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REVIEW 3 major objections 8 minor 23 references

Engineering application of physics-informed neural networks for Saint-Venant torsion

T0 review · 3 major / 8 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Physics-informed neural networks can compute Saint-Venant torsion constants and twist angles without meshes, with errors as low as 0.1 percent on simple cross-sections.

desk verdict A clean application of existing PINN variants to a classical problem, but the 2D validation table rests on analytical constants that don't match textbook formulas, so the headline accuracy claim is not yet supported. read the letter →

arxiv 2505.12389 v1 pith:P3O4UYWL submitted 2025-05-18 cs.LG

Su Yeong Jo , Sanghyeon Park , Seungchan Ko , Jongcheon Park , Hosung Kim , Sangseung Lee , Joongoo Jeon This is my paper · ORCID
classification cs.LG
keywords Saint-Venanttorsionphysics-informedneuralnetworksPrandtlstressfunctiontorsionalconstantvariable-scalingPINNparametricmesh-freePDEsolversPoissonequation
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 paper sets out to show that physics-informed neural networks (PINNs) can serve as practical solvers for Saint-Venant torsion problems, replacing the mesh generation and numerical integration of conventional finite-element workflows. It builds three solver variants: a standard PINN for the 2D Poisson equation of the Prandtl stress function, a variable-scaling PINN (VS-PINN) for shafts with sharp diameter transitions, and a parametric PINN that takes load parameters as extra inputs for real-time prediction. The reported results put the circle and square torsional constants within 0.1 percent of the paper's analytical references, the triangle within 3.0 percent, and the parametric surrogate at a $1.07\times 10^{-2}$ relative $L^2$ error over a range of forcing parameters. If those results hold, the practical payoff is a mesh-free path to torsional constants and twist angles that can be evaluated instantly for many design configurations.

What carries the argument

The central object is the scaled Prandtl stress function $\phi'$, which satisfies the Poisson equation $\nabla^2 \phi' = -2G$ with a constant (taken as zero) boundary value; the torsional constant is its integral, so the whole method rests on representing one smooth scalar field with a network and automatic differentiation. The VS-PINN mechanism is the spatial rescaling $x \mapsto x/N$, which maps a sharply varying solution profile onto a smoother one over an enlarged domain, making the residual easier for the optimizer to fit. The parametric PINN mechanism is to concatenate the parameters $(T,m,\sigma)$ with the spatial coordinate as network inputs and average the physics loss over the parameter space, so a single trained network maps any parameter set in range to its solution.

What would settle it

Recompute the torsional constant for a 0.2 m diameter circular cross-section with the classical formula $J = \pi R^4/2$, which gives about $1.57\times 10^{-4}$ m$^4$ for $R = 0.1$ m. The Table 1 analytical value is $4.63202\times 10^{-3}$; if the true analytical value is the former, then the claimed 0.1% error is measured against an unexplained reference, and the same check applied to the square and triangle entries would settle whether the benchmark is consistent.

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Extended reading notes

Core claim

The central claim is that the torsional constant $J$ of a prismatic bar can be obtained by training a neural network to solve $\nabla^2 \phi' = -2G$ for the scaled Prandtl stress function $\phi'$ with a zero boundary condition, then evaluating $J = \frac{2}{G}\int_A \phi'\,dA$; all three solvers are exercises in learning this single scalar field. For the 1D shaft with a sharp transition, the paper claims that rescaling the spatial coordinate by a factor $N$ before forming the loss function lowers the relative $L^2$ error from $9.7\times 10^{-3}$ to $1.1\times 10^{-3}$ at the same epoch count and no extra per-epoch cost. For parametric studies, treating $(T,m,\sigma)$ as extra network inputs yields a surrogate with a $1.07\times 10^{-2}$ relative $L^2$ error that predicts twist solutions for unseen parameter combinations without retraining.

Load-bearing premise

The analytical benchmark values in Table 1 are never derived or cited, and the listed circle value does not match the classical torsion-constant formula for the stated 0.2 m geometry, so the reported error percentages rest on an unverified reference solution.

