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Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space

T0 review · 1 major / 1 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Regularizing the latent-space metric to stay close to unity lets a shared physics-informed body transfer to stiff regimes, cutting RMS error from 35% to 0.46% on the flame equation.

desk verdict Impressive transfer-learning gains on stiff PINN problems, but the paper never separates its new determinant term from the Jacobian regularization it contains. read the letter →

arxiv 2501.12116 v2 pith:SWPGN3CZ submitted 2025-01-21 cs.LG cs.AIhep-thmath.AP

Pedro Tarancón-Álvarez , Pablo Tejerina-Pérez , Raul Jimenez , Pavlos Protopapas This is my paper · ORCID
classification cs.LGcs.AIhep-thmath.AP MSC 68T0765L0535R3065M70
keywords physics-informedneuralnetworksmulti-headtrainingunimodularregularizationtransferlearningstiffdifferentialequationsinverseproblemsinducedmetrictensorEinsteinfield
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

Physics-Informed Neural Networks solve differential equations by minimizing the equation's residual, but they struggle with stiff equations and need retraining for each new initial or boundary condition. This paper argues that both problems are addressed at once if the network is trained with multiple heads: a shared body learns a latent space that covers an entire family of equations, and a new head projects that latent space onto a specific solution. The new ingredient is Unimodular Regularization (UR), a penalty that holds the determinant of the latent-space metric close to unity so the latent response is smooth in all input directions. With UR, transfer learning to stiff regimes works far better than without it: RMS error drops from 35.08% to 0.46% for the flame equation, from 52.38% to 5.68% for the van der Pol oscillator, and from 64.42% to 21.65% for the Einstein Field Equations inverse problem at the stiffest tested point. A sympathetic reader would care because the method needs no precomputed solutions, only the equations and boundary conditions, and it makes inverse problems with hidden free functions tractable.

What carries the argument

The load-bearing object is the induced metric $g_{\mu\nu}$ on the latent-space hypersurface $\vec\Omega = (x^\mu, H_i(x^\mu))$, where $x^\mu$ are the inputs (independent variable plus parameters, initial conditions, or boundary conditions) and $H_i$ are the components of the shared body's output. Its determinant $g = \det g_{\mu\nu}$ measures the volume element of the latent space per coordinate differential; the UR loss $L_{\mathrm{UR}} = \lambda \sum_{\mathrm{batch}} (\sqrt{g} - 1)^2$ pushes that volume element toward unity, so the latent response is smooth in all input directions. As the paper shows explicitly, the metric contains the Frobenius norm of the Jacobian $\sum_{\mu,i}(\partial H_i/\partial x^\mu)^2$ as one term plus additional cross-derivative terms, making UR a geometric generalization of Jacobian regularization. That smoothness is what allows a frozen body to accept a new head for parameter values outside the training range.

What would settle it

Train the same multi-head architecture with the same hyperparameters but replace the unimodular loss with a pure Frobenius-norm Jacobian regularization at matched strength; if the transfer-target RMS errors stay close to the UR values, the determinant-specific part of the loss is not the active mechanism.

Watch

Extended reading notes

Core claim

The paper's central claim is that training a Physics-Informed Neural Network with multiple heads, each head projecting a shared latent space onto the solution for one member of a family of differential equations, and regularizing that latent space with Unimodular Regularization (UR) makes transfer learning to stiff regimes reliable. UR penalizes deviations of the determinant of the induced metric on the latent-space hypersurface from unity: $L_{\mathrm{UR}} = \lambda \sum_{\mathrm{batch}} (\sqrt{g} - 1)^2$, where $g = \det g_{\mu\nu}$ and $g_{\mu\nu} = \partial_\mu \vec\Omega \cdot \partial_\nu \vec\Omega$ with $\vec\Omega = (x^\mu, H_i(x^\mu))$. In the authors' experiments, the bodies trained with and without UR produce essentially identical solutions within the training range, but the frozen body transfers markedly better with UR: for the flame equation at $\delta = 0.015$, the RMS error drops from 35.08% to 0.46%; for the van der Pol oscillator at $a = 1.75$, from 52.38% to 5.68%; and for the Einstein Field Equations inverse problem at $\phi_M = 0.7$, potential recovery improves from 64.42% to 21.65% while the recovered $S(T)$ relation improves from 177.61% to 25.65%.

