example also soften the empirical case. The MNIST/Fashion results are parameter-count comparisons, not evidence about dynamics. For a discovery paper, the authors never show the learned equations, only trajectories, which makes it hard to judge what was actually recovered. The math is fine, the writing is clear, and the idea is plausible, but the validation needs a major redo: test on multiple initial conditions, report learned expressions, add repeats with error bars. This is a paper for a serious referee, not a desk reject, because the underlying approach has legs and the flaws are fixable. But I would tell the editor it needs major revision.","headline":"Plausible architecture, but the only PDE-discovery experiment is undermined by a training/test initial-condition mismatch that leaves the central claim unproven.","tokens_in":14257,"tokens_out":2010,"would_cite":false,"duration_ms":21027,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"NeuPDE recovers governing ODEs and PDEs from trajectory data by pairing a monomial dictionary with a shallow network.","keywords":["partial differential equations","ordinary differential equations","data-driven discovery","multilayer perceptrons","monomial dictionary","finite-difference kernels","image classification","reduced-order modeling"],"falsifier":"Train the Burgers model on the paper's sine-in-x initial data and evaluate on sine-in-y initial data over a horizon longer than $T=0.015$, comparing the prediction error against the error of a baseline that returns the nearest training snapshot. If the NeuPDE error is not clearly smaller, or if it grows at the same rate as the training-data spread, the model has memorized trajectories rather than inferred the differential operator. The paper's reported test MSE of 3.6, versus training MSE of 0.005, is exactly the quantity to watch, normalized by the test data variance.","tokens_in":13255,"feed_emoji":"🧮","tokens_out":11639,"duration_ms":119479,"temperature":0.7,"pith_summary":"The paper aims to establish that a neural network can learn the governing ordinary or partial differential equation of a time-dependent system from observed data, when the unknown right-hand side is represented as a shallow multilayer perceptron acting on a dictionary of monomial and derivative terms. The point of the hybrid representation is that the dictionary encodes nonlinear correlations and differential structure directly, so the network does not have to rediscover them from scratch; the resulting model is a genuine differential equation that can be integrated at arbitrary times. If this claim is right, data-driven forecasting and equation discovery become one task, and image classifiers can be made smaller by building finite-difference structure into their convolution kernels. The approach is demonstrated on the Lorenz system, a noisy nonlinear spiral, a reduced-order fluid model, Burgers' equation, and MNIST/Fashion MNIST classification.","feed_headline":"Shallow net plus monomial dictionary learns ODEs and PDEs from data","feed_subtitle":"NeuPDE extracts continuous dynamics from trajectories and beats neural-ODE and ResNet parameter counts.","key_machinery":"The load-bearing object is the dictionary-plus-MLP operator $F(D(\\cdot),\\theta)$, where $D$ is the matrix of all monomials $t^k x_1^{\\ell_1}\\cdots x_d^{\\ell_d}$ with $0