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REVIEW 3 major objections 5 minor 98 references

An approximate Riemann solver approach in Physics-Informed Neural Networks for hyperbolic conservation laws

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Roe-averaged Jacobians put shock jump conditions into PINN training

desk verdict A genuinely useful PINN tweak for shocked flows, with an overclaimed guarantee that should be softened before publication. read the letter →

arxiv 2506.11959 v2 pith:WBVJBQYM submitted 2025-06-13 physics.flu-dyn

Jorge F. Urbán , José A. Pons This is my paper · ORCID
classification physics.flu-dyn MSC 35L6565M0668T07 PACS 47.40.-i
keywords physics-informedneuralnetworksRiemannsolvershyperbolicconservationlawsRankine-HugoniotconditionsRoeaveragingshockcapturingrelativistichydrodynamicsEulerequations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that a small change inside a PINN's residual—swapping the Jacobian matrix for the Roe-averaged Jacobian only in detected compression regions—makes the network honor shock jump conditions without knowing shock speeds in advance. The authors generalize the earlier locally linearized PINNs by computing shock speeds from neighboring states with entropy conditions, and introduce LRPINNs built on Roe's approximate Riemann solver. They report that LRPINNs keep global conservation residuals nearly flat over time, reproduce sharp shocks in one-dimensional Euler and relativistic hydrodynamics, and extend to two-dimensional Riemann problems via divergence-based shock detection and dimensional splitting. A sympathetic reader cares because the method offers a path to using PINNs on genuinely discontinuous flows with conservation guarantees, matching or beating a high-order WENO solver on the benchmarked problems.

What carries the argument

The Roe-averaged Jacobian matrix $\tilde A$, defined by density-weighted averages $\tilde u$, $\tilde v$, $\tilde H$ (and the Eulderink–Mellema averaged state for relativistic hydrodynamics), satisfies the jump identity $F(U_R)-F(U_L)=\tilde A(U_R-U_L)$. The network's PDE residual $\partial_t U + \tilde A\,\partial_x U=0$ is computed with this modified matrix only inside the compression set $\Omega_s$ defined by a negative velocity gradient or negative divergence, so the shock-detection criterion decides where the modification acts. This machinery transfers the approximate Riemann solver's conservation property into the loss function.

What would settle it

On the Sod or Lax shock tube, run the LRPINN with the settings of Algorithm 2 and track $\int (U_{\mathrm{pred}}(x,t)-U_{\mathrm{pred}}(x,0))\,dx$ over time; if this conservation residual grows like the LLPINN curve in Figure 7 instead of staying near $10^{-3}$, the claim that the Roe replacement enforces jump conditions fails. Alternatively, evaluate the trained network on a fine uniform grid across the shock and check whether $F(U_R)-F(U_L) \approx s(U_R-U_L)$ with the predicted $s$; a clear violation indicates the property did not survive training.

Watch

Extended reading notes

Core claim

The central discovery is that enforcing the Roe property $F(U_R)-F(U_L)=\tilde A(U_R-U_L)$ locally, through the Roe-averaged Jacobian, is enough to turn the quasi-linear residual into a device that carries the jump conditions into the network training. Standard PINNs, even with viscous pretraining, mislocate shocks and lose contact waves; LLPINNs correctly predict shock speeds but violate conservation because their eigenvalue-substituted Jacobian does not satisfy the Roe property. LRPINNs replace the Jacobian only in regions where $\partial u/\partial x \le -M$ (or, in two dimensions, where $\nabla\cdot u \le -M$), with the Roe matrix built from neighbor states, and leave the true Jacobian elsewhere. The authors show that this yields accurate shock locations, sharp transitions, and conserved mass, momentum, and energy over time, and extends to two dimensions by applying the modification direction-by-direction with dimensional splitting.

Load-bearing premise

The load-bearing premise is that training on the quasi-linear residual with the Roe Jacobian in compression regions makes the network output satisfy the weak conservation law at the shock; the paper proves the Roe matrix has the jump property, but does not prove that residual minimization transfers this property to the network output.

Editorial extensions

If this is right

  • LRPINNs solve one-dimensional Riemann problems for Euler and relativistic Euler equations without any prior knowledge of shock speeds, matching or exceeding WENO-Z accuracy up to about 1600 grid points in the tested double-shock case.
  • Global conservation of mass, momentum, and energy stays approximately constant over time, unlike LLPINNs where conservation residuals grow with time (Figure 7).
  • The method correctly handles transonic rarefactions in a moving-frame Sod tube because the Roe replacement is limited to compression regions, avoiding the entropy-violating behavior of pure Roe solvers.
  • In two-dimensional Riemann problems, divergence-based detection plus dimensional splitting yields sharp shock and contact transitions while maintaining cylindrical symmetry (Problem 6).
  • The approach is not tied to Roe specifically: the authors state that other approximate solvers or the exact Riemann solver could be substituted by replacing eigenvalues and reconstructing the modified Jacobian.

Reading between the lines

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

  • The same selectively Roe-averaged residual principle could be transferred to other PINN formulations for conservation laws with available approximate Riemann solvers, such as shallow-water or magnetohydrodynamics, where Roe-type matrices exist; the authors gesture at this but do not test it.
  • Because the shock speed emerges from the Roe property rather than being an input, LRPINNs may be useful for inverse problems where shock trajectory or wave speed is the quantity to recover; that use is not explored here.
  • The sharper-shock-but-smoother-vortex trade-off observed in two dimensions suggests that a hybrid loss, mixing the Roe-modified residual with a spectral or multiscale term, could recover small-scale features; this is a testable design not addressed by the paper.
  • A direct comparison of conservation residual against a fully conservative PINN formulation, such as integral-form residuals, would clarify whether the local Roe replacement or the quasi-linear structure is the active ingredient.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes two modifications of locally linearized PINNs (LLPINNs) for hyperbolic conservation laws with shocks. First, a generalized LLPINN computes shock speeds from neighboring network states via Rankine-Hugoniot formulas and uses Lax entropy conditions to select which eigenvalue to replace, removing the need to prescribe shock velocities in advance. Second, LRPINNs replace the Jacobian in detected compression regions by a Roe-averaged Jacobian, aiming to enforce jump conditions and improve conservation. Both methods are extended to the 2D Euler equations using a divergence-based shock detector and direction-wise Jacobian modification. The authors test the methods on 1D Euler shock-tube problems, 1D relativistic hydrodynamics, and 2D Riemann problems, comparing against exact solutions, WENO-Z, and HLLC solvers. They report sharper shock resolution and better conservation than classical PINNs and LLPINNs, with LRPINN accuracy comparable to or better than WENO at moderate resolution.

