Pith. sign in

REVIEW 3 major objections 7 minor 108 references

An introduction to Neural Networks for Physicists

T0 review · 3 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read One pendulum can teach a physics student the four major neural-network uses: fitting, solving, compressing, and discovering.

desk verdict A useful teaching tutorial with real notebook assets, held back by a misstated theorem and an internally inconsistent SINDy success count; worth publishing after a careful correction pass. read the letter →

arxiv 2505.13042 v1 pith:7CM4CX3W submitted 2025-05-19 physics.ed-ph physics.comp-ph

G. Café de Miranda , Gubio G. de Lima , Tiago de S. Farias This is my paper · ORCID
classification physics.ed-phphysics.comp-ph
keywords NeuralnetworksPhysicseducationTutorialSimplependulumPhysics-informedAutoencodersSINDyEquationdiscovery
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

This tutorial paper sets out to show that a physicist who knows only undergraduate mechanics can learn the core ideas of neural networks through a single familiar system, the pendulum. The authors walk from the perceptron and backpropagation to deep architectures, then present four pendulum applications of increasing complexity: a one-neuron network that estimates gravitational acceleration, a physics-informed network that solves the small-angle equation, an autoencoder that measures the system's one-dimensional latent space and filters image noise, and a SINDy autoencoder that rediscovers the nonlinear pendulum equation from images. The paper's practical claim is that each of these steps is reproducible with the companion notebooks, so the reader learns both the concepts and the implementation. If the demonstrations work as described, the tutorial offers a low-cost route into physics-informed machine learning for students and researchers outside the field.

What carries the argument

The carrying object is the loss function: each application works by building the physics, or the physics to be discovered, into the cost rather than leaving the network to free-form pattern match. Example 1 uses $L = \frac{1}{M}\sum_i (\mathrm{data}_i - \theta_0 \cos(\sqrt{g'/\ell}\, t_i))^2$; Example 2 adds the ODE residual and the initial condition to the data term; Example 3 minimizes reconstruction error $\lVert x - \psi(\varphi(x))\rVert^2$ through an encoder-decoder pair; and Example 4 minimizes $L_{\mathrm{SINDy}} = L_{\mathrm{recon}} + \lambda_1 L_{\dot{x}} + \lambda_2 L_{\dot{z}} + \lambda_3 \lVert \Xi \rVert_1$, where $\Theta(z)$ is a library of candidate functions and $\Xi$ is thresholded to force sparsity. The theoretical warrant offered for why such networks can represent the needed functions is Theorem II.1, stated as a two-layer universal approximation result; the paper calls it the Kolmogorov-Arnold theorem.

What would settle it

Compare the paper's equation (15) with the statement of the Kolmogorov-Arnold theorem in the two references cited for it. The standard theorem states an exact representation on a compact cube with inner functions built for the specific target function, while the paper's version claims an approximate-and-exact equality for any continuous function on $\mathbb{R}^N$ and counts $O(N^2)$ univariate functions. A reader can settle the matter by reading the cited sources and checking whether they contain the equality and domain printed in the paper; they do not, which shows the stated theorem is not the cited theorem and the tutorial's foundation, as printed, is not sound.

Watch

Extended reading notes

Core claim

The core demonstration is that the four standard ways neural networks meet physics — supervised parameter estimation, physics-informed solution of ordinary differential equations, unsupervised latent-dimensionality estimation, and sparse model discovery — can each be taught on the same pendulum. Example 1 fits $g$ by composing the network output with $\theta_0 \cos(\sqrt{g'/\ell}\, t)$ and minimizing the squared error against simulated angle data. Example 2 adds the harmonic-oscillator residual $d^2\theta/dt^2 + (g/\ell)\theta$ and the initial condition to the loss. Example 3 trains a convolutional autoencoder to reconstruct pendulum images and finds that a latent space of dimension one is enough, with dimension two tolerating up to 40 percent added noise. Example 4 constrains the latent dynamics to lie in a library $\Theta(z)$ with sparse coefficients $\Xi$ and reports that eight of ten initializations recover an equation of the form $d^2 z/dt^2 + \sin z = 0$, with the other two recovering the linearization. The intended moral is that neural networks are not black-box substitutes for physics but tools that can be constrained by, and can recover, physical laws.

Load-bearing premise

The load-bearing premise is that neural networks can approximate any continuous function well enough for these exercises to work, and the paper grounds that premise in Theorem II.1; as stated the theorem is not correct, so even if the notebook examples run, the tutorial's conceptual foundation is unreliable.

Editorial extensions

If this is right

  • A student who follows the four notebooks can, without prior machine-learning coursework, recover a physical constant, solve an ODE, estimate a system's intrinsic dimensionality, and discover a governing equation from image data.
  • Embedding physical constraints in the loss reduces the amount of training data and epochs needed: the PINN reaches a comparable loss in about half the epochs of the pure data-fit example, and the SINDy model needs roughly 15 percent of the data other architectures would.
  • The autoencoder's latent space can act as a dimensionality detector: one latent coordinate reconstructs the pendulum's state, and enlarging the latent space to two coordinates raises the noise tolerance from 20 to 40 percent.
  • SINDy-type loss terms can yield an explicit, interpretable equation rather than a black box, since the sparse coefficient vector $\Xi$ identifies which terms in the library are active.
  • The same SINDy formulation extends to multi-variable systems such as the Lorenz attractor when cross terms are added to the library $\Theta$.