Load-bearing premise

The paper assumes that the determinant of the induced metric, not just the Frobenius norm of the Jacobian that it contains, is the ingredient that drives the improved transfer learning, since no experiment in the paper isolates those two contributions.

Editorial extensions

If this is right

  • Transfer learning to stiff regimes becomes practical: once a body is trained, a new parameter value only requires training a small head, with reported speedups of roughly 1.3x, 1.5x, and 2.9x per solution over training from scratch.
  • Inverse problems with a free function hidden in the equations can be solved from boundary-condition data alone, as demonstrated by the EFE potential recovery; no precomputed solutions are needed.
  • The method extends from ODEs to PDEs by treating extra independent variables as additional bundle inputs, at the cost of denser input sampling and larger batch sizes.
  • The UR overhead is modest in training time (5-10%) but memory-hungry; adding one head to the EFE experiment exceeded the available GPU memory, pointing to a clear scalability bottleneck.
  • Accuracy is limited to the span of the training conditions; the paper cautions that generalization may not extend to entirely unobserved regions of the input space, though the EFE boundary samples cover the region of interest.

Reading between the lines

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

  • Our inference: the same volume penalty could be applied to any latent space, not just PINN bodies, so UR may transfer to generative models or multi-task networks whose latent response needs to stay smooth in conditioning inputs.
  • Our inference: a matched-strength ablation replacing UR with pure Frobenius-norm Jacobian regularization on these three benchmarks would isolate whether the determinant-specific cross terms are the active mechanism; the paper does not run that comparison.
  • Our inference: because the metric determinant includes all pairwise input directions, UR can be viewed as an empirical Lipschitz-control scheme; measuring the Lipschitz constant of the frozen body on transfer targets would test this interpretation directly.
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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

1 major / 1 minor

Summary. The paper proposes two techniques for physics-informed neural networks (PINNs): multi-head (MH) training, in which a shared body learns a latent space of solutions for a family of differential equations and separate heads project that latent space to individual solutions, and Unimodular Regularization (UR), which penalizes deviations of the determinant of the induced metric on the latent space from one. The authors argue that UR regularizes the geometry of the frozen latent space and thereby improves transfer learning to stiffer parameter regimes. They demonstrate the approach on the flame equation, the van der Pol oscillator, and an inverse problem for the Einstein Field Equations in an AdS background, reporting large reductions in root-mean-square error for transfer learning with UR compared to the same MH setup without UR (e.g., flame 35.08% to 0.46%, van der Pol 52.38% to 5.68%, EFE potential recovery 64.42% to 21.65%). The central claim is that the determinant-based UR term, rather than generic Jacobian regularization or the multi-head architecture alone, is responsible for these improvements.

Significance. If the central claim is substantiated, MH+UR would be a practically useful tool for PINN transfer learning in stiff regimes, particularly for inverse problems where the EFE example shows a meaningful improvement. The paper is commendable for releasing code, providing detailed training configurations, and honestly acknowledging computational limitations and the implicit presence of Jacobian regularization in UR (Section 2.5 and Supplementary Note 1). However, the experiments as designed do not isolate the proposed determinant mechanism from the Jacobian-norm term that is mathematically inseparable from it, so the distinctive contribution of UR is currently unverified. The empirical comparisons are plausible and the effect sizes are large, but the lack of repeated-seed statistics further tempers the strength of the conclusions.

