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(U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A neural network that learns the numerical flux of a finite volume scheme cuts error up to tenfold versus Godunov's method while preserving exact conservation.

desk verdict A credible empirical extension of learned finite volume methods with public code, but the abstract oversells ENO/WENO wins and a missing consistency check leaves the 'solver' claim unsettled. read the letter →

arxiv 2505.23702 v1 pith:Q4NRXVV4 submitted 2025-05-29 cs.LG cs.NAmath.NA

Nathan Lichtlé , Alexi Canesse , Zhe Fu , Hossein Nick Zinat Matin , Maria Laura Delle Monache , Alexandre M. Bayen This is my paper · ORCID
classification cs.LGcs.NAmath.NA MSC 35L6565M0868T07
keywords neuralfinitevolumemethodshyperbolicconservationlawsnumericalfluxlearningweak-formresiduallossentropysolutionstrafficflowmodelingshockcapturingunsupervisedPDEsolving
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 claims that the numerical flux of a finite volume scheme can be learned instead of derived, and that doing so yields a solver that is simultaneously simpler and more accurate than hand-designed flux limiters. The authors introduce Neural Finite Volume (NFV), which replaces the flux formula of a finite volume method with a small convolutional network evaluated on a rectangular spatiotemporal stencil, while keeping the conservative update rule that makes cell-to-cell flow exactly mass-preserving. On seven scalar conservation laws, including Burgers' equation and six traffic-flow models, they report up to tenfold lower error than Godunov's method, errors below ENO and WENO on about half of the test equations, and accuracy approaching discontinuous Galerkin at much lower complexity. They further claim that the unsupervised variant, trained only on a weak-form residual of the PDE, selects the physically correct entropy solution without seeing any solution data. The same architecture, trained on one hour of Interstate 24 highway data, predicts nearly four hours of traffic density evolution more accurately than any classical scheme fitted to the same data.

What carries the argument

The object that carries the argument is the learned numerical flux $\hat{F}_{i\pm 1/2} = N(u_{i\pm 1/2}(a-1,b))$: a one-dimensional convolutional network that reads a stencil of $a$ spatial cells across $b$ past time steps and outputs the flux through a cell interface. It does the work of the flux function in a classical finite volume scheme, and the conservative update rule $u^{n+1}_i = u^n_i - \frac{\Delta t}{\Delta x}(\hat{F}_{i+1/2} - \hat{F}_{i-1/2})$ then enforces mass conservation exactly, because the flux leaving one cell enters its neighbor whatever the network outputs. The stencil classes $\mathrm{FV}^b_a$ and $\mathrm{NFV}^b_a$ organize the design space, placing Godunov's method in the 3-by-1 class and letting the authors push learned fluxes to 11-by-11 stencils that would be impractical to derive analytically. For the unsupervised variant, the training signal is a weak-form residual loss evaluated cell-wise against 250 compactly supported polynomial test functions of degree 50. The methodological claim is that this one learned object, the flux, carries the entire burden of accuracy, entropy selection, and generalization.

What would settle it

Evaluate any trained NFV at constant states: feed the network a uniform field $u$ in every stencil cell and compare its output $N(u,\ldots,u)$ to the physical flux $f(u)$. If the two disagree while the scheme still appears accurate, then measure the scheme's autoregressive error on a mesh substantially finer than the training mesh; a learned flux that is not consistent should show the error plateauing or growing as the mesh refines, whereas a genuinely consistent scheme would keep converging to the exact entropy solution.

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Extended reading notes

Core claim

The paper's central claim is that a learned numerical flux is a sufficient upgrade over classical finite volume solvers: replace the hand-derived flux $F(u^n_i, u^n_{i+1})$ with a neural network $N$ applied to an $a$-by-$b$ stencil of neighboring cells and past time steps, keep the exact update $u^{n+1}_i = u^n_i - \frac{\Delta t}{\Delta x}(\hat{F}_{i+1/2} - \hat{F}_{i-1/2})$, and accuracy improves by up to an order of magnitude while conservation remains exact by construction. The claim is made in two settings. In the supervised setting, NFV is trained on Riemann problem solutions and evaluated on harder piecewise-constant initial conditions with ten discontinuities; in the unsupervised setting, UNFV is trained purely against a weak-form residual loss with randomly sampled polynomial test functions, and the paper reports that it converges to the entropy solution across a range of flow functions. The method scales with stencil size — larger stencils such as NFV$^{11}_{11}$ add only modest parameter counts over NFV$^{1}_{3}$ yet cut error substantially — and it transfers to field data: trained on one hour of I-24 MOTION density data, NFV$^{11}_{11}$ beats the best calibrated Godunov fit both on the training day and on seven unseen days. The authors state clearly that no stability or convergence guarantees accompany these empirical results.