Significance. If the central claims were fully established, the paper would make a useful contribution: it would provide a practical way to embed approximate Riemann solver information into PINNs without a priori shock speeds, and the numerical comparisons show consistent gains over standard PINNs and the original LLPINN. The detailed hyperparameter tables and algorithm listings are a strength, and the inclusion of relativistic and 2D problems broadens applicability. However, the central theoretical claim that jump conditions are 'ensured' is not proven; the manuscript's own conservation diagnostic shows only approximate conservation. The contribution is therefore better characterized as an empirical method with heuristic motivation, whose claims need to be scaled back or supplemented with a weak-form consistency argument.

major comments (3)
  1. [Section III.C, Eq. (50), and Algorithm 2] The argument that LRPINNs 'robustly ensure' jump conditions is not established. Eq. (50) is an algebraic identity for constant states U_L and U_R, but Algorithm 2 only uses the Roe-averaged Jacobian inside the pointwise residual D[U] = ∂U/∂t + A_roe ∂U/∂x evaluated at collocation points. Minimizing the sum of squared pointwise residuals does not minimize any weak-form residual, and a smooth network can satisfy the modified quasi-linear system without satisfying the integral conservation law. The paper's own conservation check in Fig. 7 shows a residual of order 1e-3, not machine precision, so the 'ensures' language is not supported. Please either add a proof or argument connecting residual minimization to Rankine-Hugoniot satisfaction, augment the loss with an explicit weak-form or integral conservation term, or soften the claims to 'encourages' or 'approximately enforces'.
  2. [Algorithm 2, steps 7-8] The left and right states are obtained by evaluating the network at x_s ± Δx, where Δx is a hyperparameter. Since the network is smooth and Δx is small (Table I, values around 0.001 to 0.02), U_L and U_R are not the two branches of a discontinuous Riemann state; in the limit Δx→0 their Roe average tends to A(U(x_s)), so the modification degenerates to the local Jacobian. The paper should justify that these neighboring values are meaningful Riemann states for the shock and report sensitivity to Δx; otherwise the mechanism is not demonstrably acting as a Riemann solver at an interface.
  3. [Section V, Conclusions] The statement that the generalized LLPINN 'ensures that the Rankine–Hugoniot relations (22) and entropy conditions (23) are satisfied by construction' is an overstatement. Algorithm 1 replaces an eigenvalue by s_i in the quasi-linear residual, which encodes the shock speed in the differential operator but does not by itself enforce the jump condition in the network output. The conclusion should be rephrased to describe the intended design and the observed empirical behavior rather than a guaranteed property.
minor comments (5)
  1. [Section IV.A, Problem numbering] The problem numbering is inconsistent: 'Problem 4' appears twice (Sod shock tube in a moving reference frame, and relativistic Sod shock tube), and 'Problem 5' appears twice (relativistic double shock wave and, implicitly, the next problem). Please renumber the 1D and 2D problems sequentially.
  2. [Eqs. (31)-(32)] The shock-speed formulas appear to be missing a square-root symbol and closing parentheses: the expressions should read s_- = u_L - a_L * sqrt( (1/(2γ)) * ((γ+1) p_R/p_L + γ-1) ) and similarly for s_+. Please check the typesetting.
  3. [Algorithm 3 and Algorithm 4] There are small typographical errors in the residual definitions: Algorithm 3 line 12 and Algorithm 4 line 11 contain a double plus sign (∂U/∂t + Ã ∂U/∂x + + B̃ ∂U/∂y), and Algorithm 3 step 5 writes '∂_y v' without the ∂ symbol. These should be corrected.
  4. [Abstract and Section V] The abstract says the method 'applies jump conditions through entropy constraints,' but the algorithms use Lax entropy conditions only to select which eigenvalue to modify; the jump conditions enter through the Rankine-Hugoniot shock-speed formulas. Please clarify this distinction.
  5. [Figure 7] The caption says 'conserved quantities' but does not specify which component or combination is plotted, nor whether the difference is normalized. Please state whether the plotted quantity is mass, momentum, energy, or a composite, and clarify the normalization.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: LRPINN and generalized LLPINN results are validated against exact and reference solvers, nothing is fitted to the target solution, and the only self-citations are non-load-bearing optimizer references.

full rationale

The derivation is self-contained against external benchmarks, so no circular step is identified. Generalized LLPINNs compute shock speeds from the closed-form Rankine-Hugoniot expressions (31)-(32) evaluated at the network's own neighboring states; these speeds are inputs derived from the physics, not parameters fitted to the exact solution, and the exact Riemann solution, WENO-Z solver, and 1D HLLC schemes are used only for post-hoc error comparison (Tables II-III), never as training data. The Roe modification is imported from the standard defining property of the Roe average, F(U_R)-F(U_L)=A_roe(U_R-U_L) (Eq. 50, attributed to Roe 1981 and Eulderink & Mellema 1995), which is stated as a required identity rather than derived as a prediction; hence there is no renaming of a known result, and no ansatz is smuggled in through self-citation. The conclusions' claim that RH relations are satisfied 'by construction' and that LRPINNs 'robustly ensure' jump conditions overstates what pointwise residual minimization guarantees, and the paper's own Figure 7 shows conservation residuals of order 1e-3 rather than machine precision; however, this is an inferential gap or correctness risk, not circularity, because the stated identity (50) is not defined in terms of the claimed conclusion and the central results (accurate shock locations and jumps matching exact solutions) are independently demonstrated. The paper itself flags its main limitations, admitting that LLPINNs fail to satisfy property (50) and that small-scale vortex structures are smoothed in Problems 7-8, which further weighs against a charge of disguised circularity. Self-citations [59,60] (Urbán, Stefanou, Pons; Kiyani, Shukla, Urbán, Darbon, Karniadakis) support only the quasi-Newton optimizer implementation, and [94] (Aloy, Pons, Ibáñez) is cited only as an example of classical flux reformulation; none is load-bearing for the physics claim. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. Verdict: no significant circularity (score 1).