major comments (1)
  1. [Section 2.5 and Supplementary Note 1] In the EFE inverse problem, the no-UR baseline is the authors' own prior algorithm (reference [39]) rather than an independently established baseline. While this is not by itself a flaw, the paper should state explicitly which components of the EFE baseline differ from the UR version beyond the added loss term, and whether the baseline was re-run for this paper or numbers are taken from [39]. This information is needed to ensure that the improvement is not due to implementation details or to differences in training budgets.
minor comments (1)
  1. [Figure 8 caption] The discussion of computational scaling in Section 2.4 argues that MH scales as O(N) heads with Monte Carlo-style error decay N^{-1/2}; the authors correctly note that this is an assumption rather than a proof. It would be helpful to label it as a conjecture or heuristic in the main text, since the current wording may be read as an established result.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the UR transfer improvements are measured against independent numerical/theory baselines, and the Jacobian-regularization containment is a mechanistic confound, not a construction-level reduction.

full rationale

The paper's central claim is an empirical one: for the flame equation, the van der Pol oscillator, and the EFE inverse problem, a multi-head PINN whose shared body was trained with the UR loss (Eq. 9) transfers better to stiffer regimes than the same MH setup without UR (Figs. 5, 7, 10). These errors are computed against independent references—RK4 numerical integration for the ODEs and the theoretical potentials/curves of [40] for the EFE case—so the reported reductions (35.08%→0.46%, 52.38%→5.68%, 64.42%→21.65%) are not equal by construction to the UR loss. Lambda and N are per-problem hyperparameters, not fitted parameters relabeled as predictions, and the target stiff regimes are outside the body-training ranges. The self-citations [26] and [39] supply background and the EFE setup, but the UR-versus-no-UR comparison is self-contained and validated externally. Supplementary Note 1 does show that the UR loss contains the Frobenius Jacobian norm J^2 (g = 1 + J^2 + A^2), and the paper explicitly acknowledges that 'JR as a regularizing technique is implicit when performing UR.' This is a real limitation: without a JR-only ablation, the determinant-specific mechanism is not isolated, and the improvement may be attributable to generic Jacobian regularization. However, that is a confound/attribution problem about which term drives the effect, not a circular derivation in which the output is equivalent to the input by construction. No fitted value is renamed as a prediction, and no load-bearing uniqueness claim rests solely on the authors' own prior work. Hence score 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the PINN residual-loss assumption, the smoothness of the latent-space embedding, the ad hoc UR mechanism, the unproven Monte Carlo scaling of MH, and the physics assumptions of the EFE inverse problem. The two free parameters lambda and N are tuned per problem and directly control the regularizer's effect.

free parameters (2)
  • lambda_UR = 5e-7 (flame), 4e-5 (VdP), 5e-8 (EFE)
    Weight of the UR loss term, hand-tuned per problem. The paper states it must be in a narrow range or the method fails.
  • N (UR frequency) = 100 (flame, VdP), 500 (EFE)
    Number of epochs between applications of the UR loss. Chosen per problem to balance computational cost and regularization effect.
assumptions (5)
  • domain assumption Minimizing the squared residual loss (Eq. 3) yields a valid solution to the differential equation.
    Section 2.2: the loss is a sum of squared DE residuals; this assumes the optimizer can find a near-zero residual, which is the standard PINN premise.
  • domain assumption The latent space functions Hi(xmu) form a smooth hypersurface, so the induced metric and its determinant are well-defined.
    Section 2.5, Eqs. (6)-(7): the construction treats the neural network's hidden representation as a differentiable embedded hypersurface.
  • ad hoc to paper Driving det(g)^(1/2) toward 1 improves generalization to stiffer regimes.
    Section 2.5: the 'throat widening' hypothesis is presented qualitatively, with no proof or formal justification.
  • ad hoc to paper The MH approach avoids the curse of dimensionality through Monte Carlo-style scaling with the number of heads.
    Section 2.4: the authors explicitly admit this is an assumption without a rigorous proof, justified only by empirical observation.
  • domain assumption The Einstein Field Equations inverse problem has a one-to-one correspondence S(T) <-> V(phi), and the potentials belong to the family V(phi; phiM).
    Section 3.4: this physical correspondence is taken from ref [40] and underlies the whole EFE experiment.