Editorial extensions

If this is right

  • A trained PINN outputs $\phi'$ continuously on the whole cross-section, so torsional constants for arbitrary shapes can be evaluated by sampling the network instead of building a mesh and assembling stiffness matrices.
  • The VS-PINN result makes stepped shafts and other stiff-transition geometries solvable at a $1.1\times 10^{-3}$ relative $L^2$ error with the same per-epoch cost as a standard PINN, removing the main accuracy bottleneck for such cases.
  • The parametric PINN turns a torsion solver into a surrogate that can be queried in real time for any torque parameters in the training range, which is what design optimization and digital-twin workflows require.
  • Because the paper fixes the PINN collocation points to match the finite-element grid, the two approaches can be compared resolution-for-resolution rather than accuracy-for-accuracy.
  • The three variants are presented as complementary tools: the standard PINN for accurate constants, VS-PINN for stiff sections, and the parametric PINN for parameter sweeps.

Reading between the lines

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

  • The parametric PINN could be coupled with an optimizer to solve inverse torsion design problems, such as finding a torque profile that produces a target twist distribution.
  • Feeding material properties or cross-section shape descriptors as additional parameter inputs would extend the same surrogate idea from one geometry to a family of geometries.
  • The variable-scaling idea is demonstrated only on the 1D shaft equation; applying it to 2D torsion with sharp re-entrant corners is a testable extension that should show the same gradient-smoothing benefit.
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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 / 8 minor

Summary. The paper develops and tests three physics-informed neural network (PINN) approaches for Saint-Venant torsion problems: a standard PINN that solves the 2D Poisson equation for the Prandtl stress function on circular, square, triangular, and irregular cross-sections; a variable-scaling PINN (VS-PINN) for a 1D torsion problem with a sharp transition in the torsional stiffness J(x); and a parametric PINN that takes the parameters of a Gaussian body force as additional inputs. The authors report relative errors as low as 0.1% for the 2D circular and square cases, a reduction in relative L2 error from 9.7E-3 to 1.1E-3 for VS-PINN compared with the baseline PINN, and a relative L2 error of 1.07E-2 for the parametric PINN. They conclude that these methods provide accurate, mesh-free alternatives to FEM for torsional analysis.

Significance. If the reported accuracies were properly validated, this would be a useful engineering demonstration that PINN-based solvers can compute torsional constants and twist angles without meshes, and that VS-PINN improves convergence for stiff coefficient transitions. The paper's positive features include the use of analytic and FEM references for the main comparisons, the clear formulation of the 1D and parametric test problems, and the explicit statement that no ground-truth data were used during training. However, the central 2D accuracy claim is currently unsupported because the analytical benchmark values are not stated and appear inconsistent with standard formulas for the stated geometries. In addition, the reference solutions for the 1D and parametric error measurements are not defined, and all results come from single training runs. These issues prevent the reader from verifying the paper's quantitative claims.

major comments (3)
  1. [Section 5.2, Table 1] The 'Analytical Solution' column is not derived or cited, and its values contradict standard Saint-Venant torsion constants for the geometries described in Section 4.1. For the stated circle of diameter 0.2 m, the standard value is J = πR^4/2 ≈ 1.57×10^-4 m^4, whereas Table 1 lists 4.63202×10^-3 m^4 (about 29.5 times larger); for the square, 0.1406 a^4 ≈ 2.25×10^-4 m^4 versus 5.11785×10^-3 m^4; and for the equilateral triangle, √3 a^4/80 ≈ 3.46×10^-5 m^4 versus 1.71155×10^-3 m^4. The ratios are shape-dependent, so they cannot be explained by a single missing factor or unit conversion. Because the reported relative errors for both PINN and ANSYS are computed with respect to this column, the central claim of 0.1–3.0% accuracy for the 2D solver is unverifiable until the analytical formulas are supplied and reconciled with Eq. (15).
  2. [Sections 5.3 and 5.4] The reference solutions used to compute the relative L2 errors for the VS-PINN experiment (9.7E-3 vs. 1.1E-3) and for the parametric PINN experiment (1.07E-2) are never defined. For Eq. (19), the coefficient J(x) contains a sigmoid transition, so no closed-form solution is given; for Eq. (21), an exact solution in terms of error functions exists but is not stated. Without a reproducible reference, these error values cannot be checked, which undermines the quantitative claims for the two advanced methods.
  3. [Sections 5.2–5.4] All neural-network results are reported for a single training run, with no seed statistics, no variance bars, and no description of how the reported errors were selected from the training history (e.g., 'highest accuracy'). Given that PINN training is sensitive to initialization, collocation-point sampling, and loss weights, the paper's conclusion of 'accuracy and robustness' is not supported by the presented evidence. The authors should report at least the mean and standard deviation over multiple seeds, or explicitly justify why a single run is representative.
minor comments (8)
  1. [Section 3.3 after Eq. (14)] The text says 'λr, λd > 0' but the total loss is defined with λb; this appears to be a typo.
  2. [Table 1 caption] The units of the tabulated values (presumably m^4) should be stated in the table or in the caption.
  3. [Section 5.4, Figure 11] The term 'true solutions' should be defined, and the caption should state how these exact solutions were computed.
  4. [Section 4.2, Eq. (19)] The problem statement does not explain how the boundary condition ϕ'(1) = 32/(πJ(1)) is derived after the normalization, and the notation for the twist angle is inconsistent with the earlier definition θ = αz in Eq. (2).
  5. [References] References [KP25a] and [KP25b] are the same paper cited twice; the duplicate entry should be removed.
  6. [Section 5.3, paragraph 2] The phrase 'the highest accuracy achieved was the 9.7E-3 relative L2-error' should be rephrased as 'the lowest error achieved' or 'the best relative L2-error'.
  7. [Section 5.2, Figure 7 description] The phrase 'temperature fields' is a typo; the figure shows the Prandtl stress function field.
  8. [Section 1, contributions] The contribution list states 'We propose the VS-PINN formulation' and 'we introduce a parametric PINN framework', but both methods are cited to prior work [KP25a, CJL+24]; the contributions should be rephrased as applications or extensions rather than new method proposals.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the PINN results are validated against external references, with only a minor non-load-bearing self-citation.