Reading between the lines

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

  • A natural test of the tutorial's pedagogical bet is to run the companion notebooks on a different but equally familiar system — the harmonic oscillator or a falling body — and see whether the four exercises transfer without extra instruction; the paper does not run that test.
  • The reported 8-in-10 SINDy success rate suggests that reporting a seed sweep and the coefficient-pruning schedule would strengthen reproducibility, since initialization alone changes whether the exact or the linearized equation is found.
  • The tutorial's practical demonstrations would survive a corrected foundation: replacing the printed Theorem II.1 with the standard universal approximation statement (density of multilayer perceptron outputs on compact sets) would preserve every one of the four exercises, since the theorem supplies the warrant, not the code.
  • The denoising result at 20-40 percent noise could be turned into a quantitative benchmark: measure reconstruction error versus noise level and compare against principal component analysis at the same latent dimension, since the paper notes PCA would be forced to linearize.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. This paper is a tutorial aimed at physics undergraduates and engineering students. It introduces perceptrons, feed-forward neural networks, backpropagation, convolutional networks, and autoencoders, and then presents four Jupyter-notebook applications to a single physical system, the pendulum: supervised estimation of g from a known small-angle solution; a physics-informed neural network (PINN) solving the small-angle ODE; an autoencoder that compresses pendulum images into a low-dimensional latent space and denoises them; and a SINDy-autoencoder that attempts to discover the nonlinear pendulum equation from image data. The paper argues that this sequence demonstrates how neural networks can perform parameter fitting, ODE solving, latent-dimensionality estimation, and model discovery, and it directs readers to a GitHub repository for the notebooks.

Significance. If the technical content were accurate, this would be a valuable educational resource. The pedagogical structure, with four increasing-complexity tasks built around one familiar physical system, is effective, and the examples are checked against the known analytical pendulum solution rather than only against their own predictions. The availability of accompanying notebooks is a genuine strength, and the SINDy capstone is an ambitious and appropriate topic for the intended audience. The main problems are localized: a misstated mathematical theorem, a numerically inconsistent worked example, and a self-contradictory account of the SINDy results. All three are reparable, but until they are fixed the manuscript cannot serve as a dependable introduction.

major comments (3)
  1. [Section II.B, Theorem II.1 and Eq. (15)] Theorem II.1 is not a correct statement of the Kolmogorov-Arnold theorem, and it is not the universal approximation theorem that the surrounding discussion needs. The statement says a continuous function g 'can be approximated by' f(x) = sum_q phi_q(sum_p phi_{q,p}(x_p)) and then writes '= g(x)'; as written it is an exact equality, not an approximation. It also treats phi_q and phi_{q,p} as arbitrary given nonlinear functions, whereas the theorem asserts the existence of such univariate functions for a particular target g. Moreover, the Kolmogorov-Arnold representation theorem alone does not imply that a standard two-layer MLP with a fixed activation function and trainable weights can approximate every continuous function; that is the content of the universal approximation theorem (Cybenko; Hornik et al.). Because this theorem is the paper's only justification for the claim that a chain of weighted sums and activations can approximate the desired function, the mathematical foundation of the tutorial is currently incorrect. Please replace this theorem with an accurate statement and keep the two theorems clearly separated.
  2. [Section III.D, Example 4 (SINDy results)] The reported success rate for model discovery is self-contradictory. The text reads 'Among the 10 initializations we tested, 8 recovered the correct model, i.e., equation (23), while the other 2 did not recover any model' and immediately adds that in some initializations the network recovered the model of equation (24) (the small-angle linear equation) and in others recovered equations of the form z'' = xi0 sin z + xi1 z. Equation (24) is not equation (23), and a retained z term alongside sin z is also not equation (23) unless xi1 = 0 and xi0 equals the normalized coefficient. If the linear or mixed forms were successful runs, the '8 recovered the correct model' statement is wrong; if they were failures, 'did not recover any model' is wrong. This matters because Example 4 is the capstone demonstration that neural networks can discover governing equations. Please give an exact breakdown of the ten runs, define the success criterion, and provide random seeds or a pinned version of the notebook so the claim is checkable.
  3. [Section II.A.2, linear regression walkthrough] The step-by-step regression example is numerically inconsistent. Equation (11) initializes b = -0.138, but the batch computation uses b = -0.13; the first entry of wt+b is written as -0.138, which agrees with the former, not the latter, and for t = 0.25 the two choices give -0.01375 and -0.00575, respectively. After the first update the text reports w_new = 0.4906 and b_new = -0.1464, but using those updated weights the ReLU outputs are not the stated [0, 0.036, 0.178, 0.32, 0.46]; they are [0, 0, 0.0989, 0.2216, 0.3442], and the reported post-update loss 0.0643 does not follow from either set of numbers. Since the paper explicitly promises a step-by-step guide, these arithmetic errors need to be corrected and the whole walkthrough rechecked.
minor comments (7)
  1. [Abstract and Introduction] Grammar and spelling need a pass; examples include 'neural network and their basic concepts', 'Austrália' in the affiliation, and inconsistent capitalization in the enumerated list of applications.
  2. [Section II.A.1, Listing 1] The code uses variables x and y that are not defined in the snippet (the text defines x_j and y_j); as printed the code will raise a NameError. Listing 2 defines these variables correctly; the two listings should be made consistent.
  3. [Appendix A, near Table I] The sentence 'On Table I there are some of the main cost functions used in ML' refers to Table I, which actually lists activation functions; Table II is the cost-function table.
  4. [Section III.D] 'Lorentz attractors' should be 'Lorenz attractors'.
  5. [Figure 6 caption] The caption lists predictions at epochs 0, 1, 2, and 100 but refers only to red, green, and blue lines; please specify the color-to-epoch correspondence.
  6. [Section III.C and the abstract] The experiment estimates the dimensionality of the image manifold for a single simulated trajectory, but the text repeatedly says it estimates 'the dimensionality of the parameter space.' Please adjust the wording to avoid suggesting that the autoencoder is measuring the number of physical parameters of the pendulum.
  7. [Supplementary material entry [73]] The repository is cited without a commit hash or DOI; for a tutorial whose numerical claims readers are expected to reproduce, please pin the exact version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the four pendulum examples are externally validated against analytic solutions or the fitted equation itself, and no result reduces by construction to its inputs.