Load-bearing premise

The load-bearing premise is that a flux network trained on one family of stencils and equations keeps behaving like a valid numerical flux when the mesh is refined, the flux function changes, or the data is noisy — the network is never constrained to be consistent with the physical flux, and the paper offers empirical evidence but no stability or convergence proof.

Editorial extensions

If this is right

  • If the claims hold, standard first-order finite volume solvers can be swapped for a learned flux network with roughly tenfold lower error at nearly identical implementation and runtime cost.
  • Larger spatiotemporal stencils improve accuracy, which suggests that learned schemes can exploit stencil memory that manual flux design cannot feasibly use.
  • UNFV's empirical convergence to entropy solutions via weak-form residuals would mark a regime where purely residual-based training succeeds on hyperbolic equations, a setting where physics-informed neural networks are known to struggle.
  • The I-24 experiments imply that conservation structure alone, without knowledge of the exact flux function, is a strong enough inductive bias to predict real traffic congestion waves.
  • On mesh refinement below the training discretization, NFV and UNFV continue to track Godunov's error trend, which the authors read as evidence that the learned schemes converge.

Reading between the lines

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

  • I read the I-24 result as evidence that the conservation law's structure, not the specific flux law, is what carries predictive accuracy on real traffic data; a direct test would train NFV with no PDE information at all and compare against the weak-form variant to isolate the value of the equation.
  • Since the learned flux is never constrained to satisfy consistency $N(u,\ldots,u) = f(u)$, the most informative stress test would measure the network's output at constant states and check whether the fixed network remains accurate on meshes and initial-condition families far outside its training envelope.
  • The weak-form loss samples only 250 test functions per time step; enlarging or making the test-function family more oscillatory could reveal whether UNFV enforces the weak form genuinely or only on the sampled projection.
  • A natural extension the paper leaves implicit is applying the same weak-form training to non-convex or space-dependent fluxes, where entropy selection is delicate and classical flux limiters are carefully tuned.
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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

4 major / 5 minor

Summary. The paper introduces (U)NFV, a neural finite volume architecture in which a CNN learns numerical fluxes over spatiotemporal stencils while preserving conservation through the classical finite-volume update (Eq. (3)). Two training variants are proposed: supervised NFV, which minimizes an L2 error against reference solutions, and unsupervised UNFV, which minimizes a weak-form residual loss. The method is evaluated on Burgers' equation and six LWR traffic-flow models, comparing against Godunov, Lax-Friedrichs, Engquist-Osher, ENO, WENO, and discontinuous Galerkin solvers, and is also applied to experimental I-24 MOTION highway data. The paper claims up to 10x lower error than Godunov, superiority over ENO/WENO, and performance rivaling DG with less complexity, as well as the ability of UNFV to converge to entropy solutions without labeled solution data.

Significance. If substantiated, the proposed framework offers a simple, conservation-preserving, data-driven alternative to hand-designed numerical flux functions, with potential practical impact on traffic modeling and other hyperbolic PDE applications. The architecture is modular, the code and data are released, and the idea of learning extended stencils while keeping the conservative update is attractive. However, the central claims currently outrun the evidence: the abstract overstates the comparison with ENO/WENO; the learned flux is not required to be consistent with the physical flux, so the notion of convergence is not established; and the weak-form loss is not a faithful discretization of the continuous weak form. These gaps prevent the paper from currently supporting the conclusions it draws, although they are addressable with additional experiments and a more precise presentation.

