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REVIEW 3 major objections 5 minor 3 cited by

DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A deep neural network trained on sparse, noisy observations, combined with automatic differentiation and sparse regression, recovers the governing partial differential equation.

desk verdict A plausible pipeline for PDE discovery that generates meta-data from a trained network, but the paper never validates the derivative accuracy that carries the robustness claim. read the letter →

arxiv 1908.04463 v2 pith:OPQXJEL4 submitted 2019-08-13 stat.ML cs.LGphysics.comp-ph

Hao Xu , Haibin Chang , Dongxiao Zhang This is my paper · ORCID
classification stat.MLcs.LGphysics.comp-ph MSC 68T0765M3235R30
keywords data-drivendiscoverypartialdifferentialequationsdeepneuralnetworksparseregressionautomaticdifferentiationnoisydatameta-datageneration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes DL-PDE, a pipeline that discovers the partial differential equation governing a physical process from discrete, noisy observations. It trains a fully connected neural network to fit the data, uses automatic differentiation to obtain time and space derivatives from that network, generates a large synthetic meta-data set of solution points and derivatives, and then applies sparse regression (STRidge) to select the few terms in a candidate library that form the equation. The authors test it on groundwater flow, convection-diffusion, Burgers, and KdV equations, and report correct equation forms with coefficients close to the true values under noise levels up to 5–10% and with data volumes as low as 1–25% of a full grid. They also show it works when observations come only from fixed temporal or spatial monitoring points, which direct finite-difference sparse regression cannot handle.

What carries the argument

The mechanism is a trained feed-forward neural network that approximates the solution as a function of $(x,t)$, with automatic differentiation supplying the derivatives $u_t$, $u_x$, $u_{xx}$, and so on at a large set of generated meta-data points; STRidge then selects a sparse coefficient vector $\xi$ from the candidate library $\Theta(u)$ to satisfy $U_t = \Theta(U)\xi$. The meta-data generation step is the load-bearing addition: it multiplies the number of equation-fitting points well beyond the original observations, which is what makes the sparse regression stable under noise and small data.

What would settle it

On a clean synthetic Burgers solution with known coefficients, add 5% uniform noise to a sparse training subset and run DL-PDE; if the automatic-differentiation derivatives from the noisy-trained network differ from a high-order finite-difference reference on a clean grid by more than the finite-difference error, the recovered equation will pick up spurious terms, falsifying the noise-robustness claim.

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Extended reading notes

Core claim

The central claim is that a trained deep neural network can stand in for the unknown solution of a PDE well enough that its automatic differentiation yields accurate derivatives, and that sparse regression on those derivatives recovers the correct governing equation even when the original data are noisy and sparse. The novelty is the meta-data step: instead of evaluating derivatives only at the observed points, DL-PDE uses the trained network to generate a much larger set of points with their derivatives, which stabilizes the sparse regression. The paper demonstrates this on four classical equations and reports that the method is robust to uniform noise without any smoothing, outperforming direct STRidge and a version of DL-PDE without meta-data.

Load-bearing premise

The method assumes that a neural network trained on noisy and sparse observations yields, through automatic differentiation, derivatives of the true underlying solution that are accurate enough for sparse regression to recover the correct equation; no error bounds or convergence guarantees are provided.

Editorial extensions

If this is right

  • For groundwater flow, convection-diffusion, Burgers, and KdV equations, the correct PDE form is recovered with coefficients within a few percent of the true values, even under 5–10% uniform noise and with as little as 1–25% of the original data.
  • The method works in engineering settings where observations are taken only at fixed temporal or spatial monitoring points, a regime where finite-difference-based sparse regression cannot compute derivatives.
  • Generating meta-data from the trained network is essential: the ablation shows that without it, recovered coefficients degrade sharply under noise (for example, the groundwater coefficient falls to about 0.026 at 1% noise instead of 0.971).
  • On identical gridded data, DL-PDE maintains higher accuracy than direct STRidge as data volume shrinks and stays stable under noise, whereas direct STRidge selects an incorrect equation form at 5% noise.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same pipeline should extend to PDEs with smoothly varying coefficients in space or time, because the neural network can represent the coefficient fields as additional outputs and automatic differentiation can still supply the needed derivatives.
  • A practical sanity check suggested by the method's reliance on the network's derivatives is to compare them against finite-difference derivatives on a small clean holdout set before trusting the sparse regression result.
  • Running the discovery step across multiple network initializations and keeping only terms that survive every run would give a rough, data-driven measure of confidence in the recovered equation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes DL-PDE, a data-driven method for discovering governing partial differential equations from discrete and noisy observations. The pipeline trains a feedforward deep neural network to fit the observed field u(x,t), uses automatic differentiation of the trained network to obtain the derivatives appearing in a candidate library, generates a large set of meta-data by querying the trained network on a denser grid, and finally applies sequential threshold ridge regression (STRidge) to identify a sparse PDE. The method is demonstrated on four model equations: groundwater flow, convection-diffusion, Burgers, and KdV. Experiments vary the data volume and noise level, include fixed-temporal and fixed-spatial monitoring settings, and compare against direct STRidge and against a no-meta-data ablation.