full rationale

The paper is a tutorial whose central demonstrations are known-answer tests rather than derivations from first principles. Example 1 fits g through the known small-angle solution (Eq. 25) and compares the fitted curve to the data; this is parameter estimation, not a circular prediction. Example 2 solves the harmonic-oscillator ODE with a PINN by minimizing the ODE residual (Eq. 28) plus initial and boundary terms, then checks against the analytic solution (Eq. 25); the validation target is external to the training loss. Example 3 trains an autoencoder on images generated from the pendulum ODE and checks that a one-dimensional latent space reconstructs them; the expected dimensionality is known in advance, so the exercise is a controlled demonstration rather than a self-referential derivation. Example 4 uses the standard SINDy-autoencoder loss (Eqs. 39-44) on data generated from Eq. (23), with a library that contains sin(z) and z; the recovered sparse equation is compared with the known pendulum equation. That is a benchmark of the method, not a prediction equivalent to its fitted inputs. The SINDy paragraph is internally inconsistent ('8 recovered the correct model' while also reporting recovered linear and mixed sin/z models), which is a reproducibility/correctness problem but not circularity. Section II's 'Universal Approximation (Kolmogorov-Arnold)' theorem (Eq. 15) is misstated as written, conflating exact Kolmogorov-Arnold representation with approximate MLP universality; again this is a mathematical correctness issue, not circular reasoning. No load-bearing self-citations appear: the only author self-citation ([69]) is for an adapted figure, and the SINDy architecture and pendulum example are attributed to external works [17,106]. The notebooks [73] are offered as reproducibility support, not as the justification of any derived claim. Because no claim in the paper reduces by construction to its own inputs, the circularity score is 0.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities. Its central demonstrations rest on standard calculus, the pendulum ODE, and a misstated universal approximation theorem. The free parameters are ordinary ML hyperparameters and fitted quantities from the examples; none are claimed as new physics.

free parameters (7)
  • Learning rate eta (perceptron example) = 0.01
    Chosen by hand in Section II A.1; used in the Iris classification update rule.
  • Learning rate eta (linear regression example) = 0.05
    Chosen by hand in Section II A.2 for the velocity-time fit.
  • Loss weights lambda_1, lambda_2, lambda_3 (SINDy) = lambda_1=5e-4, lambda_2=5e-5, lambda_3=1e-5
    Chosen by hand in Section III D to balance reconstruction, derivative, and sparsity terms.
  • Sparsity threshold = 0.1
    Coefficients with |xi|<0.1 are pruned in Section III D.
  • Latent space dimensions (autoencoder) = 1 and 2
    Chosen in Section III C to test dimensionality of the pendulum data.
  • Gravity g' (Example 1) = Not reported
    The neural network outputs a fitted value for gravitational acceleration in Section III A; the text does not give the recovered number or its error.
  • SINDy coefficients Xi = Recovered equation (23) in 8/10 runs
    The coefficients of the function library are fitted to the image data in Section III D; the paper reports that some runs instead yield the linearized equation (24) or fail.
assumptions (6)
  • standard math The chain rule and calculus of differentiable functions apply to network training (backpropagation).
    Used throughout Section II C to derive weight updates; standard.
  • ad hoc to paper A two-layer neural network with enough neurons can approximate any continuous function (Theorem II.1).
    The theorem is stated in Section II B but is mathematically inaccurate as written (conflates approximation with exact representation and misapplies Kolmogorov-Arnold); the tutorial uses this to motivate neural network expressivity.
  • domain assumption The simple pendulum ODE d^2 theta/dt^2 + (g/l) sin theta = 0 describes the physical system.
    Equation (23) is assumed to be the true model for all four examples; this is a textbook assumption.
  • domain assumption For small angles, sin(theta) approximately equals theta.
    Used in Examples 1 and 2 (Equations 24 and 28) to linearize the pendulum ODE.
  • domain assumption The pendulum state can be described by a single variable theta(t), so one latent dimension should suffice for the autoencoder.
    Section III C states this expectation; it follows from the pendulum ODE but is presented as motivation for the latent dimensionality test.
  • domain assumption Images of the pendulum are generated from theta(t) and derivatives can be computed numerically from smooth images.
    Section III D assumes images x(t) are smooth and that numerical differentiation is valid, and that the encoder outputs a latent variable z that is an affine transformation of theta.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An introduction to Neural Networks for Physicists." pith.science (2026). https://pith.science/paper/7CM4CX3W

@misc{pith2026250513042,
  author       = {Pith},
  title        = {Pith review of: An introduction to Neural Networks for Physicists},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CM4CX3W}},
  note         = {Machine review of arXiv:2505.13042}
}
read the original abstract

Machine learning techniques have emerged as powerful tools to tackle various challenges. The integration of machine learning methods with Physics has led to innovative approaches in understanding, controlling, and simulating physical phenomena. This article aims to provide a practical introduction to neural network and their basic concepts. It presents some perspectives on recent advances at the intersection of machine learning models with physical systems. We introduce practical material to guide the reader in taking their first steps in applying neural network to Physics problems. As an illustrative example, we provide four applications of increasing complexity for the problem of a simple pendulum, namely: parameter fitting of the pendulum's ODE for the small-angle approximation; Application of Physics-Inspired Neural Networks (PINNs) to find solutions of the pendulum's ODE in the small-angle regime; Autoencoders applied to an image dataset of the pendulum's oscillations for estimating the dimensionality of the parameter space in this physical system; and the use of Sparse Identification of Non-Linear Dynamics (SINDy) architectures for model discovery and analytical expressions for the nonlinear pendulum problem (large angles).

Figures

Figures reproduced from arXiv: 2505.13042 by the authors.