full rationale

The claimed derivation chain is not circular. The 2D torsional-constant solver solves the Poisson equation ∇²ϕ′ = −2G with ϕ′ = 0 on the boundary and computes J = (2/G)∫ϕ′ dA; the reference values are external analytical and ANSYS FEM results, and the paper explicitly states that ground truth was not used in training. The 1D VS-PINN section applies a published variable-scaling transformation from the authors' earlier JCP paper [KP25a, KP25b]; although this is a self-citation by two co-authors, the improvement claim is re-demonstrated in this paper's own experiments against a baseline PINN, so the citation is not load-bearing in a circular sense. The parametric PINN is presented as an application of cited prior work [CJL+24] and is tested within the trained parameter range, which is interpolation rather than extrapolation and does not reduce the result to its inputs. Two concerns fall outside the circularity definition: Table 1's 'Analytical Solution' values are never derived and appear inconsistent with textbook torsional constants for the stated 0.2 m geometries, and the phrase 'highest accuracy achieved' suggests test-based checkpoint selection; both are benchmarking and correctness risks, not circular reductions. Because the only self-citation is minor and non-load-bearing, the circularity score is 2.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim depends on the Saint-Venant formulation, the zero boundary condition for simply connected sections, hand-tuned loss weights and scaling factors, and unverified analytical reference values. No new physical entity is postulated.

free parameters (5)
  • boundary loss weight λb = 100,000 (λr=1)
    Section 4.1, Eq. (16); chosen by hand to enforce the zero boundary condition; two orders of magnitude above the residual weight.
  • VS-PINN scaling factor N = 2 and 4
    Section 5.3; the scaling factor is a problem-specific choice; N=1 is the standard PINN baseline.
  • VS-PINN loss weights = 1/N^4 for residual, 20 for boundary
    Section 5.3; hand-tuned to rebalance the scaled residual and boundary terms.
  • Parametric PINN loss weights = λr=4, λb=1
    Section 5.4; chosen without sensitivity analysis.
  • Parametric domain ranges = (T,m,σ) in [1,10]×[0.5,0.9]×[0.2,1]
    Section 5.4; arbitrary ranges defining the surrogate training space.
assumptions (6)
  • domain assumption Saint-Venant assumptions: constant rate of twist and identical free warping across cross-sections
    Section 2.1; the displacement field Eq. (1) and the Poisson equation rely on this.
  • domain assumption Homogeneous, isotropic, linear-elastic material with constant G
    Section 2.1; stress-strain relations and the constant -2G source term assume this.
  • domain assumption The Prandtl stress function can be set to zero on the whole boundary
    Section 4.1; valid for simply connected sections with one boundary; fails for multiply connected sections, which are not tested.
  • ad hoc to paper The normalized 1D problem Eq. (19) with J(x)=(r^4-(r-0.2)^4) and sigmoid transition faithfully models a sharp diameter transition
    Section 5.3; the sigmoid width, radii, and normalization are problem-specific choices.
  • domain assumption The analytical torsional constants used as references in Table 1 are correct
    Table 1; the formulas are not given, and the absolute values appear inconsistent with textbook values for the stated geometry sizes.
  • standard math Standard existence and uniqueness of the Poisson equation with Dirichlet boundary conditions
    Invoked implicitly in Sections 2 and 4 when posing the boundary-value problem.