major comments (4)
  1. [Abstract; Section 5.4 and Table 1] The abstract states that (U)NFV "outperforms ENO/WENO," but Table 1 contains direct counterexamples: for Burgers' equation, NFV1_3 reports L2 error 8.5e-4 versus WENO's 1.0e-4, and for Underwood, NFV1_3 reports 3.8e-4 versus WENO's 9.8e-5. The body text in Section 5.4 correctly qualifies the claim as "surpass ENO/WENO schemes on about half of the equations," which is at odds with the unqualified abstract language. The abstract should be revised to reflect the equation-dependent nature of the comparison, or the evaluation should be reported only for the settings where the claim holds.
  2. [Section 4 and Figure 5] The learned flux network N is defined without any requirement that it be consistent with the physical flux, i.e., N(u,...,u) = f(u), nor is it constrained to be monotone or scale-aware. The conservative update (3) is a discretization of Eq. (1) only if the numerical flux satisfies such a consistency condition. Without it, the mesh-refinement behavior in Figure 5 (error decreasing as Δt and Δx are refined) does not establish convergence in the numerical-analysis sense; it could reflect that the training and evaluation distributions are both sampled from Riemann-type piecewise constant data, making the model a curve-fit surrogate rather than a solver for (1). Section 7 explicitly concedes the absence of stability and convergence guarantees. The authors should add a mesh-refinement study on smooth initial data (e.g., a sine wave) and on initial conditions outside the Riemann training family, reporting the error at multiple resolutions; until this is done, the 'convergence' claim in Figure 5 is not supported.
  3. [Section 4.1.2 (weak-form loss)] The unsupervised loss Lw is not a faithful discretization of the continuous weak form of (1). The flux term f(û_i^n)[φ] at cell boundaries uses a single cell-centered value of f; when summed over adjacent cells, the interface contributions do not cancel because neighboring cells generally have different values of f(û), so the loss does not correspond to a conservative finite-volume weak residual. In addition, the time derivative is approximated by a one-point backward difference rather than by integrating the test function over the time interval, so the loss is not a weak form in time. Consequently, the assertion in Section 4.1.2 that UNFV "converges to the entropy solution" is not justified by the loss construction. The authors should validate this claim on a standard convex-flux problem with a known entropy solution, verifying the discrete entropy inequality (e.g., a Krushkov-entropy check), and either repair the loss discretization or present it as an ad hoc heuristic with strong empirical support, not as a weak-form residual.
  4. [Section 5.4 and Table 2] The claim that (U)NFV "rivals discontinuous Galerkin solvers" is only partially supported by Table 2. On Burgers' equation, NFV5_5 has L2 error 2.2e-4 versus DG's 1.0e-4, a factor of two difference; on Greenshields and Triangular the errors are close (4.6e-5 vs 4.2e-5 and 2.9e-4 vs 2.7e-4), but the comparison is missing a computational cost analysis. The phrase "far less complexity" in the abstract is not backed by any runtime, parameter-count, or memory comparison with DG or WENO. Please state the comparison in more precise terms (e.g., 'within a factor of two of DG on these equations') or provide the missing complexity measurements.
minor comments (5)
  1. [Table 2 caption and text] The table header in the text above Table 2 contains the typo 'NVF' instead of 'NFV' (e.g., 'the computational complexity of NVF').
  2. [Section 4.1.2] The definition of the test functions Φ is incomplete: the reader is told they are 'compactly supported polynomials of degree 50 over the spatial domain,' but the formula for Lw uses only φ and its boundary values; it is unclear whether φ is a function of x only and how the boundary values are evaluated. Please clarify the notation, and write the flux term explicitly, e.g., f(û_i^n)(φ(x_{i+1/2}) - φ(x_{i-1/2})).
  3. [Section 5.3] The phrase 'a more complicated set of a several hundred initial conditions' is ungrammatical; please rephrase to 'a set of several hundred initial conditions' and report the exact number used for evaluation.
  4. [Figure 5] The caption should state explicitly that the convergence plot is for the evaluation set of piecewise-constant initial conditions (Riemann-like data), not for smooth initial data; otherwise the reader may infer classical convergence behavior that does not apply to this setting.
  5. [Section 6.2] In the sentence 'NFV11 11 adds only 1728 parameters over NFV1 3', the subscript notation is inconsistent with the earlier NFV^b_a convention; please use consistent notation, e.g., NFV_{11}^{11} vs NFV_1^3 or spell out the parameter counts.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the learned flux is trained against held-out solutions or the PDE residual itself, and the reference solutions are external benchmarks.