Figure 1
Figure 1. Diagram illustrating the intersection of Artificial Intel [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the perceptron’s computational steps. The [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. A sample from the Iris dataset, showing the first five rows. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Graph of the parameter space (w1, w2) (on the plane) for a loss function L(w1, w2) (vertical axis). When minimizing a loss function, an initial parameter value is chosen, and at each iteration, the gradient descent method determines the direction to follow until the di…
Figure 5
Figure 5. Figure 5: Evolution of the perceptron’s decision boundaries on the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Linear regression fitting behavior using a perceptron over [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: Pictorial representation of a Convolutional Neural Net [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Examples of convolution uses for edge detection. (a) Orig [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: Result of Example 1. a) Evolution of the cost function [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: Result of Example 2 - solving the differential equation [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: Pictorial representation of a neural network [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: Comparison between the original image and the recon [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: Comparison between the original image with added noise [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 15
Figure 15. Figure 15: 3D visualization in a) and 2D in b) of the latent space [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: Generalization of Equation (33) for a multivariable sys￾tem. The SINDy autoencoder performs a search among various func￾tions in a library Θ, where the derivative of the latent space z is a linear combination of the library with coefficients given by Ξ. This equation …
Figure 17
Figure 17. Figure 17: Multiple loss functions of the SINDy autoencoder model [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
Figure 18
Figure 18. Figure 18: Jupyter Notebook image comparing the input and output [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]
Figure 19
Figure 19. Figure 19: Activation functions (above) cited in Table [PITH_FULL_IMAGE:figures/full_fig_p024_19.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

108 extracted references · 55 canonical work pages

  1. [1]

    i.e., each elementMij is the average of the elements of a2×2 submatrix ofB 4×4

    Given a block of pixels from a larger image,B 4×4, i.e., a submatrix of the input data matrix, then 12 pooling(B4×4) =M 2×2, whereMij = 1 4 i+2,j+2X k=i,l=j Bkl. i.e., each elementMij is the average of the elements of a2×2 submatrix ofB 4×4

  2. [2]

    Note that after such an operation, the size of the processed input data is now com- pressed into a smaller matrix

    From the same block of pixelsB 4×4, taking the largest of the elements of a2x2submatrixM ij = maxi+2,j+2 k=i,l=j (Bkl), provides a different kind of pooling. Note that after such an operation, the size of the processed input data is now com- pressed into a smaller matrix. Reducing the dimensionality of the data decreases the network’s processing time and ...

  3. [3]

    ∆l′ = (y[n] l′ −ˆyl′) d dz[n] l′ fl′(z[n] l′ )(19)

    Start by calculating the deviation of thelossfunction from the true value with respect to an outputland its derivative but applying the chain rule only once. ∆l′ = (y[n] l′ −ˆyl′) d dz[n] l′ fl′(z[n] l′ )(19)

  4. [4]

    (18) ∆l′←∆ l′ ˆM d dw[k] il z[k] l (20) 13

    Now, since the derivatives d dz[k] l′ fl′(z[k] l′ )were calcu- lated during thefeedforwardstep where we computed fl′(z[k] l′ ), retrieve them from memory and update delta with the matrix composition ˆMfrom the equation. (18) ∆l′←∆ l′ ˆM d dw[k] il z[k] l (20) 13

  5. [5]

    03- Example 1,

    Repeat steps 1 and 2 for alll ′. Now take the average of the resulting deltas∆, and update the weightw [k] il as we did in the equation.(5) e (6) w[k] il,new←w [k] il,old−η∆(21) b[k] l,new←b [k] l,old−η∆(22) Thebackpropagationalgorithm has limitations that can compromise its performance and applicability in deep NNs. One of the main limitations is the pro...

  6. [6]

    Géron, A. (2017). Hands-on Machine Learning with Scikit- Learn, Keras, and TensorFlow. O’Reilly Media. ISBN: 978-1- 4920-3264-9

    2017

  7. [7]

    Brunton, S. L. and Kutz, J. N. Data Driven Science and Engi- neering. (2017)

    2017

  8. [8]

    Reviews of Modern Physics.91, 045002 (2019)

    Carleo, Giuseppe et al, Machine learning and the physical sci- ences. Reviews of Modern Physics.91, 045002 (2019)

    2019

Show all 108 references
  1. [9]

    Schmidhuber, J. (2015). Deep learning in neural networks: An overview. Neural Networks, 61, 85-117. ISSN 0893-6080

    2015

  2. [10]

    Uddin, S., Haque, I., Lu, H. et al. Comparative performance analysis of K-nearest neighbour (KNN) algorithm and its dif- ferent variants for disease prediction. Sci Rep 12, 6256 (2022)

    2022

  3. [11]

    k-Nearest neighbour classifiers: (with Python examples)

    Cunningham, Padraig, and Sarah Jane Delany. "k-Nearest neighbour classifiers: (with Python examples)." arXiv preprint arXiv:2004.04523 (2020)

    2020 arXiv

  4. [12]

    An Introduction to Logistic Regression Analysis and Reporting

    Peng, Chao-Ying Joanne, et al. “An Introduction to Logistic Regression Analysis and Reporting.” The Journal of Educa- tional Research, vol. 96, no. 1, pp. 3–14,(2002)

    2002

  5. [13]

    Logistic Regression: From Art to Sci- ence

    Bertsimas, D., King, A. Logistic Regression: From Art to Sci- ence. Statistical Science, 32, 367-384, (2017)

    2017

  6. [14]

    What is a support vector machine?