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

The method relies on standard conservation-law theory plus several per-problem hyperparameters and two unproved modeling assumptions: that the Roe-modified residual enforces jump conditions, and that neighboring network states provide valid shock-side states. No new physical entities are introduced.

free parameters (6)
  • Shock detection threshold M (and Mx, My in 2D) = 0.05 to 0.1 in 1D; Mx and My from 1e-5 to 0.7
    Determines which residual points get the modified Jacobian. Values are chosen per problem in Table I and justified only qualitatively.
  • Neighbor spacing Delta x (and Delta y) = 0.001 to 0.7 across problems
    Defines the left and right states used to compute shock speeds and Roe averages. No sensitivity study is provided.
  • Viscous pretraining coefficient nu = 1e-3 to 3e-3 per problem
    Initializes the network for the inviscid phase and is central to convergence, but is set per problem without a stated criterion.
  • Viscous pretraining iterations = 1500, 2000, or 3000 depending on problem
    Length of the viscous warm-up phase, chosen per problem.
  • Gradient-annihilation constant a = not reported
    Appears in the residual homogenization factor in Eqs. (81) and (82); the value is not given in Table I or the text.
  • Network architecture (layers x neurons) = 7L20N, 6L20N, 6L30N
    Chosen per problem without ablations or sensitivity analysis.
assumptions (5)
  • domain assumption The solution satisfies the ideal-gas equation of state p=(gamma-1) rho epsilon with fixed gamma.
    Used throughout in Eqs. (25), (40), and the shock-speed formulas (31)-(32).
  • standard math Weak solutions satisfy the Rankine-Hugoniot condition (22) and Lax entropy inequalities (23) select physical shocks.
    These are classical results from conservation-law theory invoked in Section III.A.
  • ad hoc to paper Replacing the true Jacobian by the Roe-averaged Jacobian only in the detected compression region makes the PINN residual an adequate surrogate for the weak conservation law.
    This is the conceptual leap behind Algorithm 2. The paper proves the Roe property (50) but does not prove that residual minimization transfers this property to the network output. Discussed in Section III.C.
  • ad hoc to paper The network's values at x_s +/- Delta x during training are valid left and right states across the shock, so the computed shock speeds and Roe averages are meaningful.
    Required for Algorithms 1 through 4. No convergence analysis of this coupled estimation during training is given.
  • ad hoc to paper Pretraining with viscosity nu Laplacian U and then setting nu=0 leads to convergence to the inviscid solution.
    Introduced in Section IV and used for all problems. There is no guarantee that this warm start lands in the correct basin.

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Cite this review

Pith. "Pith review of An approximate Riemann solver approach in Physics-Informed Neural Networks for hyperbolic conservation laws." pith.science (2026). https://pith.science/paper/WBVJBQYM

@misc{pith2026250611959,
  author       = {Pith},
  title        = {Pith review of: An approximate Riemann solver approach in Physics-Informed Neural Networks for hyperbolic conservation laws},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WBVJBQYM}},
  note         = {Machine review of arXiv:2506.11959}
}
read the original abstract

This study enhances the application of Physics-Informed Neural Networks (PINNs) for modeling discontinuous solutions in both hydrodynamics and relativistic hydrodynamics. Conventional PINNs, trained with partial differential equation residuals, frequently face convergence issues and lower accuracy near discontinuities. To address these issues, we build on the recently proposed locally linearized PINNs (LLPINNs), which improve shock detection by modifying the Jacobian matrix resulting from the linearization of the equations, only in regions where the velocity field exhibits strong compression. However, the original LLPINN framework required a priori knowledge of shock velocities, limiting its practical utility. We present a generalized LLPINN method that dynamically computes shock speeds using neighboring states and applies jump conditions through entropy constraints. Additionally, we introduce locally Roe PINNs (LRPINNs), which incorporate an approximate Roe Riemann solver to improve shock resolution and conservation properties across discontinuities. These methods are adapted to two-dimensional Riemann problems by using a divergence-based shock detection combined with dimensional splitting, delivering precise solutions. Compared to a high-order weighted essentially non-oscillatory solver, our method produces sharper shock transitions but smoother solutions in areas with small-scale vortex structures. Future research will aim to improve the resolution of these small-scale features without compromising the model's ability to accurately capture shocks.

Figures

Figures reproduced from arXiv: 2506.11959 by the authors.

Figure 1
Figure 1. FIG. 1: ( [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Final predictions for each algorithm (PINN, LLPINN and LRPINN) for the double shock problem (Problem 1). The [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: ( [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Same as Figure 2 for the Sod shock tube problem. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: LLPINN and LRPINN final predictions for all the variables (left and central panels, respectively), with a zoom in the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Difference between the conserved quantities and their [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Same as Figure 6 for the Sod shock tube problem in a moving reference frame. [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Same as Figure 6 for the relativistic Sod shock tube problem. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Same as Figure 6 for the relativistic double shock problem. [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Predictions of the density, velocity and pressure profiles of the cylindrical Sod shock tube problem. Here, [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Colormaps with overimposed contours of the LRPINN predictions of the density, the radial velocity and the pressure. [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Colormaps with overimposed contours of the LRPINN (left) and a numerical 2D Riemann solver (right) of the density [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Same as Figure 13 for Problem 8. [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Points obtained from a test set (a 300 [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]