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Cite this review

Pith. "Pith review of Engineering application of physics-informed neural networks for Saint-Venant torsion." pith.science (2026). https://pith.science/paper/P3O4UYWL

@misc{pith2026250512389,
  author       = {Pith},
  title        = {Pith review of: Engineering application of physics-informed neural networks for Saint-Venant torsion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3O4UYWL}},
  note         = {Machine review of arXiv:2505.12389}
}
read the original abstract

The Saint-Venant torsion theory is a classical theory for analyzing the torsional behavior of structural components, and it remains critically important in modern computational design workflows. Conventional numerical methods, including the finite element method (FEM), typically rely on mesh-based approaches to obtain approximate solutions. However, these methods often require complex and computationally intensive techniques to overcome the limitations of approximation, leading to significant increases in computational cost. The objective of this study is to develop a series of novel numerical methods based on physics-informed neural networks (PINN) for solving the Saint-Venant torsion equations. Utilizing the expressive power and the automatic differentiation capability of neural networks, the PINN can solve partial differential equations (PDEs) along with boundary conditions without the need for intricate computational techniques. First, a PINN solver was developed to compute the torsional constant for bars with arbitrary cross-sectional geometries. This was followed by the development of a solver capable of handling cases with sharp geometric transitions; variable-scaling PINN (VS-PINN). Finally, a parametric PINN was constructed to address the limitations of conventional single-instance PINN. The results from all three solvers showed good agreement with reference solutions, demonstrating their accuracy and robustness. Each solver can be selectively utilized depending on the specific requirements of torsional behavior analysis.

Figures

Figures reproduced from arXiv: 2505.12389 by the authors.

Figure 1
Figure 1. Comparison of schematic diagrams of PINN, VS-PINN and Parametric PINN. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Case study of four shapes. The Poisson equation with the Dirichlet boundary condition is [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Sensitivity study for grid size determination. The unified grid size was determined to be 0.005 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: 0.005 m grid modeling for each shape. The spatial coordinates of the grid generated in ANSYS [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Training loss by epoch. Training was terminated when the loss converged in all shapes. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Prandtl stress function field over epochs. It was observed that the field converged as learning [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Comparison of Prandtl stress function field: ANSYS vs PINN. [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Training dynamics for the 1D torsion problem using VS-PINN with the scales [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Comparison of the solution predictions of VS-PINN and standard PINN after training up to a [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: , where both the relative L 2 -error and the total loss were measured during training. Note that in this case, the errors are measure with respect to both physical and parametric variables. To further highlight the efficiency of our proposed method, we compare the pre…
Figure 11
Figure 11. Figure 11: Comparison between the true solutions usol(x; T, m, σ) (red) and the predictions ub(x; T, m, σ) (green) obtained by the parametric PINN across various parameter sets (T, m, σ). 16 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

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Works this paper leans on

23 extracted references · 15 canonical work pages

  1. [1]

    Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind

    Atilim Gunes Baydin, Barak A. Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res. , 18:153:1--153:43, 2017

    work page 2017

  2. [2]

    Saint venant’s torsion by the finite-volume method

    H Chen. Saint venant’s torsion by the finite-volume method. Master of Science Thesis, University of Virginia , 2019

    work page 2019

  3. [3]

    Parameterized physics-informed neural networks for parameterized pdes

    Woojin Cho, Minju Jo, Haksoo Lim, Kookjin Lee, Dongeun Lee, Sanghyun Hong, and Noseong Park. Parameterized physics-informed neural networks for parameterized pdes. arXiv preprint arXiv:2408.09446 , 2024

    arXiv 2024

  4. [4]

    Prandtl’s formulation for the saint--venant’s torsion of homogeneous piezoelectric beams

    Istv \'a n Ecsedi and Attila Baksa. Prandtl’s formulation for the saint--venant’s torsion of homogeneous piezoelectric beams. International Journal of Solids and Structures , 47(22-23):3076--3083, 2010

    work page 2010

  5. [5]

    Some analytical solutions for saint-venant torsion of non-homogeneous cylindrical bars

    Istv \'a n Ecsedi. Some analytical solutions for saint-venant torsion of non-homogeneous cylindrical bars. European Journal of Mechanics-A/Solids , 28(5):985--990, 2009

    work page 2009

  6. [6]