    Noble, W. What is a support vector machine?. Nat Biotechnol 24, 1565–1567 (2006)

    2006

  7. [15]

    C., de Carvalho, A

    Lorena, A. C., de Carvalho, A. C. P. L. F. Uma Introdução às Support Vector Machines. Revista De Informática Teórica E Aplicada, 14(2), 43–67,(2007)

    2007

  8. [16]

    Decision trees: a recent overview

    Kotsiantis, S.B. Decision trees: a recent overview. Artif Intell Rev 39, 261–283 (2013)

    2013

  9. [17]

    What are decision trees? Nat Biotechnol

    Kingsford C, Salzberg SL. What are decision trees? Nat Biotechnol. Sep;26(9):1011-3 (2008)

    2008

  10. [18]

    Introduction to machine learning for the sciences

    Neupert, Titus, et al. "Introduction to machine learning for the sciences." arXiv preprint arXiv:2102.04883 (2021)

    2021 arXiv

  11. [19]

    Machine Learning Algorithms: A ref- erence guide to popular algorithms for data science and ma- chine learning

    Giuseppe Bonaccorso. Machine Learning Algorithms: A ref- erence guide to popular algorithms for data science and ma- chine learning. Packt Publishing, (2017)

    2017

  12. [20]

    Machine learning and quantum devices

    Marquardt, F. Machine learning and quantum devices. SciPost Physics Lecture Notes (2021)

    2021

  13. [21]

    and Courville, A

    Goodfellow, I., Bengio, Y . and Courville, A. Deep Learning. MIT Press (2016). Available at: http://www.deeplearningbook.org

    2016

  14. [22]

    Champion, K., Lusch, B., Kutz, J. N. and Brunton, S. L. 22 Data-driven discovery of coordinates and governing equa- tions. Proc. Natl. Acad. Sci. U.S.A. 116, 22445–22451 (2019)

    2019

  15. [23]

    Silver, D., Huang, A., Maddison, C. et al. Mastering the game of Go with deep neural networks and tree search. Nature 529, 484–489 (2016). https://doi.org/10.1038/nature16961

    2016 doi

  16. [24]

    Mnih, V ., Kavukcuoglu, K., Silver, D. et al. Human-level control through deep reinforcement learning. Nature 518, 529–533 (2015). https://doi.org/10.1038/nature14236

    2015 doi

  17. [25]

    Solving Rubik’s Cube with a Robot Hand

    OpenAI, et al. Solving Rubik’s Cube with a Robot Hand. arXiv:1910.07113, 2019. https://doi.org/10.48550/arXiv.1910.07113

    1910 doi arXiv

  18. [26]

    Gligorov, V V and Williams, M, Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree, Journal of Instrumentation.02, vol 8

  19. [27]

    and Whiteson, D

    Baldi, P., Sadowski, P. and Whiteson, D. Searching for exotic particles in high-energy physics with deep learning. Nat Com- mun 5, 4308 (2014)

    2014

  20. [28]

    A living re- view of machine learning for particle physics

    Feickert, Matthew, and Benjamin Nachman. "A living re- view of machine learning for particle physics." arXiv preprint arXiv:2102.02770 (2021)

    2021 arXiv

  21. [29]

    Machine learning phases of matter

    Carrasquilla, J., Melko, R. Machine learning phases of matter. Nature Phys 13, 431–434 (2017)

    2017

  22. [30]

    and Huber, S

    Van Nieuwenburg, E., Liu, YH. and Huber, S. Learning phase transitions by confusion. Nature Phys 13, 435–439 (2017)

    2017

  23. [31]

    S., Isayev, O., Roitberg, A

    Smith, J. S., Isayev, O., Roitberg, A. E. (2017). ANI-1: an ex- tensible neural network potential with DFT accuracy at force field computational cost. Chemical Science, 8(4), 3192-3203. The Royal Society of Chemistry

    2017

  24. [32]

    Physics-informed machine learning: A survey on problems, methods and applications

    Hao, Zhongkai, et al. "Physics-informed machine learning: A survey on problems, methods and applications." arXiv preprint arXiv:2211.08064 (2022)

    2022 arXiv

  25. [33]

    Physics-Informed machine learning,

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-Informed machine learning,” Nature Reviews Physics, vol. 3, no. 6, pp. 422–440, 2021

    2021

  26. [34]

    When physics meets machine learn- ing: A survey of physics-informed machine learning

    Meng, Chuizheng, et al. "When physics meets machine learn- ing: A survey of physics-informed machine learning." arXiv preprint arXiv:2203.16797 (2022)

    2022 arXiv

  27. [35]

    Scientific machine learning through physics–informed neural networks: Where we are and what’s next

    Cuomo, Salvatore, et al. "Scientific machine learning through physics–informed neural networks: Where we are and what’s next." Journal of Scientific Computing 92.3 (2022): 88

    2022

  28. [36]

    Physics-guided, physics-informed, and physics-encoded neural networks in scientific comput- ing

    Faroughi, Salah A., et al. "Physics-guided, physics-informed, and physics-encoded neural networks in scientific comput- ing." arXiv preprint arXiv:2211.07377 (2022)

    2022 arXiv

  29. [37]

    Physics informed neural networks (pinns) for fluid mechan- ics: A review,

    S. Cai, Z. Mao, Z. Wang, M. Yin, and G. E. Karniadakis, “Physics informed neural networks (pinns) for fluid mechan- ics: A review,” arXiv preprint arXiv:2105.09506, 2021

    2021 arXiv

  30. [38]

    A Review of Physics-Informed Machine Learning in Fluid Mechanics

    Sharma, P.; Chung, W.T.; Akoush, B.; Ihme, M. A Review of Physics-Informed Machine Learning in Fluid Mechanics. Energies 2023, 16, 2343

    2023

  31. [39]

    Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons,

    A. F. Psaros, X. Meng, Z. Zou, L. Guo, and G. E. Kar- niadakis, “Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons,” arXiv preprint arXiv:2201.07766, 2022

    2022 arXiv

  32. [40]

    Physics-guided deep learning for dynam- ical systems: A survey,

    R.WangandR.Yu,“Physics-guided deep learning for dynam- ical systems: A survey,” arXiv preprint arXiv:2107.01272, 2021

    2021 arXiv

  33. [41]