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Reference graph

Works this paper leans on

98 extracted references · 42 canonical work pages

  1. [1]

    merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  2. [2]

    merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  3. [3]

    merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  4. [4]

    Andreopoulos , author J

    author author Y. Andreopoulos , author J. H. \ Agui , \ and\ author G. Briassulis ,\ title title Shock wave—turbulence interactions , \ https://doi.org/10.1146/annurev.fluid.32.1.309 journal journal Annual Review of Fluid Mechanics \ volume 32 ,\ pages 309--345 ( year 2000 ) NoStop

    work page doi:10.1146/annurev.fluid.32.1.309

  5. [5]

    author author P. A. \ Durbin ,\ title title Some recent developments in turbulence closure modeling , \ https://doi.org/10.1146/annurev-fluid-122316-045020 journal journal Annual Review of Fluid Mechanics \ volume 50 ,\ pages 77--103 ( year 2018 ) NoStop

    doi:10.1146/annurev-fluid-122316-045020

  6. [6]

    Saurel \ and\ author C

    author author R. Saurel \ and\ author C. Pantano ,\ title title Diffuse-interface capturing methods for compressible two-phase flows , \ https://doi.org/10.1146/annurev-fluid-122316-050109 journal journal Annual Review of Fluid Mechanics \ volume 50 ,\ pages 105--130 ( year 2018 ) NoStop

    work page doi:10.1146/annurev-fluid-122316-050109

  7. [7]

    author author S. K. \ Godunov ,\ title title A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics , \ @noop journal journal Matematicheskii Sbornik \ volume 47 ,\ pages 271–306 ( year 1959 ) NoStop

    1959

  8. [8]

    Van Leer ,\ title title Towards the ultimate conservative difference scheme

    author author B. Van Leer ,\ title title Towards the ultimate conservative difference scheme. iv. a new approach to numerical convection , \ https://doi.org/10.1016/0021-9991(77)90095-X journal journal Journal of Computational Physics \ volume 23 ,\ pages 276--299 ( year 1977 ) NoStop

    doi:10.1016/0021-9991(77)90095-x

Show all 98 references
  1. [9]

    van Leer ,\ title title Towards the ultimate conservative difference scheme

    author author B. van Leer ,\ title title Towards the ultimate conservative difference scheme. v. a second-order sequel to godunov's method , \ https://doi.org/10.1016/0021-9991(79)90145-1 journal journal Journal of Computational Physics \ volume 32 ,\ pages 101--136 ( year 197...

    doi

  2. [10]

    Harten , author B

    author author A. Harten , author B. Engquist , author S. Osher , \ and\ author S. R. \ Chakravarthy ,\ title title Uniformly high order accurate essentially non-oscillatory schemes, iii , \ https://doi.org/10.1016/0021-9991(87)90031-3 journal journal Journal of Computational P...

    doi

  3. [11]

    \ Liu , author S

    author author X.-D. \ Liu , author S. Osher , \ and\ author T. Chan ,\ title title Weighted essentially non-oscillatory schemes , \ https://doi.org/10.1006/jcph.1994.1187 journal journal Journal of Computational Physics \ volume 115 ,\ pages 200--212 ( year 1994 ) NoStop

    1994

  4. [12]

    author author E. Toro ,\ https://books.google.es/books?id=SqEjX0um8o0C title Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction \ ( publisher Springer Berlin Heidelberg ,\ year 2009 ) NoStop

    2009

  5. [13]

    author author R. J. \ LeVeque ,\ @noop title Finite Volume Methods for Hyperbolic Problems ,\ Cambridge Texts in Applied Mathematics\ ( publisher Cambridge University Press ,\ year 2002 ) NoStop

    2002

  6. [14]

    author author C. G. \ Canuto , author M. Y. \ Hussaini , author A. M. \ Quarteroni , \ and\ author T. A. \ Zang ,\ @noop title Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation) \ ( publisher Springer-Verlag ,\ address...

    2007

  7. [15]

    author author D. J. \ Mavriplis ,\ title title Unstructured grid techniques , \ https://doi.org/10.1146/annurev.fluid.29.1.473 journal journal Annual Review of Fluid Mechanics \ volume 29 ,\ pages 473--514 ( year 1997 ) NoStop

    1997 doi

  8. [16]

    author author T. Colonius ,\ title title Modeling artificial boundary conditions for compressible flow , \ https://doi.org/10.1146/annurev.fluid.36.050802.121930 journal journal Annual Review of Fluid Mechanics \ volume 36 ,\ pages 315--345 ( year 2004 ) NoStop

  9. [17]

    Lagaris , author A

    author author I. Lagaris , author A. Likas , \ and\ author D. Fotiadis ,\ title title Artificial neural networks for solving ordinary and partial differential equations , \ 10.1109/72.712178 journal journal IEEE Transactions on Neural Networks \ volume 9 ,\ pages 987--1000 ( y...

    doi

  10. [18]

    Raissi , author P

    author author M. Raissi , author P. Perdikaris , \ and\ author G. E. \ Karniadakis ,\ title title Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , \ @noop journal journal J...

    2019

  11. [19]

    Sharma , author W

    author author P. Sharma , author W. T. \ Chung , author B. Akoush , \ and\ author M. Ihme ,\ title title A review of physics-informed machine learning in fluid mechanics , \ @noop journal journal Energies \ volume 16 ,\ pages 2343 ( year 2023 ) NoStop

    2023

  12. [20]

    author author S. A. \ Faroughi , author N. M. \ Pawar , author C. Fernandes , author M. Raissi , author S. Das , author N. K. \ Kalantari , \ and\ author S. Kourosh Mahjour ,\ title title Physics-guided, physics-informed, and physics-encoded neural networks and operators in sc...

    2024

  13. [21]

    Ji , author J

    author author W. Ji , author J. Chang , author H.-X. \ Xu , author J. R. \ Gao , author S. Gr \"o blacher , author H. P. \ Urbach , \ and\ author A. J. \ Adam ,\ title title Recent advances in metasurface design and quantum optics applications with machine learning, physics-in...