full rationale

The paper's derivation chain is self-contained. NFV's flux network N is trained either with supervised MSE against solution data (Section 4.1.1) or with the weak-form residual loss Lw (Section 4.1.2), and the quantities being predicted—Lax-Hopf exact solutions on a finer grid, held-out piecewise-constant initial conditions with ten discontinuities, and unseen I-24 days—are external to the fitted weights. The Lax-Hopf references [4,5] and the I-24 dataset references [42,43] include paper co-authors, but they are parameter-free or externally collected benchmarks used to generate ground truth and do not themselves justify the model's predictions; hence they are not load-bearing self-citations. The manuscript explicitly concedes the absence of stability/convergence guarantees (Section 7) and that the weak-form loss alone does not guarantee entropy-solution convergence (Section 4.1.2), so the convergence and entropy claims are empirical claims about held-out data rather than consequences of the training objective. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors, and no known result is repackaged under new coordinates. Therefore no circular step is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central method is an empirical fit: the network weights, loss hyperparameters, and baseline calibration parameters are all fitted to data. The mathematical structure (conservation update) is standard, but the entropy-selection and generalization claims rely on unproven assumptions about the learned flux. No new physical entities are introduced.

free parameters (4)
  • Neural network weights (NFV and UNFV models) = Not reported for synthetic models; field data models have 1105 + 16*((a-1)*b + 1) parameters
    All reported accuracy comes from weights fitted to training data or residuals; this is the model itself, not a small ad hoc constant.
  • Weak-form test function count and degree = 250 randomly sampled polynomials of degree 50
    These hyperparameters are chosen by hand; the unsupervised loss and its entropy selection behavior depend on them.
  • Training horizon schedule and learning rates = Horizon increased 10 to 250 steps (synthetic) or 10 to 100 (field); LR decayed 1e-4 to 1e-8 (synthetic), 1e-3 to 1e-4…
    Chosen to stabilize autoregressive training; performance is sensitive to these schedules.
  • Calibrated Godunov flow parameters on I-24 = Optimized via broad search, values not listed in text
    The baselines in Tables 3-4 are fitted to training data; comparison fairness and conclusions depend on this fit.
assumptions (5)
  • standard math Entropy solutions exist and are unique for the considered scalar conservation laws and are computed exactly by the Lax-Hopf algorithm on a fine grid.
    Used to generate training and reference solutions (Section 5.3, Section E); the validity of the benchmark depends on these exact solutions.
  • domain assumption The CNN with kernel size a-1 and five 1x1 conv layers can represent adequate numerical flux functions for the tested stencils.
    No expressivity or approximation-bound analysis provided; success of training presupposes this.
  • ad hoc to paper The discretized weak-form loss (Section 4.1.2) with one-point time quadrature is an acceptable surrogate for the PDE residual.
    The loss is not the exact continuous weak form; no quadrature error or convergence analysis is given, yet the paper uses it to claim entropy-solution selection.
  • domain assumption The learned flux remains stable and accurate under long autoregressive rollout (hundreds of steps) for initial conditions outside the training class.
    Training uses up to 250 steps and simple Riemann problems; evaluation uses up to 1000 steps and piecewise constant ICs with 10 discontinuities.
  • ad hoc to paper For field data, boundary values and initial cell are sufficient to predict interior densities, ignoring lane changes and source terms.
    Section D.2 sets one boundary cell per side; real traffic has merges, exits, and noise, which the conservation law does not model.

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Cite this review

Pith. "Pith review of (U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs." pith.science (2026). https://pith.science/paper/Q4NRXVV4

@misc{pith2026250523702,
  author       = {Pith},
  title        = {Pith review of: (U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q4NRXVV4}},
  note         = {Machine review of arXiv:2505.23702}
}
read the original abstract

We introduce (U)NFV, a modular neural network architecture that generalizes classical finite volume (FV) methods for solving hyperbolic conservation laws. Hyperbolic partial differential equations (PDEs) are challenging to solve, particularly conservation laws whose physically relevant solutions contain shocks and discontinuities. FV methods are widely used for their mathematical properties: convergence to entropy solutions, flow conservation, or total variation diminishing, but often lack accuracy and flexibility in complex settings. Neural Finite Volume addresses these limitations by learning update rules over extended spatial and temporal stencils while preserving conservation structure. It supports both supervised training on solution data (NFV) and unsupervised training via weak-form residual loss (UNFV). Applied to first-order conservation laws, (U)NFV achieves up to 10x lower error than Godunov's method, outperforms ENO/WENO, and rivals discontinuous Galerkin solvers with far less complexity. On traffic modeling problems, both from PDEs and from experimental highway data, (U)NFV captures nonlinear wave dynamics with significantly higher fidelity and scalability than traditional FV approaches.

Figures

Figures reproduced from arXiv: 2505.23702 by the authors.