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Cite this review

Pith. "Pith review of Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space." pith.science (2026). https://pith.science/paper/SWPGN3CZ

@misc{pith2026250112116,
  author       = {Pith},
  title        = {Pith review of: Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWPGN3CZ}},
  note         = {Machine review of arXiv:2501.12116}
}
read the original abstract

Non-linear differential equations are a fundamental tool to describe different phenomena in nature. However, we still lack a well-established method to tackle stiff differential equations. Here we present a machine learning framework to facilitate the solution of nonlinear multiscale differential equations and, especially, inverse problems using Physics-Informed Neural Networks (PINNs). This framework is based on what is called \textit{multi-head} (MH) training, which involves training the network to learn a general space of all solutions for a given set of equations with certain variability, rather than learning a specific solution of the system. This setup is used with a second novel technique that we call Unimodular Regularization (UR) of the latent space of solutions. We show that the multi-head approach, combined with Unimodular Regularization, significantly improves the efficiency of PINNs by facilitating the transfer learning process thereby enabling the finding of solutions for nonlinear, coupled, and multiscale differential equations.

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Works this paper leans on

41 extracted references · 12 canonical work pages

  1. [39]

    JHEP 07, 087 (2024) https://doi.org/10.1007/JHEP07(2024)087 arXiv:2403.14763 [hep-th]

    Bea, Y., Jimenez, R., Mateos, D., Liu, S., Protopapas, P., Taranc´ on-´Alvarez, P., Tejerina-P´ erez, P.: Gravitational duals from equations of state. JHEP 07, 087 (2024) https://doi.org/10.1007/JHEP07(2024)087 arXiv:2403.14763 [hep-th]

    arXiv 2024

  2. [1]

    Pinney, E.: Nonlinear differential equations. Bull. New Ser. Am. Math. Soc.61(5), 373–388 (1955) https://doi.org/10.1090/s0002-9904-1955-09934-8 32

    work page doi:10.1090/s0002-9904-1955-09934-8 1955

  3. [2]

    Mathematical Problems in Engineering 2014, 1–1 (2014) https://doi.org/10.1155/2014/787412

    Khalique, M., Abudiab, M., Mahomed, F., Jafari, H., Wu, G.-C.: Recent advances on methods and applications of nonlinear differential equations. Mathematical Problems in Engineering 2014, 1–1 (2014) https://doi.org/10.1155/2014/787412

    work page doi:10.1155/2014/787412 2014

  4. [3]

    Pure & Applied Mathematics S

    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. Pure & Applied Mathematics S. McGraw-Hill Education, London, England (1984)

    work page 1984

  5. [4]

    https://doi.org/10.48550/arXiv.2006

    Flamant, C., Protopapas, P., Sondak, D.: Solving Differential Equations Using Neural Network Solution Bundles (2020). https://doi.org/10.48550/arXiv.2006. 14372

    doi:10.48550/arxiv.2006 2020

  6. [5]

    Journal of Open Source Software 5(46), 1931 (2020) https: //doi.org/10.21105/joss.01931

    Chen, F., Sondak, D., Protopapas, P., Mattheakis, M., Liu, S., Agarwal, D., Giovanni, M.D.: Neurodiffeq: A python package for solving differential equations with neural networks. Journal of Open Source Software 5(46), 1931 (2020) https: //doi.org/10.21105/joss.01931

    doi:10.21105/joss.01931 2020

  7. [6]