    2023

  14. [22]

    Luna , author J

    author author R. Luna , author J. Calder \'o n Bustillo , author J. J. \ Seoane Mart \' nez , author A. Torres-Forn \'e , \ and\ author J. A. \ Font ,\ title title Solving the Teukolsky equation with physics-informed neural networks , \ 10.1103/PhysRevD.107.064025 journal jour...

    arXiv

  15. [23]

    Mishra \ and\ author R

    author author S. Mishra \ and\ author R. Molinaro ,\ title title Physics informed neural networks for simulating radiative transfer , \ 10.1016/j.jqsrt.2021.107705 journal journal Journal of Quantitative Spectroscopy and Radiative Transfer \ volume 270 ,\ eid 107705 ( year 202...

    2021

  16. [24]

    Cuomo , author V

    author author S. Cuomo , author V. S. \ Di Cola , author F. Giampaolo , author G. Rozza , author M. Raissi , \ and\ author F. Piccialli ,\ title title Scientific machine learning through physics--informed neural networks: Where we are and what’s next , \ @noop journal journal ...

    2022

  17. [25]

    Farea , author O

    author author A. Farea , author O. Yli-Harja , \ and\ author F. Emmert-Streib ,\ title title Understanding physics-informed neural networks: Techniques, applications, trends, and challenges , \ 10.3390/ai5030074 journal journal AI \ volume 5 ,\ pages 1534--1557 ( year 2024 ) NoStop

    doi

  18. [26]

    author author J. D. \ Toscano , author V. Oommen , author A. J. \ Varghese , author Z. Zou , author N. Ahmadi Daryakenari , author C. Wu , \ and\ author G. E. \ Karniadakis ,\ title title From pinns to pikans: Recent advances in physics-informed machine learning , \ @noop jour...

    2025

  19. [27]

    Lu , author P

    author author L. Lu , author P. Jin , author G. Pang , author Z. Zhang , \ and\ author G. E. \ Karniadakis ,\ title title Learning nonlinear operators via deeponet based on the universal approximation theorem of operators , \ @noop journal journal Nature machine intelligence \...

    2021

  20. [28]

    Li , author N

    author author Z. Li , author N. Kovachki , author K. Azizzadenesheli , author B. Liu , author K. Bhattacharya , author A. Stuart , \ and\ author A. Anandkumar ,\ title title Fourier neural operator for parametric partial differential equations , \ @noop journal journal arXiv p...

    2010 arXiv

  21. [29]

    Liu , author Y

    author author Z. Liu , author Y. Wang , author S. Vaidya , author F. Ruehle , author J. Halverson , author M. Solja c i \'c , author T. Y. \ Hou , \ and\ author M. Tegmark ,\ title title Kan: Kolmogorov-arnold networks , \ @noop journal journal arXiv preprint arXiv:2404.19756 ...

    2024 arXiv

  22. [30]

    author author G. Cybenko ,\ title title Approximation by superpositions of a sigmoidal function , \ 10.1007/BF02551274 journal journal Mathematics of Control, Signals, and Systems \ volume 2 ,\ pages 303--314 ( year 1989 ) NoStop

    1989 doi

  23. [31]

    Hornik , author M

    author author K. Hornik , author M. Stinchcombe , \ and\ author H. White ,\ title title Multilayer feedforward networks are universal approximators , \ https://doi.org/10.1016/0893-6080(89)90020-8 journal journal Neural Networks \ volume 2 ,\ pages 359--366 ( year 1989 ) NoStop

    doi

  24. [32]

    author author A. Pinkus ,\ title title Approximation theory of the mlp model in neural networks , \ 10.1017/S0962492900002919 journal journal Acta Numerica \ volume 8 ,\ pages 143–195 ( year 1999 ) NoStop

    1999 doi

  25. [33]

    Liu , author S

    author author L. Liu , author S. Liu , author H. Xie , author F. Xiong , author T. Yu , author M. Xiao , author L. Liu , \ and\ author H. Yong ,\ title title Discontinuity computing using physics-informed neural networks , \ @noop journal journal Journal of Scientific Computin...

    2024

  26. [34]

    Cai , author J

    author author Z. Cai , author J. Chen , \ and\ author M. Liu ,\ title title Least-squares relu neural network (lsnn) method for linear advection-reaction equation , \ https://doi.org/10.1016/j.jcp.2021.110514 journal journal Journal of Computational Physics \ volume 443 ,\ pag...

    2021

  27. [35]

    Cai , author J

    author author Z. Cai , author J. Choi , \ and\ author M. Liu ,\ title title Least-squares neural network (lsnn) method for linear advection-reaction equation: Discontinuity interface , \ 10.1137/23M1568107 journal journal SIAM Journal on Scientific Computing \ volume 46 ,\ pag...

    doi

  28. [36]

    Michoski , author M

    author author C. Michoski , author M. Milosavljević , author T. Oliver , \ and\ author D. R. \ Hatch ,\ title title Solving differential equations using deep neural networks , \ https://doi.org/10.1016/j.neucom.2020.02.015 journal journal Neurocomputing \ volume 399 ,\ pages 1...

    2020 doi

  29. [37]

    author author R. G. \ Patel , author I. Manickam , author N. A. \ Trask , author M. A. \ Wood , author M. Lee , author I. Tomas , \ and\ author E. C. \ Cyr ,\ title title Thermodynamically consistent physics-informed neural networks for hyperbolic systems , \ https://doi.org/1...

  30. [38]

    Ferrer-S \'a nchez , author J

    author author A. Ferrer-S \'a nchez , author J. D. \ Mart \' n-Guerrero , author R. R. \ de Austri-Bazan , author A. Torres-Forn \'e , \ and\ author J. A. \ Font ,\ title title Gradient-annihilated pinns for solving riemann problems: Application to relativistic hydrodynamics ,...