Figure 1
Figure 1. Prediction of entropy solutions of hyperbolic PDEs. Top: NFV5 5 prediction vs. the Godunov scheme for Burgers’ equation at a fixed time. Mid: Entropic solution u(t, x) for Burgers’ equation over domain (t, x) ∈ [0, 1]2 , and its corresponding initial condition. Bottom: Entropic solution u(t, x) for the LWR equation with different fluxes sharing the same initial condition. rely on weak solutions, Closed-form solution… view at source ↗
Figure 2
Figure 2. Example stencil for FV2 5 , which takes in a rectangular stencil of 2 time steps times 5 space cells to compute the next cell average (in red) using Equation (3) (specifically, both the in-flow and out-flow are computed using the illustrated 2x4 sub-stencils). Motivated by these limitations, our proposed NFV approach learns local update rules directly from data or PDE residuals. By preserving the fundamental conserv… view at source ↗
Figure 3
Figure 3. Comparison of numerical schemes across flow functions. Each cell shows the proportion of the evaluation set on which the row scheme outperforms the column scheme. DG, the only FEM tested, is rarely beaten. NFV1 3 and UNFV1 3 outperform other first-order schemes and rival higher-order ones, making them strong choices depending on the equation. 5.2 Equations The Lighthill-Whitham-Richards model [27, 28], known as LWR,… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Comparison of the final density of the Burgers’ equation (left) and LWR triangular [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Convergence plots on Greenshields’ flux. The L2 error is computed against the exact solution on the evaluation set for different mesh discretizations. We report both error average and standard deviation, on a log-log scale. The dashed vertical line illustrates the disc…
Figure 6
Figure 6. Figure 6: Autoregressive prediction of NFV11 11 (right) compared to the ground truth (left). Full results are shown in [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Flow models for LWR. We consider six different variants of the LWR PDE with the flows illustrated here, each mapping road density (veh/m) to traffic flow (veh/s). B Variants of LWR We consider six different LWR PDEs variants, each consisting of a different fundamental …
Figure 8
Figure 8. Figure 8: Exact solution for two Riemann problems (left, middle) and one piecewise-constant [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: I-24 MOTION illustration. High-definition camera poles are mounted along a portion of I-24 at regular intervals. This generates massive amounts of video data, which is processed through a software stack. The resulting data for a single day is shown in the time-space di…
Figure 10
Figure 10. Figure 10: Time-space diagrams of car trajectories extracted from the video, colour-coded [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Boundary conditions used by NFV during training and evaluation on the I-24 dataset. The figures show the input provided to the model: the initial condition at t = 0 on the left, and boundary conditions at x = 0 (bottom) and x = xmax (top). The model must then predict …
Figure 12
Figure 12. Figure 12: Colorbar showing density scale for all I-24 data heatmaps. [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]
Figure 13
Figure 13. Figure 13: Predictions of FV methods and trained NFV. Corresponding metrics are reported in [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: Predictions of best FV fit and trained NFV11 11. Godunov is derived by fitting a flow function on the prediction and comparing it against the ground truth; we keep the fitted Trapezoidal flow as it performed best (see [PITH_FULL_IMAGE:figures/full_fig_p023_14.png]

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Reference graph

Works this paper leans on

53 extracted references · 42 canonical work pages · cited by 2 Pith papers

  1. [1]

    American Mathematical Society, 2022

    Lawrence C Evans.Partial differential equations, volume 19. American Mathematical Society, 2022

    2022

  2. [2]

    Cam- bridge university press, 2002

    Randall J LeVeque.Finite volume methods for hyperbolic problems, volume 31. Cam- bridge university press, 2002

    work page 2002

  3. [3]

    Hyperbolic systems of conservation laws ii.Communications on Pure and Applied Mathematics, 10(4):537–566, 1957

    PD Lax. Hyperbolic systems of conservation laws ii.Communications on Pure and Applied Mathematics, 10(4):537–566, 1957

    work page 1957

  4. [4]

    Lax–hopf based incorporation of internal boundary conditions into hamilton–jacobi equation

    Christian G Claudel and Alexandre M Bayen. Lax–hopf based incorporation of internal boundary conditions into hamilton–jacobi equation. part i: Theory.IEEE Transactions on Automatic Control, 55(5):1142–1157, 2010

    work page 2010

  5. [5]

    Lax–hopf based incorporation of internal boundary conditions into hamilton-jacobi equation