    SIAM Review 63, 208–228 (2021) https://doi.org/ 10.1137/19M1274067

    Lu, L., Meng, X., Mao, Z., Karniadakis, G.: Deepxde: A deep learning library for solving differential equations. SIAM Review 63, 208–228 (2021) https://doi.org/ 10.1137/19M1274067

    doi:10.1137/19m1274067 2021

  8. [7]

    Nature Reviews Physics 3(6), 422–440 (2021) https://doi.org/10.1038/s42254-021-00314-5

    Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L.: Physics-informed machine learning. Nature Reviews Physics 3(6), 422–440 (2021) https://doi.org/10.1038/s42254-021-00314-5

    doi:10.1038/s42254-021-00314-5 2021

Show all 41 references
  1. [8]

    Nature Machine Intelligence 3(3), 218–229 (2021) https://doi.org/10.1038/ s42256-021-00302-5

    Lu, L., Jin, P., Pang, G., Zhang, Z., Karniadakis, G.E.: Learning nonlinear operators via deeponet based on the universal approximation theorem of opera- tors. Nature Machine Intelligence 3(3), 218–229 (2021) https://doi.org/10.1038/ s42256-021-00302-5

    2021

  2. [9]

    https://doi.org/10.48550/arXiv.2010.08895

    Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., Anandkumar, A.: Fourier Neural Operator for Parametric Partial Differential Equations (2021). https://doi.org/10.48550/arXiv.2010.08895

    2021 doi arXiv

  3. [10]

    Stiff and Differential-Algebraic Problems vol

    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems vol. 14, (1996). https://doi.org/10.1007/ 978-3-662-09947-6

    1996

  4. [11]

    Seiler, E., Lei, W., Protopapas, P.: Stiff transfer learning for physics-informed neural networks (2025) https://doi.org/10.48550/arXiv.2501.17281 [cs.LG]

    2025 doi arXiv

  5. [12]

    John Wiley & Sons, Inc., USA (1991)

    Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: the Initial Value Problem. John Wiley & Sons, Inc., USA (1991)

    1991

  6. [13]

    Journal of Computational Physics 378, 686–707 (2019) https://doi.org/10.1016/j.jcp.2018.10.045 33

    Raissi, M., Perdikaris, P., Karniadakis, G.E.: 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) https://doi.org/10.1016/...

    2019 doi

  7. [14]

    IEEE Trans- actions on Neural Networks and Learning Systems 31(4), 1410–1416 (2020) https://doi.org/10.1109/TNNLS.2019.2919705

    Kaul, P., Lall, B.: Riemannian curvature of deep neural networks. IEEE Trans- actions on Neural Networks and Learning Systems 31(4), 1410–1416 (2020) https://doi.org/10.1109/TNNLS.2019.2919705

    2020

  8. [15]

    https://doi.org/10.48550/arXiv

    Chen, J., Huang, T., Chen, W., Liu, Y.: Thoughts on the Consistency between Ricci Flow and Neural Network Behavior (2022). https://doi.org/10.48550/arXiv. 2111.08410

    2022 doi arXiv

  9. [16]

    Glass, S., Spasov, S., Li` o, P.: RicciNets: Curvature-guided pruning of high- performance neural networks using ricci flow (2020) https://doi.org/10.48550/ arXiv.2007.04216 2007.04216 [cs.LG]

    2020 doi arXiv

  10. [17]

    558–567 (2023)

    Sun, L., Huang, Z., Wu, H., Ye, J., Peng, H., Yu, Z., Yu, P.: Deepricci: Self- supervised graph structure-feature co-refinement for alleviating over-squashing, pp. 558–567 (2023). https://doi.org/10.1109/ICDM58522.2023.00065

    2023

  11. [18]

    17053 arXiv:2310.17053 [math-ph]

    Arora, S., Bihlo, A., Valiquette, F.: Invariant physics-informed neural networks for ordinary differential equations (2023) https://doi.org/10.48550/ARXIV.2310. 17053 arXiv:2310.17053 [math-ph]

    2023 doi arXiv

  12. [19]