    2024

  31. [39]

    Liu , author S

    author author J. Liu , author S. Zheng , author X. Song , \ and\ author D. Xu ,\ title title Locally linearized physics-informed neural networks for riemann problems of hyperbolic conservation laws , \ 10.1063/5.0238865 journal journal Physics of Fluids \ volume 36 ,\ pages 11...

    doi

  32. [40]

    author author P. L. \ Roe ,\ title title Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , \ 10.1006/jcph.1997.5705 journal journal Journal of Computational Physics \ volume 135 ,\ pages 250--258 ( year 1997 ) NoStop

    1997

  33. [41]

    author author A. G. \ Baydin , author B. A. \ Pearlmutter , author A. A. \ Radul , \ and\ author J. M. \ Siskind ,\ title title Automatic differentiation in machine learning: a survey , \ http://jmlr.org/papers/v18/17-468.html journal journal Journal of Machine Learning Resear...

    2018

  34. [42]

    Dawood \ and\ author N

    author author H. Dawood \ and\ author N. Megahed ,\ title title Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation , \ 10.7717/peerj-cs.1301 journal journal PeerJ. Computer science \ volume...

    doi

  35. [43]

    Abadi , author P

    author author M. Abadi , author P. Barham , author J. Chen , author Z. Chen , author A. Davis , author J. Dean , author M. Devin , author S. Ghemawat , author G. Irving , author M. Isard , et al. ,\ title title Tensorflow: a system for large-scale machine learning , \ in\ @noo...

    2016

  36. [44]

    Ruder ,\ title title An overview of gradient descent optimization algorithms , \ @noop journal journal arXiv preprint arXiv:1609.04747 \ ( year 2016 ) NoStop

    author author S. Ruder ,\ title title An overview of gradient descent optimization algorithms , \ @noop journal journal arXiv preprint arXiv:1609.04747 \ ( year 2016 ) NoStop

    2016 arXiv

  37. [45]

    author author D. Soydaner ,\ title title A comparison of optimization algorithms for deep learning , \ 10.1142/S0218001420520138 journal journal International Journal of Pattern Recognition and Artificial Intelligence \ volume 34 ,\ pages 2052013 ( year 2020 ) NoStop

    doi

  38. [46]

    Abdulkadirov , author P

    author author R. Abdulkadirov , author P. Lyakhov , \ and\ author N. Nagornov ,\ title title Survey of optimization algorithms in modern neural networks , \ @noop journal journal Mathematics \ volume 11 ,\ pages 2466 ( year 2023 ) NoStop

    2023

  39. [47]

    He , author X

    author author K. He , author X. Zhang , author S. Ren , \ and\ author J. Sun ,\ title title Deep residual learning for image recognition , \ in\ @noop booktitle Proceedings of the IEEE conference on computer vision and pattern recognition \ ( year 2016 )\ pp.\ pages 770--778 NoStop

    2016

  40. [48]

    Krizhevsky , author I

    author author A. Krizhevsky , author I. Sutskever , \ and\ author G. E. \ Hinton ,\ title title Imagenet classification with deep convolutional neural networks , \ in\ https://proceedings.neurips.cc/paper_files/paper/2012/file/c399862d3b9d6b76c8436e924a68c45b-Paper.pdf booktit...

    2012

  41. [49]

    Devlin , author M.-W

    author author J. Devlin , author M.-W. \ Chang , author K. Lee , \ and\ author K. Toutanova ,\ title title Bert: Pre-training of deep bidirectional transformers for language understanding , \ in\ @noop booktitle Proceedings of the 2019 conference of the North American chapter ...

    2019

  42. [50]

    author author D. P. \ Kingma \ and\ author J. Ba ,\ title title Adam: A Method for Stochastic Optimization , \ 10.48550/arXiv.1412.6980 journal journal arXiv e-prints \ ,\ eid arXiv:1412.6980 ( year 2014 ) ,\ http://arxiv.org/abs/1412.6980 arXiv:1412.6980 [cs.LG] NoStop

    2014 doi arXiv

  43. [51]

    Alacaoglu , author Y

    author author A. Alacaoglu , author Y. Malitsky , author P. Mertikopoulos , \ and\ author V. Cevher ,\ title title A new regret analysis for A dam-type algorithms , \ in\ https://proceedings.mlr.press/v119/alacaoglu20b.html booktitle Proceedings of the 37th International Confe...

    2020

  44. [52]

    Reyad , author A

    author author M. Reyad , author A. M. \ Sarhan , \ and\ author M. Arafa ,\ title title A modified adam algorithm for deep neural network optimization , \ 10.1007/s00521-023-08568-z journal journal Neural Comput. Appl. \ volume 35 ,\ pages 17095–17112 ( year 2023 ) NoStop

    doi

  45. [53]

    author author R. M. \ Schmidt , author F. Schneider , \ and\ author P. Hennig ,\ title title Descending through a crowded valley-benchmarking deep learning optimizers , \ in\ @noop booktitle International Conference on Machine Learning \ ( organization PMLR ,\ year 2021 )\ pp....

    2021

  46. [54]

    Marin , author A

    author author I. Marin , author A. Kuzmanic Skelin , \ and\ author T. Grujic ,\ title title Empirical evaluation of the effect of optimization and regularization techniques on the generalization performance of deep convolutional neural network , \ 10.3390/app10217817 journal j...

    doi

  47. [55]

    Wang , author S

    author author S. Wang , author S. Sankaran , \ and\ author P. Perdikaris ,\ title title Respecting causality for training physics-informed neural networks , \ https://doi.org/10.1016/j.cma.2024.116813 journal journal Computer Methods in Applied Mechanics and Engineering \ volu...

    2024

  48. [56]

    author author L. D. \ McClenny \ and\ author U. M. \ Braga-Neto ,\ title title Self-adaptive physics-informed neural networks , \ https://doi.org/10.1016/j.jcp.2022.111722 journal journal Journal of Computational Physics \ volume 474 ,\ pages 111722 ( year 2023 ) NoStop

    2022

  49. [57]

    author author S. J. \ Anagnostopoulos , author J. D. \ Toscano , author N. Stergiopulos , \ and\ author G. E. \ Karniadakis ,\ title title Residual-based attention in physics-informed neural networks , \ https://doi.org/10.1016/j.cma.2024.116805 journal journal Computer Method...