    Christian G Claudel and Alexandre M Bayen. Lax–hopf based incorporation of internal boundary conditions into hamilton-jacobi equation. part ii: Computational methods. IEEE Transactions on Automatic Control, 55(5):1158–1174, 2010

    work page 2010

  6. [6]

    A discontinuous galerkin finite element method for hamilton–jacobi equations.SIAM Journal on Scientific computing, 21(2):666–690, 1999

    Changqing Hu and Chi-Wang Shu. A discontinuous galerkin finite element method for hamilton–jacobi equations.SIAM Journal on Scientific computing, 21(2):666–690, 1999

    work page 1999

  7. [7]

    The initial value problem for nonlinear hyperbolic equations in two independent variables.Ann

    PD Lax. The initial value problem for nonlinear hyperbolic equations in two independent variables.Ann. Math. Studies, 33(21):1–229, 1954. 11

    work page 1954

  8. [8]

    A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics.Sbornik: Mathematics, 47(8-9):357–393, 1959

    SK Godunov. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics.Sbornik: Mathematics, 47(8-9):357–393, 1959

    work page 1959

Show all 53 references
  1. [9]

    High order eno and weno schemes for computational fluid dynamics

    Chi-Wang Shu. High order eno and weno schemes for computational fluid dynamics. In High-order methods for computational physics, pages 439–582. Springer, 1999

    1999

  2. [10]

    Weighted essentially non-oscillatory schemes.Journal of computational physics, 115(1):200–212, 1994

    Xu-Dong Liu, Stanley Osher, and Tony Chan. Weighted essentially non-oscillatory schemes.Journal of computational physics, 115(1):200–212, 1994

    1994

  3. [11]

    The local discontinuous galerkin method for time-dependent convection-diffusion systems.SIAM Journal on Numerical Analysis, 35 (6):2440–2463, 1998

    Bernardo Cockburn and Chi-Wang Shu. The local discontinuous galerkin method for time-dependent convection-diffusion systems.SIAM Journal on Numerical Analysis, 35 (6):2440–2463, 1998

    1998

  4. [12]

    Fourier neural operator for para- metric partial differential equations.arXiv preprint arXiv:2010.08895, 2020

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhat- tacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for para- metric partial differential equations.arXiv preprint arXiv:2010.08895, 2020

    2010 arXiv

  5. [13]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators.Nature machine intelligence, 3(3):218–229, 2021

    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

  6. [14]

    Convolutional networks for images, speech, and time series.The handbook of brain theory and neural networks, 3361(10):1995, 1995

    Yann LeCun, Yoshua Bengio, et al. Convolutional networks for images, speech, and time series.The handbook of brain theory and neural networks, 3361(10):1995, 1995

    1995

  7. [15]

    Geometric deep learning: going beyond euclidean data.IEEE Signal Processing Magazine, 34(4):18–42, 2017

    Michael M Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Van- dergheynst. Geometric deep learning: going beyond euclidean data.IEEE Signal Processing Magazine, 34(4):18–42, 2017

    2017

  8. [16]

    Characterizing possible failure modes in physics-informed neural networks

    Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W Mahoney. Characterizing possible failure modes in physics-informed neural networks. Advances in neural information processing systems, 34:26548–26560, 2021

    2021

  9. [17]

    Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations.arXiv preprint arXiv:1711.10561, 2017

    Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations.arXiv preprint arXiv:1711.10561, 2017

    2017 arXiv

  10. [18]

    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

  11. [19]

    Ameya D Jagtap, Ehsan Kharazmi, and George Em Karniadakis. Conservative physics- informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems.Computer Methods in Applied Mechanics and Engineering, 365:113028, 2020

    2020

  12. [20]

    Understanding the behaviour of contrastive loss

    Feng Wang and Huaping Liu. Understanding the behaviour of contrastive loss. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 2495–2504, 2021

    2021

  13. [21]

    Limitations of physics informed machine learning for nonlinear two-phase transport in porous media.Journal of Machine Learning for Modeling and Computing, 1(1), 2020

    Olga Fuks and Hamdi A Tchelepi. Limitations of physics informed machine learning for nonlinear two-phase transport in porous media.Journal of Machine Learning for Modeling and Computing, 1(1), 2020

    2020

  14. [22]

    wpinns: Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws.SIAM Journal on Numerical Analysis, 62(2):811–841, 2024

    Tim De Ryck, Siddhartha Mishra, and Roberto Molinaro. wpinns: Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws.SIAM Journal on Numerical Analysis, 62(2):811–841, 2024. 12

