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REVIEW 3 major objections 5 minor 41 references

NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data

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

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Pith's one-line read NeuPDE recovers governing ODEs and PDEs from trajectory data by pairing a monomial dictionary with a shallow network.

desk verdict Plausible architecture, but the only PDE-discovery experiment is undermined by a training/test initial-condition mismatch that leaves the central claim unproven. read the letter →

arxiv 1908.03190 v1 pith:K6KHZPNR submitted 2019-08-08 cs.LG stat.ML

Yifan Sun , Linan Zhang , Hayden Schaeffer This is my paper · ORCID
classification cs.LGstat.ML
keywords partialdifferentialequationsordinarydata-drivendiscoverymultilayerperceptronsmonomialdictionaryfinite-differencekernelsimageclassificationreduced-ordermodeling
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 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.

What carries the argument

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 $0<k+\sum_i \ell_i\le p$, plus, for PDEs, the same construction over $t$, $x$, $u$, and finite-difference approximations of spatial derivatives. $F$ is a two-layer MLP with a smooth activation; the ODE/PDE constraint $\dot{x}=F(D(N(x)),\theta)$ is enforced by integrating the equation with a Runge-Kutta scheme inside the loss, and temporal mini-batches of short trajectories are used with Adam. In the image-classification variant, the same idea becomes a semi-discrete PDE whose $3\times 3$ convolution kernels lie in the span of six finite-difference kernels (identity, $D_x$, $D_y$, $D_{xx}$, $D_{xy}$, $D_{yy}$), so each layer is an explicit differential operator rather than a free kernel.

What would settle it

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.

Watch

Extended reading notes

Core claim

The central claim is that the forcing function of a neural ODE/PDE should not be a generic MLP on raw inputs, but an MLP applied to a dictionary of the time variable, state components, and, in the PDE case, finite-difference approximations of spatial derivatives. In the paper's notation the model is $\dot{x}=F(D(N(x)),\theta)$, with $F(z,\theta)=A_2\sigma(A_1 z+b_1)+b_2$; training minimizes trajectory misfit plus an $\ell^1$ weight penalty and an $\ell^2$ penalty on the time derivative, while the state sequence is constrained to be the output of a Runge-Kutta solver. The authors demonstrate that this parameterization identifies accurate dynamics where sparse polynomial methods miss a time-dependent term, improves an eight-mode DMD reduced model of the von Kármán vortex sheet, and, when convolution kernels are restricted to linear combinations of finite-difference operators, matches or improves image-classification accuracy with fewer parameters than the baselines.

Load-bearing premise

The learned differential operator is assumed to transfer to initial conditions outside the training set, so the model must have inferred the dynamics rather than memorized the particular trajectories it was trained on.

Editorial extensions

If this is right

  • Learned models are continuous differential equations, so after training they can be queried and integrated at arbitrary time resolutions, not only at the observed time stamps.
  • In the noisy spiral example, the method recovered the time-dependent z-dynamics that SINDy and LASSO failed to recover, indicating the dictionary-MLP parameterization can handle time-dependent terms and noise better than fixed sparse dictionaries.
  • The low-rank fluid example shows a learned nonlinear closure term can improve an eight-mode DMD reduced model, cutting terminal relative $L^2$ error from 0.060 to 0.049; more modes would make DMD accurate, but the closure provides an alternative.
  • On MNIST, restricting kernels to finite-difference operators preserves 0.51% error while reducing parameters from ODE-Net's 0.22M to 0.18M; on Fashion MNIST, 0.38M parameters outperform ResNet18's 2.78M parameters at 7.6% versus 8.0% error.
  • Using a quadratic dictionary for Lorenz, 263 parameters track the trajectory as well as a degree-one dictionary with 703 parameters, so the dictionary level is a direct control on model capacity.

Reading between the lines

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

  • The paper leaves dictionary degree and MLP width as hand-set hyperparameters; a natural next step is to add sparsity over dictionary entries so the method selects its own monomials.
  • The paper tests fixed-horizon errors only; feeding the model's own predictions back as initial conditions over many cycles would reveal whether the learned operator is stable under iteration.
  • The finite-difference kernel basis suggests a broader design principle: constraining convolution layers to learned linear combinations of differential operators may transfer parameter savings to larger image and video tasks.
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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 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.

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 (3)
  1. [Sec. 3.1, Eq. (3.3)] 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.
  2. [Sec. 2.3] 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.
  3. [Sec. 2.1, Fig. 2.1] 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.
minor comments (5)
  1. [Sec. 2, definition of ELU] 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.
  2. [Sec. 2, text] There is a typo: "we purpose a different parameterization" should read "we propose a different parameterization."
  3. [Sec. 3.2, text] The word "semi-discete" should be "semi-discrete" in the description of the PDE block.
  4. [Eq. (2.1) and Eq. (2.2)] 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.
  5. [Fig. 3.1 caption] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: NeuPDE is a data-fitting architecture whose predictions are evaluated on held-out time stamps and initial conditions; the self-citations are background, and the Burgers train/test gap is a generalization concern, not a circular dependency.