Significance. If the central claim holds, DL-PDE would be a useful addition to the PDE-discovery toolbox because it avoids direct numerical differentiation of noisy data, handles irregularly spaced and limited observations, and does not require explicit smoothing. The paper's experimental design has genuine strengths: it validates the learned PDE by solving it and comparing with the true solution using a relative-error metric, and it includes a direct comparison with STRidge on a common regular grid as well as an ablation that isolates the effect of meta-data generation. These elements make the empirical claims substantially more credible than a bare table of recovered coefficients. The main gaps are that the accuracy of the automatic-differentiation derivatives is never directly assessed, the role of meta-data is asserted rather than mechanistically demonstrated, and no uncertainty quantification or sensitivity analysis is provided for the hand-chosen STRidge threshold. These are fixable with additional experiments and analysis, so the paper is a solid candidate after major revision.

major comments (3)
  1. [II.B, II.C, and Eq. (13)] The central mechanism of the method—that a neural network trained on noisy, sparse observations yields, through automatic differentiation, derivatives accurate enough for the sparse regression in Eq. (4)—is never directly tested. The training loss in Eq. (8) contains only pointwise function-value matching, with no derivative penalty or smoothing, and Eq. (13) validates the solution of the learned PDE against the true solution, not the derivatives actually fed into STRidge. A flexible deep network can fit noisy data by oscillating, in which case its AD derivatives are biased estimates of the true derivatives. Generating meta-data from the same trained network adds more rows with the same systematic bias and cannot remove it. I request a direct assessment of derivative accuracy, for example error norms between NN-AD derivatives and the true derivatives as a function of noise level and data volume, or a PDE-residual check on the derivatives, to support the claimed robustness without smoothing.
  2. [III.A.1 and Table IX] The meta-data ablation in Table IX is informative but also exposes a weakness in the interpretation. Without meta-data, the groundwater coefficient collapses from 1.110 (clean data) to 0.026 at 1% noise and 0.00024 at 5% noise, while with meta-data it remains 0.971 and 0.906. This dramatic difference shows that the raw AD derivatives from the noisy-data-trained network are highly unreliable. The paper attributes the improvement to meta-data enhancing the stability of sparse regression, but no mechanism is given for why averaging over the same biased derivative estimates removes systematic bias. The authors should either provide a variance-reduction argument with supporting evidence, or acknowledge that the recovered coefficients at higher noise levels (e.g., 0.906 vs. 1.0 for groundwater at 5% noise) remain nontrivial offsets that need explanation and uncertainty quantification.
  3. [III.A.1, III.A.4, and II.C] The paper reports a single run for each configuration and does not report the STRidge threshold tol, the neural-network training hyperparameters (learning rate, epochs, batch size, initialization, normalization, early stopping), or any sensitivity analysis. Because the method's output is a single discovered equation whose form and coefficients depend on the threshold and on the trained network, the reader cannot determine whether coefficient deviations such as 0.940 vs. 1.0 for the KdV equation at 10% noise are typical or artifacts of one seed and one threshold. I ask for repeated independent runs (different NN initializations and noise draws), reported tol values, and a small sensitivity sweep over tol; without these, the 'satisfactory' and 'robust' conclusions are not yet fully supported.
minor comments (5)
  1. [II.C and Refs. [24]] The statement 'According to the universal approximation theorem of neural network, functions of any complexity can be approximated by a neural network with any precision' cites Ref. [24], which is a paper on numerical differentiation and regularization, not the universal approximation theorem. Moreover, universal approximation results apply to function values, not to derivatives of arbitrary order. Please correct the citation and qualify the statement.
  2. [II.B and III.A] The network architecture is described per case (e.g., nine-layer tanh for most cases, five-layer sin for KdV and engineering settings), but training details such as learning rate, number of epochs, batch size, data normalization, and early stopping are omitted. These details are needed for reproducibility.
  3. [III.A.4] The text says DL-PDE is 'robust to 10% noise' for the KdV equation, but Table VII shows the coefficient of u_xxx is 0.940 at 10% noise, a 6% deviation. Please define 'robust' quantitatively or soften the wording to match the reported accuracy.
  4. [Eq. (13)] The relative error definition divides by the range of the true solution, which can be small or zero for solutions that change sign, such as Burgers or KdV waves. Please specify the evaluation domains and confirm that the denominator does not make the metric misleading.
  5. [III.B and Appendix B] The fixed-temporal and fixed-spatial monitoring experiments are only presented for the convection-diffusion and Burgers equations; the main text should state this scope explicitly rather than implying that all four test cases were evaluated in engineering settings.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: DL-PDE fits a neural-network surrogate to observations, then discovers a PDE from that surrogate via automatic differentiation and sparse regression, with the true PDE used only as a benchmark.