    Neural networks in quantum many-body physics: a hands-on tutorial

    Carrasquilla, Juan, and Giacomo Torlai. "Neural networks in quantum many-body physics: a hands-on tutorial." arXiv preprint arXiv:2101.11099 (2021)

    2021 arXiv

  34. [42]

    Artificial neural networks for photonic applications—from algorithms to implementation: tutorial,

    Pedro Freire, Egor Manuylovich, Jaroslaw E. Prilepsky, and Sergei K. Turitsyn, "Artificial neural networks for photonic applications—from algorithms to implementation: tutorial," Adv. Opt. Photon. 15, 739-834 (2023)

    2023

  35. [43]

    Froim, S.; Yosef, Erez ; Giryes, Raja ;Bahabad, Alon

    Hadad, B. ; Froim, S.; Yosef, Erez ; Giryes, Raja ;Bahabad, Alon. Deep learning in optics - a tutorial. Journal of Optics

  36. [44]

    A Review of Optical Neural Networks

    Zhang, D.; Tan, Z. A Review of Optical Neural Networks. Appl. Sci. 2022, 12, 5338

    2022

  37. [45]

    Quantum Computing and Ma- chine Learning on an Integrated Photonics Platform

    Zhu, H.; Lin, H.; Wu, S.; Luo, W.; Zhang, H.; Zhan, Y .; Wang, X.; Liu, A.; Kwek, L.C. Quantum Computing and Ma- chine Learning on an Integrated Photonics Platform. Informa- tion 2024, 15, 95

    2024

  38. [46]

    Palmieri, A.M., Kovlakov, E., Bianchi, F. et al. Experimental neural network enhanced quantum tomography. npj Quantum Inf 6, 20 (2020)

    2020

  39. [47]

    Adaptive quantum state tomogra- phy with neural networks

    Quek, Y ., Fort, S.; Ng, H.K. Adaptive quantum state tomogra- phy with neural networks. npj Quantum Inf 7, 105 (2021)

    2021

  40. [49]

    Three ways to solve par- tial differential equations with neural networks—a review,

    J. Blechschmidt and O. G. Ernst, “Three ways to solve par- tial differential equations with neural networks—a review,” GAMM-Mitteilungen, vol. 44, no. 2, p. e202100006,( 2021)

    2021

  41. [50]

    In- tegrating physics-based modeling with machine learning: A survey,

    J. Willard, X. Jia, S. Xu, M. Steinbach, and V . Kumar, “In- tegrating physics-based modeling with machine learning: A survey,” arXiv preprint arXiv:2003.04919, vol. 1, no. 1, pp. 1–34, 2020

    2003 arXiv

  42. [51]

    Adv.6,eaay2631(2020)

    Silviu-Marian Udrescu, Max Tegmark ,AI Feynman: A physics-inspired method for symbolic regression.Sci. Adv.6,eaay2631(2020)

    2020

  43. [53]

    Li, Z., Kovachki, N.B., Azizzadenesheli, K., Liu, B., Bhat- tacharya, K., Stuart, A.M., and Anandkumar, A. (2020). Fourier Neural Operator for Parametric Partial Differential Equations. ArXiv, abs/2010.08895

    2020 arXiv

  44. [54]

    Optical machine learning with inco- herent light and a single-pixel detector

    Jiao, Shuming, et al. "Optical machine learning with inco- herent light and a single-pixel detector." Optics letters 44.21 (2019): 5186-5189

    2019

  45. [55]

    Arti- ficial intelligence and machine learning in optics: tutorial,

    Ksenia Yadav, Serge Bidnyk, and Ashok Balakrishnan, "Arti- ficial intelligence and machine learning in optics: tutorial," J. Opt. Soc. Am. B 41, 1739-1753 (2024)

    2024

  46. [56]

    Neuralpde: Automating physics- informed neural networks (pinns) with error approximations

    Zubov, Kirill, et al. "Neuralpde: Automating physics- informed neural networks (pinns) with error approximations." arXiv preprint arXiv:2107.09443 (2021)

    2021 arXiv

  47. [57]

    Adv.6,eaay2631(2020).DOI:10.1126/sciadv.aay2631

    Silviu-Marian Udrescu, Max Tegmark, AI Feynman: A physics-inspired method for symbolic regression.Sci. Adv.6,eaay2631(2020).DOI:10.1126/sciadv.aay2631

    2020 doi

  48. [58]

    PhyCV: the first physics-inspired com- puter vision library

    Zhou, Yiming, et al. "PhyCV: the first physics-inspired com- puter vision library." arXiv preprint arXiv:2301.12531 (2023)

    2023 arXiv

  49. [59]

    Physics-informed computer vision: A review and perspectives

    Banerjee, Chayan, et al. "Physics-informed computer vision: A review and perspectives." ACM Computing Surveys (2024)

    2024

  50. [60]

    ChatGPT: A comprehensive review on background, applications, key challenges, bias, ethics, limita- tions and future scope

    Ray, Partha Pratim. “ChatGPT: A comprehensive review on background, applications, key challenges, bias, ethics, limita- tions and future scope.” Internet of Things and Cyber-Physical Systems (2023)

    2023

  51. [61]

    Convolutional neu- ral networks for image classification,

    N. Jmour, S. Zayen and A. Abdelkrim, "Convolutional neu- ral networks for image classification," 2018 International Con- ference on Advanced Systems and Electric Technologies (IC ASET), Hammamet, Tunisia, pp. 397-402,(2018)

    2018

  52. [62]

    The Application and Im- provement of Deep Neural Networks in Environmental Sound Recognition

    Lin, Y .-K.; Su, M.-C.; Hsieh, Y .-Z. The Application and Im- provement of Deep Neural Networks in Environmental Sound Recognition. Appl. Sci., 10, 5965, (2020)

    2020

  53. [63]

    Natural Language Processing with Improved Deep Learning Neural Networks

    YiTao Zhou, "Natural Language Processing with Improved Deep Learning Neural Networks", Scientific Programming, 8 23 pages, (2022)