    Engineering Applications of Artificial Intelligence 127, 107324 (2024) https://doi.org/10.1016/j.engappai.2023.107324

    Sahli Costabal, F., Pezzuto, S., Perdikaris, P.: Delta-pinns: Physics-informed neural networks on complex geometries. Engineering Applications of Artificial Intelligence 127, 107324 (2024) https://doi.org/10.1016/j.engappai.2023.107324

    2024

  13. [20]

    Neural Networks 181, 106826 (2025) https://doi.org/10.1016/j.neunet.2024.106826

    Cardoso-Bihlo, E., Bihlo, A.: Exactly conservative physics-informed neural net- works and deep operator networks for dynamical systems. Neural Networks 181, 106826 (2025) https://doi.org/10.1016/j.neunet.2024.106826

    2025

  14. [21]

    https://doi.org/10.48550/arXiv.2301.02152

    Zou, Z., Karniadakis, G.E.: L-HYDRA: Multi-Head Physics-Informed Neural Networks (2023). https://doi.org/10.48550/arXiv.2301.02152

    2023 doi

  15. [22]

    https://doi.org/10.48550/ arXiv.2110.11286

    Desai, S., Mattheakis, M., Joy, H., Protopapas, P., Roberts, S.: One-Shot Transfer Learning of Physics-Informed Neural Networks (2022). https://doi.org/10.48550/ arXiv.2110.11286

    2022 doi arXiv

  16. [23]

    https://doi.org/10

    Zhou, W., Xu, Y.F.: Data-Guided Physics-Informed Neural Networks for Solv- ing Inverse Problems in Partial Differential Equations (2024). https://doi.org/10. 48550/arXiv.2407.10836

    2024 doi arXiv

  17. [24]

    Computer Methods in Applied Mechanics and Engineering 435, 117628 (2025) https://doi.org/10.1016/j.cma

    Berardi, M., Difonzo, F.V., Icardi, M.: Inverse physics-informed neural net- works for transport models in porous materials. Computer Methods in Applied Mechanics and Engineering 435, 117628 (2025) https://doi.org/10.1016/j.cma. 2024.117628

    2025

  18. [25]

    arXiv e-prints, 2311–14931 (2023) https://doi.org/10.48550/arXiv.2311

    Lei, W., Protopapas, P., Parikh, J.: One-Shot Transfer Learning for Nonlinear ODEs. arXiv e-prints, 2311–14931 (2023) https://doi.org/10.48550/arXiv.2311. 14931 [cs.LG] 34

    2023 doi

  19. [26]

    https://doi.org/10.48550/arXiv.2211.00214

    Pellegrin, R., Bullwinkel, B., Mattheakis, M., Protopapas, P.: Transfer Learn- ing with Physics-Informed Neural Networks for Efficient Simulation of Branched Flows (2022). https://doi.org/10.48550/arXiv.2211.00214

    2022 doi arXiv

  20. [27]

    IEEE Transactions on Magnetics 59(2), 1–5 (2023) https://doi.org/10.1109/CEFC55061.2022.9940890

    Baldan, M., Di Barba, P., Lowther, D.A.: Physics-informed neural networks for inverse electromagnetic problems. IEEE Transactions on Magnetics 59(2), 1–5 (2023) https://doi.org/10.1109/CEFC55061.2022.9940890

    2023

  21. [28]

    communications in Numerical Methods in Engineering 10(3), 195–201 (1994) https://doi.org/10.1002/cnm.1640100303

    Dissanayake, M., Phan-Thien, N.: Neural-network-based approximations for solv- ing partial differential equations. communications in Numerical Methods in Engineering 10(3), 195–201 (1994) https://doi.org/10.1002/cnm.1640100303

    1994 doi

  22. [29]

    IEEE Transactions on Neural Networks 11(5), 1041–1049 (2000) https://doi.org/10.1109/72.870037