    Torsion of a non-circular bar

    Jan Francu, Petra Nov \'a ckov \'a , and Premysl Jan \' cek. Torsion of a non-circular bar. Eng. Mech , 19(1):45--60, 2012

    work page 2012

  7. [7]

    Cross-sectional analysis of beams subjected to saint-venant torsion using the green’s theorem and the finite difference method, 2022

    Valentin Fogang. Cross-sectional analysis of beams subjected to saint-venant torsion using the green’s theorem and the finite difference method, 2022

    work page 2022

  8. [8]

    Nvidia simnet : An ai-accelerated multi-physics simulation framework

    Oliver Hennigh, Susheela Narasimhan, Mohammad Amin Nabian, Akshay Subramaniam, Kaustubh Tangsali, Zhiwei Fang, Max Rietmann, Wonmin Byeon, and Sanjay Choudhry. Nvidia simnet : An ai-accelerated multi-physics simulation framework. In Maciej Paszynski, Dieter Kranzlm \"u ller, Valeria V. Krzhizhanovskaya, Jack J. Dongarra, and Peter M.A. Sloot, editors, Com...

    work page 2021

Show all 23 references
  1. [9]

    Galerkin solutions for the saint-venant torsion of prismatic bars with rectangular crosssections

    C Ike. Galerkin solutions for the saint-venant torsion of prismatic bars with rectangular crosssections. Advances in Modelling and Analysis A , 56(1):13--20, 2019

    2019

  2. [10]

    Solving saint venant torsion problems for rectangular beams using single finite fourier sine transform method

    Charles Chinwuba Ike. Solving saint venant torsion problems for rectangular beams using single finite fourier sine transform method. Journal of Mechatronics and Artificial Intelligence in Engineering , 5(1):31--41, 2024

    2024

  3. [11]

    Double finite sine transform method for saint-venant torsional analysis of beams with rectangular cross-section

    Charles Chinwuba Ike and Onyedikachi Aloysius Oguaghamba. Double finite sine transform method for saint-venant torsional analysis of beams with rectangular cross-section. In Proceedings of Sustainable Engineering and Industrial Technology Conference , pages 290--296, 2023

    2023

  4. [12]

    Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations

    Ameya D Jagtap and George Em Karniadakis. Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Communications in Computational Physics , 28(5), 2020

    2020

  5. [13]

    A finite element method for the saint-venant torsion and bending problems for prismatic beams

    CS Jog and Imrankhan Sherkhan Mokashi. A finite element method for the saint-venant torsion and bending problems for prismatic beams. Computers & Structures , 135:62--72, 2014

    2014

  6. [14]

    Physics-informed machine learning

    George Em Karniadakis, Ioannis G Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning. Nature Reviews Physics , 3(6):422--440, 2021

    2021

  7. [15]

    Vs-pinn: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving pdes with stiff behavior

    Seungchan Ko and Sanghyeon Park. Vs-pinn: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving pdes with stiff behavior. Journal of Computational Physics , 529:113860, 2025

    2025

  8. [16]

    The deep ritz method for parametric p -dirichlet problems

    Alex Kaltenbach and Marius Zeinhofer. The deep ritz method for parametric p -dirichlet problems. arXiv preprint arXiv:2207.01894 , 2022

    2022 arXiv

  9. [17]

    hp-vpinns: Variational physics-informed neural networks with domain decomposition

    Ehsan Kharazmi, Zhongqiang Zhang, and George Em Karniadakis. hp-vpinns: Variational physics-informed neural networks with domain decomposition. Computer Methods in Applied Mechanics and Engineering , 374:113547, 2021

    2021

  10. [18]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence , 3(3):218--229, 2021

    2021

  11. [19]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. 2021

    2021

  12. [20]

    Reduced-pinn: An integration-based physics-informed neural networks for stiff odes

    Pouyan Nasiri and Roozbeh Dargazany. Reduced-pinn: An integration-based physics-informed neural networks for stiff odes. arXiv preprint arXiv:2208.12045 , 2022

    2022 arXiv

  13. [21]

    Automatic differentiation in pytorch

    Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017

    2017

  14. [22]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics , 378:686--707, 2019

    2019

  15. [23]

    A-pinn: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations

    Lei Yuan, Yi-Qing Ni, Xiang-Yun Deng, and Shuo Hao. A-pinn: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations. Journal of Computational Physics , 462:111260, 2022

    2022

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