    2024

  15. [23]

    Ppinn: Parareal physics-informed neural network for time-dependent pdes.Computer Methods in Applied Mechanics and Engineering, 370:113250, 2020

    Xuhui Meng, Zhen Li, Dongkun Zhang, and George Em Karniadakis. Ppinn: Parareal physics-informed neural network for time-dependent pdes.Computer Methods in Applied Mechanics and Engineering, 370:113250, 2020

    2020

  16. [24]

    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

  17. [25]

    Enhanced fifth order weno shock-capturing schemes with deep learning.Results in Applied Mathematics, 12: 100201, 2021

    Tatiana Kossaczká, Matthias Ehrhardt, and Michael Günther. Enhanced fifth order weno shock-capturing schemes with deep learning.Results in Applied Mathematics, 12: 100201, 2021

    2021

  18. [26]

    Roenet: Predicting discontinuity of hyperbolic systems from continuous data.International Journal for Numerical Methods in Engineering, 125(6):e7406, 2024

    Yunjin Tong, Shiying Xiong, Xingzhe He, Shuqi Yang, Zhecheng Wang, Rui Tao, Runze Liu, and Bo Zhu. Roenet: Predicting discontinuity of hyperbolic systems from continuous data.International Journal for Numerical Methods in Engineering, 125(6):e7406, 2024

    2024

  19. [27]

    On kinematic waves i

    Michael James Lighthill and G Be Whitham. On kinematic waves i. flood movement in long rivers.Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229(1178):281–316, 1955

    1955

  20. [28]

    P. I. Richards. Shock waves on the highway.Operations Research, 4:42–51, 1956

    1956

  21. [29]

    A study of traffic capacity

    Bruce D Greenshields, J Rowland Bibbins, WS Channing, and Harvey H Miller. A study of traffic capacity. InHighway research board proceedings, volume 14, pages 448–477. Washington, DC, 1935

    1935

  22. [30]

    Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings.Transportation Research Part B: Methodological, 42(9):759–770, 2008

    Nikolas Geroliminis and Carlos F Daganzo. Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings.Transportation Research Part B: Methodological, 42(9):759–770, 2008

    2008

  23. [31]

    Properties of a well-defined macroscopic fundamental diagram for urban traffic.Transportation Research Part B: Methodological, 45(3): 605–617, 2011

    Nikolas Geroliminis and Jie Sun. Properties of a well-defined macroscopic fundamental diagram for urban traffic.Transportation Research Part B: Methodological, 45(3): 605–617, 2011

    2011

  24. [32]

    An analysis of traffic flow.Operations research, 7(1):79–85, 1959

    Harold Greenberg. An analysis of traffic flow.Operations research, 7(1):79–85, 1959

    1959

  25. [33]

    Speed, volume and density relationships.Quality and theory of traffic flow, 1961

    Robin T Underwood. Speed, volume and density relationships.Quality and theory of traffic flow, 1961

    1961

  26. [34]

    JM Burgers. Math. examples illustrating relations occuring in the theory of turbulent fluid motion. akademie van wetenschappen.Amsterdam. 1. Sect., 17:2, 1939

    1939

  27. [35]

    A fractional burgers equation arising in nonlinear acoustics: theory and numerics.IFAC Proceedings Volumes, 46(23): 406–411, 2013

    Bruno Lombard, Denis Matignon, and Yann Le Gorrec. A fractional burgers equation arising in nonlinear acoustics: theory and numerics.IFAC Proceedings Volumes, 46(23): 406–411, 2013

    2013

  28. [36]

    DE Panayotounakos and D Drikakis. On the closed-form solutions of the wave, diffusion and burgers equations in fluid mechanics.ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 75(6):437–447, 1995

    1995

  29. [37]

    Traffic current fluctuation and the burgers equation.Japanese journal of applied physics, 17(5):811, 1978

    Toshimitsu Musha and Hideyo Higuchi. Traffic current fluctuation and the burgers equation.Japanese journal of applied physics, 17(5):811, 1978

    1978

  30. [38]

    Notes on the burgers equation.University of Marylan, 2011

    MARIA Cameron. Notes on the burgers equation.University of Marylan, 2011

    2011

  31. [39]

    A. Bressan. Leggi di conservazione.La Matematica nella Societa e nella Cultura, 2003

    2003

  32. [40]

    second order

    AATM Aw and Michel Rascle. Resurrection of" second order" models of traffic flow. SIAM journal on applied mathematics, 60(3):916–938, 2000. 13