    Lagaris, I.E., Likas, A.C., Papageorgiou, D.G.: Neural-network methods for boundary value problems with irregular boundaries. IEEE Transactions on Neural Networks 11(5), 1041–1049 (2000) https://doi.org/10.1109/72.870037 . Conference Name: IEEE Transactions on Neural Networks

    2000 doi

  23. [30]

    Mattheakis, M., Sondak, D., Dogra, A.S., Protopapas, P.: Hamiltonian neural networks for solving equations of motion. Phys. Rev. E 105, 065305 (2022) https: //doi.org/10.1103/PhysRevE.105.065305

    2022 doi

  24. [31]

    Journal of Computational Physics 375, 1339–1364 (2018) https://doi.org/10.1016/j.jcp.2018.08.029

    Sirignano, J., Spiliopoulos, K.: Dgm: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics 375, 1339–1364 (2018) https://doi.org/10.1016/j.jcp.2018.08.029

    2018 doi

  25. [32]

    Journal of Computational Physics 394, 56–81 (2019) https://doi

    Zhu, Y., Zabaras, N., Lu, L., Perdikaris, P.: Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. Journal of Computational Physics 394, 56–81 (2019) https://doi. org/10.1016/j.jcp.2019.05.048

    2019 doi

  26. [33]

    Choudhary, A., Lindner, J.F., Holliday, E.G., Miller, S.T., Sinha, S., Ditto, W.L.: Physics enhanced neural networks predict order and chaos. Phys. Rev. E 101, 062207 (2020) 101(6), 062207 (2019) https://doi.org/10.1103/physreve.101. 062207 arXiv:1912.01958 [physics.comp-ph]

    2020 arXiv

  27. [34]

    Choudhary, A., Lindner, J.F., Holliday, E.G., Miller, S.T., Sinha, S., Ditto, W.L.: Forecasting hamiltonian dynamics without canonical coordinates (2020) https: //doi.org/10.48550/ARXIV.2010.15201 arXiv:2010.15201 [cs.LG]

    2020 doi arXiv

  28. [35]

    Greydanus, S., Dzamba, M., Yosinski, J.: Hamiltonian neural networks (2019) https://doi.org/10.48550/ARXIV.1906.01563 arXiv:1906.01563 [cs.NE]

    2019 doi arXiv

  29. [36]

    https: //doi.org/10.1007/978-981-19-3079-9 15-1

    Alvarez, E., Velasco-Aja, E.: A Primer on Unimodular Gravity, (2023). https: //doi.org/10.1007/978-981-19-3079-9 15-1

    2023 doi

  30. [37]

    Carballo-Rubio, R., Garay, L.J., Garc ´ ıa-Moreno, G.: Unimodular gravity vs general relativity: a status report. Class. Quant. Grav. 39(24), 243001 (2022) 35 https://doi.org/10.1088/1361-6382/aca386 arXiv:2207.08499 [gr-qc]

    2022 arXiv

  31. [38]

    https://doi.org/10.48550/arXiv.1908.02729

    Hoffman, J., Roberts, D.A., Yaida, S.: Robust Learning with Jacobian Regular- ization (2019). https://doi.org/10.48550/arXiv.1908.02729

    2019 doi arXiv

  32. [40]

    JHEP 08, 034 (2018) https://doi.org/10.1007/JHEP08(2018)034 arXiv:1805.01806 [hep-th]

    Bea, Y., Mateos, D.: Heating up Exotic RG Flows with Holography. JHEP 08, 034 (2018) https://doi.org/10.1007/JHEP08(2018)034 arXiv:1805.01806 [hep-th]

    2018 arXiv

  33. [41]

    Gubser, S.S., Nellore, A.: Mimicking the QCD equation of state with a dual black hole. Phys. Rev. D 78, 086007 (2008) https://doi.org/10.1103/PhysRevD. 78.086007 arXiv:0804.0434 [hep-th] 36

    2008 arXiv

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