full rationale

The paper's derivation chain is an empirical fitting procedure: it parameterizes the unknown ODE/PDE right-hand side as F(D(...),theta) with a monomial dictionary and an MLP, then minimizes a data-misfit loss (Eqs. 2.1, 2.2, 2.3, 3.3) over the parameters. The learned model is then used to generate trajectories from initial data, and these are compared with held-out data. Nothing in the construction defines the target output as an input to the model: the dictionary terms are candidate features, the finite-difference kernels are fixed numerical derivative approximations, and the train/test splits are genuine. Baselines (SINDy, LASSO, DMD, RNN, ODE-Net) are external. The paper includes several self-citations (e.g., refs. 32-35, 40-41), but they are used for background on sparse dictionary methods, for convergence theory of the SINDy baseline, and for ResNet stability; none is invoked as the justification that NeuPDE's model is correct or unique. The Burgers experiment in Sec. 3.1 reports train MSE 0.005 and test MSE 3.6 when the test initial condition is sine-in-y rather than sine-in-x; this is evidence that the fitted model may not have learned a generalizable differential operator, and it is a correctness/validation concern rather than a circular argument. Every claimed prediction is produced by forward integration of the trained right-hand side, not by a quantity that was fitted into the loss. Therefore the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper does not introduce any new physical entity or force. The free parameters are architectural hyperparameters and regularization weights, all chosen by hand or fixed to constants. The key assumptions are that the dynamics are first-order in time, that the dictionary-MLP representation is sufficiently expressive, and that finite-difference derivative estimates are reliable.

free parameters (3)
  • Dictionary degree p = 1, 2, or 4 depending on experiment
    The order of monomials in the dictionary is chosen by hand for each problem. It controls the expressiveness of the feature set and directly affects the model's ability to capture nonlinear interactions.
  • Hidden units per layer = 20, 38, 50, 64, 100, 128 depending on experiment
    The width of the MLP layers is an architecture choice that influences capacity and parameter count. The paper does not perform a systematic study, and the values vary across examples.
  • Regularization weights beta1 and beta2 = beta1=1e-4, beta2=1e-5 (claimed no tuning)
    These hyperparameters control the L1 penalty on weights and the derivative penalty. They are fixed to the stated values for all regression examples, but the choice is not derived from data.
assumptions (5)
  • domain assumption The time-series data is governed by a first-order ODE of the form dx/dt = f(t,x).
    The entire method is built on the premise that the underlying dynamics are first-order in time. Systems that do not fit this form (e.g., higher-order or delay differential equations) would not be modeled correctly.
  • domain assumption For spatio-temporal data, the evolution is first-order in time and can be written as u_t = G(t,x,u,Du,...).
    The PDE formulation in Section 3 assumes this general form, which excludes some types of dynamics (e.g., wave equations with second-order time derivatives).
  • domain assumption The unknown right-hand side f or G can be accurately approximated by an MLP over the monomial dictionary.
    This represents a universal approximation style assumption. If the true function has features that the finite dictionary and MLP cannot capture with reasonable parameter counts, the learned model will be inaccurate.
  • domain assumption The finite-difference approximations to spatial derivatives are sufficiently accurate at the available grid resolution.
    In Section 3.1, spatial derivatives are computed with finite differences. If the data is under-resolved, the derivative estimates will be noisy or biased, which would degrade the learned PDE.
  • standard math The numerical ODE solver (Runge-Kutta) advances the state accurately enough, and backpropagation through the solver is stable.
    The training relies on discretize-then-optimize with an explicit Runge-Kutta solver. This assumes the solver's error remains small over the integration horizon and that gradients flow through the solver without vanishing or exploding.

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Cite this review

Pith. "Pith review of NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data." pith.science (2026). https://pith.science/paper/K6KHZPNR

@misc{pith2026190803190,
  author       = {Pith},
  title        = {Pith review of: NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K6KHZPNR}},
  note         = {Machine review of arXiv:1908.03190}
}
read the original abstract

We propose a neural network based approach for extracting models from dynamic data using ordinary and partial differential equations. In particular, given a time-series or spatio-temporal dataset, we seek to identify an accurate governing system which respects the intrinsic differential structure. The unknown governing model is parameterized by using both (shallow) multilayer perceptrons and nonlinear differential terms, in order to incorporate relevant correlations between spatio-temporal samples. We demonstrate the approach on several examples where the data is sampled from various dynamical systems and give a comparison to recurrent networks and other data-discovery methods. In addition, we show that for MNIST and Fashion MNIST, our approach lowers the parameter cost as compared to other deep neural networks.

Figures

Figures reproduced from arXiv: 1908.03190 by the authors.

Figure 2.1
Figure 2.1. Time-series data generated by the 3d Lorenz system and the corresponding learned [PITH_FULL_IMAGE:figures/full_fig_p005_2_1.png] view at source ↗
Figure 2
Figure 2. (a)) but requires more than double the number of parameters. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 2.2
Figure 2.2. Extracting and modeling of a nonlinear time-dependent spiral. Noisy data is given in [PITH_FULL_IMAGE:figures/full_fig_p006_2_2.png] view at source ↗
Figures from the paper (4 more)
Figure 2
Figure 2. Figure 2: (c), a nonlinear dictionary can fit the data and extract the correct pattern (the ‘squared’- [PITH_FULL_IMAGE:figures/full_fig_p007_2.png]
Figure 2.3
Figure 2.3. Figure 2.3: Comparing dynamics between the true (green) time-series, the solution generated by [PITH_FULL_IMAGE:figures/full_fig_p008_2_3.png]
Figure 2.4
Figure 2.4. Figure 2.4: Learning reduced-order dynamics from fluid simulations. [PITH_FULL_IMAGE:figures/full_fig_p010_2_4.png]
Figure 3.1
Figure 3.1. Figure 3.1: Burgers’ Equation Example. The surfaces at the terminal time simulated by the [PITH_FULL_IMAGE:figures/full_fig_p012_3_1.png]

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