full rationale

The derivation chain is self-contained. The neural network is trained with the data-mismatch loss in Eq. (8), which contains no PDE term, so the PDE is not encoded in the training objective. The trained network is then used to generate meta-data and derivatives by automatic differentiation, and STRidge solves Eq. (4) over the candidate library of Eq. (2). The true PDE enters only when generating synthetic training data for the numerical experiments and when validating the learned PDE through the relative error in Eq. (13); it is not an input to the loss function or the sparse regression. Meta-data are evaluations of the trained surrogate, so the discovered PDE is a parsimonious description of the surrogate; this is a modeling pipeline, not a reasoning loop, because the surrogate fit does not presuppose the target PDE. Self-citations [7] and [8] are used for comparison with earlier methods and for remarks about possible future extensions, not as load-bearing premises for the correctness of DL-PDE. The unreported STRidge threshold and the lack of direct validation of automatic-differentiation derivatives are tuning and robustness concerns, not circularity. No equation is shown to be equivalent to an input by construction, and no fitted parameter is renamed as a prediction. Therefore the paper exhibits no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method is empirical and rests on standard machine-learning assumptions: rich candidate library, accurate neural-network derivatives, constant coefficients, and a valid sparse-regression threshold. The free parameters are the threshold and meta-data settings, which are not reported or justified. No new physical entities are introduced.

free parameters (3)
  • STRidge threshold tol
    Section II.C describes a threshold that retains coefficients larger than tol and omits smaller ones. No values are given for any experiment, and the sparse output likely depends on this hand-chosen parameter.
  • Meta-data generation domain and grid = varies per case (e.g., groundwater: x* in [4,105], t* in [0,200], N'=101000)
    The subdomain and number of synthetic points are chosen per test case without justification; these choices affect regression stability and accuracy.
  • Neural network training hyperparameters = not reported
    Layer count and activation are stated, but learning rate, epochs, loss tolerance, and initialization are omitted. These affect derivative accuracy and the final discovered coefficients.
assumptions (4)
  • domain assumption The candidate library is sufficiently rich, containing all terms of the true PDE.
    Stated in Section II.A: Phi(u) is supposed to be sufficiently rich. The libraries in Eq. (12) include the true terms by construction.
  • domain assumption A feedforward neural network trained on finite noisy data provides accurate function and derivative approximations.
    Section II.B invokes the universal approximation theorem, but does not establish that finite noisy training yields accurate derivatives for the subsequent regression.
  • domain assumption The underlying PDE has coefficients that are constant in space and time.
    The authors state this limitation in Section IV: coefficients are assumed constant in space and time.
  • domain assumption STRidge with iterative thresholding reliably recovers the correct sparse support from the noisy derivative matrix.
    Section II.C adopts STRidge from Rudy et al. [2]; the correctness of the final support depends on the threshold and the accuracy of the matrix, which is not quantified.

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Pith. "Pith review of DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data." pith.science (2026). https://pith.science/paper/OPQXJEL4

@misc{pith2026190804463,
  author       = {Pith},
  title        = {Pith review of: DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OPQXJEL4}},
  note         = {Machine review of arXiv:1908.04463}
}
read the original abstract

In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deep-learning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions. In the DL-PDE, a neural network is first trained, and then a large amount of meta-data is generated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method is tested with physical processes, governed by groundwater flow equation, convection-diffusion equation, Burgers equation and Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed method achieves satisfactory results when data are noisy and limited.

Figures

Figures reproduced from arXiv: 1908.04463 by the authors.

Figure 1
Figure 1. The neural network comprises an input layer, an output layer, and one or several layer(s) between the input and output layers that are termed hidden layer(s). Each hidden layer is composed of multiple neurons. Two adjacent layers are connected as follows: ( 1 ), 1,..., 1 l l l l z W z b = + = −  − lL (5) where l denotes the layer index; W denotes the weight matrix; b denotes the bias vector; and  denotes the activ… view at source ↗

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Forward citations

Cited by 3 Pith papers

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  1. Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models

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  2. LLT: Local Linear Transformer for PDE Operator Learning

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    Local Linear Transformer learns PDE operators by combining linear global attention with local spatial mixing, achieving competitive accuracy and lower training-step cost than prior transformers.

  3. A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systems

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    A DNN-based pipeline with Gauss-Jacobi quadrature and alternating sparse/global optimization discovers fractional differential equations from sparse, noisy data.

Reference graph

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