  50. [58]

    Wang , author B

    author author S. Wang , author B. Li , author Y. Chen , \ and\ author P. Perdikaris ,\ title title Piratenets: Physics-informed deep learning with residual adaptive networks , \ @noop journal journal Journal of Machine Learning Research \ volume 25 ,\ pages 1--51 ( year 2024 ) NoStop

    2024

  51. [59]

    Robbins \ and\ author S

    author author H. Robbins \ and\ author S. Monro ,\ title title A stochastic approximation method , \ @noop journal journal The Annals of Mathematical Statistics \ volume 22 ,\ pages 400--407 ( year 1951 ) NoStop

    1951

  52. [60]

    author author G. E. \ Hinton ,\ @noop title Lecture 6e - rmsprop: Divide the gradient by a running average of its recent magnitude , \ howpublished http://www.cs.toronto.edu/ tijmen/csc321/slides/lecture_slides_lec6.pdf ( year 2012 ),\ note coursera: Neural Networks for Machin...

    2012

  53. [61]

    Duchi , author E

    author author J. Duchi , author E. Hazan , \ and\ author Y. Singer ,\ title title Adaptive subgradient methods for online learning and stochastic optimization , \ http://jmlr.org/papers/v12/duchi11a.html journal journal Journal of Machine Learning Research \ volume 12 ,\ pages...

    2011

  54. [62]

    author author J. F. \ Urb \'a n , author P. Stefanou , \ and\ author J. A. \ Pons ,\ title title Unveiling the optimization process of physics informed neural networks: How accurate and competitive can pinns be? \ @noop journal journal Journal of Computational Physics \ volume...

    2025

  55. [63]

    Kiyani , author K

    author author E. Kiyani , author K. Shukla , author J. F. \ Urb \'a n , author J. Darbon , \ and\ author G. E. \ Karniadakis ,\ title title Which optimizer works best for physics-informed neural networks and kolmogorov-arnold networks? \ @noop journal journal arXiv preprint ar...

    2025

  56. [64]

    Wang , author A

    author author S. Wang , author A. K. \ Bhartari , author B. Li , \ and\ author P. Perdikaris ,\ title title Gradient alignment in physics-informed neural networks: A second-order optimization perspective , \ @noop journal journal arXiv preprint arXiv:2502.00604 \ ( year 2025 ) NoStop

    2025

  57. [65]

    author author C. G. \ Broyden ,\ title title The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations , \ 10.1093/imamat/6.1.76 journal journal IMA Journal of Applied Mathematics \ volume 6 ,\ pages 76--90 ( year 1970 ) NoStop

    doi

  58. [66]

    author author R. Fletcher ,\ title title A new approach to variable metric algorithms , \ 10.1093/comjnl/13.3.317 journal journal The Computer Journal \ volume 13 ,\ pages 317--322 ( year 1970 ) NoStop

    1970 doi

  59. [67]

    author author D. Goldfarb ,\ title title A family of variable-metric methods derived by variational means , \ https://api.semanticscholar.org/CorpusID:790344 journal journal Mathematics of Computation \ volume 24 ,\ pages 23--26 ( year 1970 ) NoStop

    1970

  60. [68]

    author author D. F. \ Shanno ,\ title title Conditioning of quasi-newton methods for function minimization , \ https://api.semanticscholar.org/CorpusID:7977144 journal journal Mathematics of Computation \ volume 24 ,\ pages 647--656 ( year 1970 ) NoStop

    1970

  61. [69]

    Al-Baali \ and\ author H

    author author M. Al-Baali \ and\ author H. Khalfan ,\ title title Wide interval for efficient self-scaling quasi-newton algorithms , \ @noop journal journal Optimization Methods and Software \ volume 20 ,\ pages 679--691 ( year 2005 ) NoStop

    2005

  62. [70]

    Al-Baali , author E

    author author M. Al-Baali , author E. Spedicato , \ and\ author F. Maggioni ,\ title title Broyden's quasi-newton methods for a nonlinear system of equations and unconstrained optimization: a review and open problems , \ 10.1080/10556788.2013.856909 journal journal Optimizatio...

  63. [71]

    Contreras \ and\ author R

    author author M. Contreras \ and\ author R. A. \ Tapia ,\ title title Sizing the bfgs and dfp updates: Numerical study , \ @noop journal journal J. Optim. Theory Appl. \ volume 78 ,\ pages 93–108 ( year 1993 ) NoStop

    1993

  64. [72]

    Dennis Jr \ and\ author R

    author author J. Dennis Jr \ and\ author R. B. \ Schnabel ,\ @noop title Numerical Methods for Unconstrained Optimization and Nonlinear Equations \ ( publisher SIAM ,\ year 1996 ) NoStop

    1996

  65. [73]

    author author R. J. \ LeVeque , author D. Mihalas , author E. A. \ Dorfi , \ and\ author W. Müller ,\ 10.1007/978-3-642-60543-7 title Computational Methods for Astrophysical Fluid Flow ,\ edited by\ editor R. J. \ LeVeque , editor D. Mihalas , editor E. A. \ Dorfi , \ and\ edi...

    doi

  66. [74]

    author author W. J. M. \ Rankine ,\ title title On the thermodynamic theory of waves of finite longitudinal disturbance , \ 10.1098/rstl.1870.0015 journal journal Philosophical Transactions of the Royal Society of London \ volume 160 ,\ pages 277--288 ( year 1870 ) NoStop

  67. [75]

    author author P.-H. \ Hugoniot ,\ title title Mémoire sur la propagation du mouvement dans les corps et spécialement dans les gaz parfaits (première partie) , \ @noop journal journal Journal de l'École Polytechnique \ volume 57 ,\ pages 3--97 ( year 1887 ) NoStop