    2000

  33. [41]

    Structural properties of solutions arising from a nonequilibrium traffic flow theory.Transportation Research Part B: Methodological, 34(7):583–603, 2000

    HM Zhang. Structural properties of solutions arising from a nonequilibrium traffic flow theory.Transportation Research Part B: Methodological, 34(7):583–603, 2000

    2000

  34. [42]

    I-24 motion: An instrument for freeway traffic science.Transportation Research Part C: Emerging Technologies, 155: 104311, 2023

    Derek Gloudemans, Yanbing Wang, Junyi Ji, Gergely Zachar, William Barbour, Eric Hall, Meredith Cebelak, Lee Smith, and Daniel B Work. I-24 motion: An instrument for freeway traffic science.Transportation Research Part C: Emerging Technologies, 155: 104311, 2023

    2023

  35. [43]

    Derek Gloudemans, Gergely Zachár, Yanbing Wang, Junyi Ji, Matt Nice, Matt Bunting, William Barbour, Jonathan Sprinkle, Benedetto Piccoli, Maria Laura Monache, Alexan- dre Bayen, Benjamin Seibold, and Daniel B. Work. So you think you can track?arXiv preprint arXiv:2309.07268, 2023

    2023 arXiv

  36. [44]

    Automatic vehicle trajectory data reconstruction at scale

    Yanbing Wang, Derek Gloudemans, Zi Nean Teoh, Lisa Liu, Gergely Zachár, William Barbour, and Daniel Work. Automatic vehicle trajectory data reconstruction at scale. arXiv preprint arXiv:2212.07907, 2022

    2022 arXiv

  37. [45]

    Afinitedifferencemethodforthenumericalcomputationofdiscontinuous solutions of the equations of fluid dynamics.Mathematicheckii Sbornik, 47:271–290, 1959

    S.K.Godunov. Afinitedifferencemethodforthenumericalcomputationofdiscontinuous solutions of the equations of fluid dynamics.Mathematicheckii Sbornik, 47:271–290, 1959

    1959

  38. [46]

    One-sided difference approximations for nonlinear conservation laws.Mathematics of Computation, 36(154):321–351, 1981

    Björn Engquist and Stanley Osher. One-sided difference approximations for nonlinear conservation laws.Mathematics of Computation, 36(154):321–351, 1981

    1981

  39. [47]

    second order

    A. Aw and M. Rascle. Resurrection of "second order" models of traffic flow.SIAM Journal on Applied Mathematics, 60:916–938, 2000

    2000

  40. [48]

    A non-equilibrium traffic model devoid of gas-like behavior.Trans- portation Research Part B: Methodological, 36(3):275–290, 2002

    H Michael Zhang. A non-equilibrium traffic model devoid of gas-like behavior.Trans- portation Research Part B: Methodological, 36(3):275–290, 2002

    2002

  41. [49]

    Parsing gigabytes of json per second.The VLDB Journal, 28(6):941–960, 2019

    Geoff Langdale and Daniel Lemire. Parsing gigabytes of json per second.The VLDB Journal, 28(6):941–960, 2019. 14 A Finite Volume Methods Several finite volume-based numerical schemes are studied in this work. They include the following common classical first-order schemes: God...

    2019

  42. [50]

    Evaluate the smoothness indicators: δ− =|f + i −f + i−1|, δ + =|f + i+1 −f + i |

  43. [51]

    •Otherwise, choose the stencil{f + i−1, f+ i }

    Select the stencil that minimizes the smoothness indicator: •Ifδ + < δ−, choose the stencil{f+ i , f+ i+1}. •Otherwise, choose the stencil{f + i−1, f+ i }

  44. [52]

    A similar approach is applied to computeˆf − i+ 1 2 using the right-biased stencil

    Perform linear interpolation to compute the numerical flux: ˆf + i+ 1 2 =f + i + 1 2 δ+ whereδ + is the difference between the selected stencil elements. A similar approach is applied to computeˆf − i+ 1 2 using the right-biased stencil. The final numerical flux at the interfa...

    2022

  45. [53]

    Each time step is represented as a separate input channel, for a total ofb input channels

    Using a CNN enables efficient vectorized computation over all stencils, which is equivalent to sliding a fully connected network along the input but significantly faster. Each time step is represented as a separate input channel, for a total ofb input channels. The output is a...

    2022

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.