    2022

  54. [64]

    Fisher,R. A.. (1988). Iris. UCI Machine Learning Repository. https://doi.org/10.24432/C56C76

    1988 doi

  55. [65]

    Scikit-learn: Machine learning in Python

    Pedregosa, Fabian, et al. "Scikit-learn: Machine learning in Python." the Journal of machine Learning research 12: 2825- 2830, (2011)

    2011

  56. [66]

    Understanding the dif- ficulty of training deep feedforward neural networks

    Glorot, Xavier, and Yoshua Bengio. "Understanding the dif- ficulty of training deep feedforward neural networks." Pro- ceedings of the thirteenth international conference on artificial intelligence and statistics. JMLR Workshop and Conference Proceedings, (2010)

    2010

  57. [67]

    Review and comparison of commonly used activation functions for deep neural networks

    Szandała, Tomasz. "Review and comparison of commonly used activation functions for deep neural networks." Bio- inspired neurocomputing: 203-224, (2021)

    2021

  58. [68]

    Implicit neural representations with periodic activation functions

    Sitzmann, Vincent, et al. "Implicit neural representations with periodic activation functions." Advances in neural information processing systems 33: 7462-7473. (2020)

    2020

  59. [69]

    Dawid, A. et al. Modern applications of machine learning in quantum sciences. arXiv:2204.04198 (2022)

    2022

  60. [70]

    A review on applications of genetic algorithm for artificial neural network

    Al Tobi, Maamer Ali Saud, et al. "A review on applications of genetic algorithm for artificial neural network." International Journal of Advance Computational Engineering and Network- ing 4.9 (2016): 50-54

    2016

  61. [71]

    Neural network structure optimization by simulated annealing

    Kuo, Chun Lin, Ercan Engin Kuruoglu, and Wai Kin Victor Chan. "Neural network structure optimization by simulated annealing." Entropy 24.3 (2022): 348

    2022

  62. [72]

    An Overview of Stochastic Gradient De- scent in Machine Learning

    del Rosario, Mason. "An Overview of Stochastic Gradient De- scent in Machine Learning." (2019)

    2019

  63. [73]

    Survey of optimization algorithms in modern neural net- works

    Abdulkadirov, Ruslan, Pavel Lyakhov, and Nikolay Nagornov. "Survey of optimization algorithms in modern neural net- works." Mathematics 11.11 (2023): 2466

    2023

  64. [74]

    Pulse modulation for enhanced control in NMR-based Quantum Devices

    Café de Miranda, G. Pulse modulation for enhanced control in NMR-based Quantum Devices. Universidade Federal do ABC, Janeiro 2024

    2024

  65. [75]

    Grid search, random search, genetic algorithm: a big comparison for NAS

    Liashchynskyi, Petro, and Pavlo Liashchynskyi. "Grid search, random search, genetic algorithm: a big comparison for NAS." arXiv preprint arXiv:1912.06059 (2019)

    2019 arXiv

  66. [76]

    An overview of stochastic gradient-based methods

    Gower, Robert, and Telecom Paris. "An overview of stochastic gradient-based methods." (2019)

    2019

  67. [77]

    Kingma and Jimmy Ba

    Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. arXiv:1412.6980, 2014

    2014 arXiv

  68. [78]

    Café de Miranda, G., Lima, G. G. Notebooks acompan- hando o artigo. Repositório GitHub,https://github. com/Coffee4MePlz/Notebooks_NN_Physics

  69. [79]

    Introduction to Neural Network

    Safrin. Introduction to Neural Network. Medium, https://medium.com/@safrin1128/ introduction-to-neural-network-9fdff448f43f. Acessado em: 15 de fevereiro de 2025

    2025

  70. [80]

    Learning internal representations by error propagation,

    D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning internal representations by error propagation,” California Uni- vSan Diego La Jolla Inst for Cognitive Science, Tech. Rep., (1985)

    1985

  71. [81]

    Hinton, G

    LeCun, Y ., Bengio, Y . Hinton, G. Deep learning. Nature 521, 436–444 (2015)

    2015

  72. [82]

    Long short-term memory,

    S. Hochreiter and J. Schmidhuber, “Long short-term memory,” Neural computation, vol. 9, no. 8, pp. 1735–1780, (1997)

    1997

  73. [83]

    A survey of transformers

    Lin, Tianyang, et al. "A survey of transformers." AI open 3: 111-132 (2022)

    2022

  74. [84]

    An introduction to autoencoders

    Michelucci, Umberto. "An introduction to autoencoders." arXiv preprint arXiv:2201.03898 (2022)

    2022 arXiv

  75. [85]

    Electronics 8(3), (2019)

    Alom MZ, Taha TM, Yakopcic C, et al, A state-of-the-art survey on deep learning theory and architectures. Electronics 8(3), (2019)

    2019

  76. [86]

    Neural Network Architectures

    Herberg, Evelyn. "Neural Network Architectures." arXiv preprint arXiv:2304.05133 (2023)

    2023 arXiv

  77. [87]

    Hopfield networks is all you need

    Ramsauer, Hubert, et al. "Hopfield networks is all you need." arXiv preprint arXiv:2008.02217 (2020)

    2020 arXiv

  78. [88]

    Liu, Z. et al. KAN: Kolmogorov-Arnold Networks. Preprint at http://arxiv.org/abs/2404.19756 (2024)

    2024 arXiv

  79. [89]

    Pytorch: An imperative style, high- performance deep learning library

    Paszke, Adam, et al. "Pytorch: An imperative style, high- performance deep learning library." Advances in neural infor- mation processing systems 32 (2019)

    2019

  80. [90]

    Tensor- Flow: Large-scale machine learning on heterogeneous sys- tems, Software available from tensorflow.org

    Martín Abadi, Ashish Agarwal, Paul Barham, et al. Tensor- Flow: Large-scale machine learning on heterogeneous sys- tems, Software available from tensorflow.org. (2015)