  68. [76]

    author author P.-H. \ Hugoniot ,\ title title Mémoire sur la propagation du mouvement dans les corps et spécialement dans les gaz parfaits (deuxième partie) , \ @noop journal journal Journal de l'École Polytechnique \ volume 58 ,\ pages 1--125 ( year 1889 ) NoStop

  69. [77]

    author author P. D. \ Lax ,\ title title Hyperbolic systems of conservation laws ii , \ https://doi.org/10.1002/cpa.3160100406 journal journal Communications on Pure and Applied Mathematics \ volume 10 ,\ pages 537--566 ( year 1957 ) ,\ http://arxiv.org/abs/https://onlinelibra...

    doi

  70. [78]

    note In this case, we work in c=1 units NoStop

  71. [79]

    author author J. M. \ Marti \ and\ author E. Muller ,\ title title Analytical solution of the Riemann problem in relativistic hydrodynamics , \ 10.1017/S0022112094003344 journal journal Journal of Fluid Mechanics \ volume 258 ,\ pages 317--333 ( year 1994 ) NoStop

    doi

  72. [80]

    note An exception is made for the Burgers equation, where the shock speed is calculated based on the left and right states of the velocity. Stop

  73. [81]

    Eulderink \ and\ author G

    author author F. Eulderink \ and\ author G. Mellema ,\ title title General relativistic hydrodynamics with a ROE solver. \ 10.48550/arXiv.astro-ph/9411056 journal journal Astronomy and Astrophysics Supplement \ volume 110 ,\ pages 587 ( year 1995 ) ,\ http://arxiv.org/abs/astr...

    doi arXiv

  74. [82]

    When this happens, we change the training set and continue the training process

    note We say ``at most'' because the optimization process for a given set of points may be interrupted before completing 500 iterations if the optimizer does not find a proper value of step-length _k . When this happens, we change the training set and continue the training proc...

  75. [83]

    Omang , author S

    author author M. Omang , author S. Børve , \ and\ author J. Trulsen ,\ title title Sph in spherical and cylindrical coordinates , \ https://doi.org/10.1016/j.jcp.2005.08.023 journal journal Journal of Computational Physics \ volume 213 ,\ pages 391--412 ( year 2006 ) NoStop

    2005 doi

  76. [84]

    Bernard-Champmartin , author J.-P

    author author A. Bernard-Champmartin , author J.-P. \ Braeunig , \ and\ author J.-M. \ Ghidaglia ,\ title title An eulerian finite volume solver for multi-material fluid flows with cylindrical symmetry , \ https://doi.org/10.1016/j.compfluid.2012.09.014 journal journal Compute...

    doi

  77. [85]

    note The solution closely resembles that presented in OMANG2006391 (see their Figure 11). Stop

  78. [86]

    @noop title hydrofv , \ howpublished https://github.com/alvinng4/hydroFV ( year 2025 ),\ note gitHub repository NoStop

    2025

  79. [87]

    Kurganov \ and\ author E

    author author A. Kurganov \ and\ author E. Tadmor ,\ title title Solution of two-dimensional riemann problems for gas dynamics without riemann problem solvers , \ https://doi.org/10.1002/num.10025 journal journal Numerical Methods for Partial Differential Equations \ volume 18...

    doi

  80. [88]

    @noop title Hypar , \ howpublished https://github.com/debog/hypar ( year 2023 ),\ note gitHub repository NoStop

    2023

  81. [89]

    Wang , author H

    author author S. Wang , author H. Wang , \ and\ author P. Perdikaris ,\ title title On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks , \ @noop journal journal Computer Methods in Applied Mech...

    2021

  82. [90]

    Moseley , author A

    author author B. Moseley , author A. Markham , \ and\ author T. Nissen-Meyer ,\ title title Finite basis physics-informed neural networks (fbpinns): a scalable domain decomposition approach for solving differential equations , \ @noop journal journal Advances in Computational ...

    2023

  83. [91]

    Dolean , author A

    author author V. Dolean , author A. Heinlein , author S. Mishra , \ and\ author B. Moseley ,\ title title Multilevel domain decomposition-based architectures for physics-informed neural networks , \ https://doi.org/10.1016/j.cma.2024.117116 journal journal Computer Methods in ...

  84. [92]

    Harten , author P

    author author A. Harten , author P. D. \ Lax , \ and\ author B. v. \ Leer ,\ title title On upstream differencing and godunov-type schemes for hyperbolic conservation laws , \ 10.1137/1025002 journal journal SIAM Review \ volume 25 ,\ pages 35--61 ( year 1983 ) ,\ http://arxiv...

    doi

  85. [93]

    author author B. Einfeldt ,\ title title On godunov-type methods for gas dynamics , \ 10.1137/0725021 journal journal SIAM Journal on Numerical Analysis \ volume 25 ,\ pages 294--318 ( year 1988 ) ,\ http://arxiv.org/abs/https://doi.org/10.1137/0725021 https://doi.org/10.1137/...

    1988 doi

  86. [94]

    author author E. F. \ Toro , author M. Spruce , \ and\ author W. Speares ,\ title title Restoration of the contact surface in the hll-riemann solver , \ https://api.semanticscholar.org/CorpusID:119972653 journal journal Shock Waves \ volume 4 ,\ pages 25--34 ( year 1994 ) NoStop

    1994

  87. [95]

    Engquist \ and\ author S

    author author B. Engquist \ and\ author S. Osher ,\ title title One-sided difference approximations for nonlinear conservation laws , \ http://www.jstor.org/stable/2007646 journal journal Mathematics of Computation \ volume 36 ,\ pages 321--351 ( year 1981 ) NoStop

    1981

  88. [96]

    Osher \ and\ author F

    author author S. Osher \ and\ author F. Solomon ,\ title title Upwind difference schemes for hyperbolic systems of conservation laws , \ http://www.jstor.org/stable/2007275 journal journal Mathematics of Computation \ volume 38 ,\ pages 339--374 ( year 1982 ) NoStop

    1982

  89. [97]

    merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number orga...

    2010

  90. [98]

    merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number orga...

    2010

Pith tools

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