    2015

  81. [91]

    JAX: composable transformations of Python+ NumPy programs

    Bradbury, James, et al. "JAX: composable transformations of Python+ NumPy programs." (2018)

    2018

  82. [92]

    Automatic differentiation in ma- chine learning: a survey

    Baydin, Atilim Gunes, et al. "Automatic differentiation in ma- chine learning: a survey." Journal of machine learning re- search 18.153 (2018): 1-43

    2018

  83. [93]

    Artigos no Medium

    Pham, K. Artigos no Medium. Medium,https:// medium.com/@khang.pham.exxact. Acessado em: 15 de fevereiro de 2025

    2025

  84. [94]

    Im- ageNet Classification with Deep Convolutional Neural Net- works

    Alex Krizhevsky, Ilya Sutskever, Geoffrey E Hinton. "Im- ageNet Classification with Deep Convolutional Neural Net- works". In:Advances in Neural Information Processing Sys- tems, vol. 25, ed. by F. Pereira et al., Curran Associates, Inc., 2012

    2012

  85. [95]

    VisualBackProp: efficient visualization of CNNs

    Mariusz Bojarski, Anna Choromanska, Krzysztof Choroman- ski, Bernhard Firner, Larry Jackel, Urs Muller, Karol Zieba. "VisualBackProp: efficient visualization of CNNs". (2017). arXiv preprint arXiv:1611.05418

    2017 arXiv

  86. [96]

    Feature Detection with Automatic Scale Selec- tion

    Lindeberg, T. Feature Detection with Automatic Scale Selec- tion. Int. J. Comput. Vis. 30, 79–116 (1998)

    1998

  87. [97]

    and Murphey, Y .L

    Park, J.M. and Murphey, Y .L. (2008). Edge Detection in Grayscale, Color, and Range Images. In Wiley Encyclopedia of Computer Science and Engineering, B.W. Wah (Ed.)

    2008

  88. [98]

    Detecting salient blob-like image structures and their scales with a scale-space primal sketch: A method for focus-of-attention

    Lindeberg, T. Detecting salient blob-like image structures and their scales with a scale-space primal sketch: A method for focus-of-attention. Int. J. Comput. Vis. 11, 283–318 (1993)

    1993

  89. [99]

    Neocognitron: A self-organizing neural net- work model for a mechanism of pattern recognition unaffected by shift in position

    Fukushima, K. Neocognitron: A self-organizing neural net- work model for a mechanism of pattern recognition unaffected by shift in position. Biol. Cybernetics 36, 193–202 (1980)

    1980

  90. [100]

    Basodi, C

    S. Basodi, C. Ji, H. Zhang and Y . Pan. Gradient amplification: An efficient way to train deep neural networks. Big Data Min- ing and Analytics, vol. 3, no. 3, pp. 196-207, Sept. 2020, doi: 10.26599/BDMA.2020.9020004

    2020

  91. [101]

    Bashir, D., Montañez, G.D., Sehra, S., Segura, P.S., Lauw, J. (2020). An Information-Theoretic Perspective on Overfitting and Underfitting. In: Gallagher, M., Moustafa, N., Lakshika, E. (eds) AI 2020: Advances in Artificial Intelligence. AI 2020. Lecture Notes in Computer Scie...

    2020

  92. [102]

    Addressing the loss-metric mismatch with adaptive loss alignment

    Huang, Chen, et al. "Addressing the loss-metric mismatch with adaptive loss alignment." International conference on ma- chine learning. PMLR, 2019

    2019

  93. [103]

    Plateau phenomenon in gradient descent training of RELU networks: Explanation, quantification, and avoidance

    Ainsworth, Mark, and Yeonjong Shin. "Plateau phenomenon in gradient descent training of RELU networks: Explanation, quantification, and avoidance." SIAM Journal on Scientific Computing 43.5 (2021): A3438-A3468

    2021

  94. [104]

    Which neural net architectures give rise to ex- ploding and vanishing gradients?

    Hanin, Boris. "Which neural net architectures give rise to ex- ploding and vanishing gradients?." Advances in neural infor- mation processing systems 31 (2018)

    2018

  95. [105]

    Lusch, B., Kutz, J. N. and Brunton, S. L. Deep learning for universal linear embeddings of nonlinear dynamics. Nat Com- mun 9, 4950 (2018)

    2018

  96. [106]

    Gamelin, T. W. Complex Analysis. Springer New York, New York, NY (2001). doi:10.1007/978-0-387-21607-2. 24

    2001 doi

  97. [108]

    and Manzagol, P.-A

    Vincent, P., Larochelle, H., Bengio, Y . and Manzagol, P.-A. Extracting and composing robust features with denoising au- toencoders. in Proceedings of the 25th international confer- ence on Machine learning - ICML ’08 1096–1103 ACM Press (2008). doi:10.1145/1390156.1390294

    2008

  98. [109]

    C Gheller, F Vazza, Convolutional deep denoising au- toencoders for radio astronomical images, Monthly Notices of the Royal Astronomical Society, V ol- ume 509, Issue 1, January 2022, Pages 990–1009, https://doi.org/10.1093/mnras/stab3044

    2022 doi

  99. [110]

    and Petruccione, F

    Schuld, M. and Petruccione, F. Machine Learning with Quan- tum Computers (2021)

    2021

  100. [111]

    SINDy Pendulum

    Sillano, Pietro. SINDy Pendulum. Github repositoryhttps: //github.com/pietro-sillano/SindyPendulum APPENDIX A - CALCULA TION OF THE COST FUNCTION Activation Function f(x) Derivative Sigmoid 1 1+e−x f′(x) =f(x)(1−f(x)) Tanh tanh(x) f′(x) = 1−f(x) 2 ReLU max(0,x) f′(x) = ( 1ifx>...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.