arxiv: 2604.22862 · v2 · submitted 2026-04-23 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Physics-Informed Neural Networks for Solving Two-Flavor Neutrino Oscillations in Vacuum and Matter Environments for Atmospheric and Reactor Neutrinos

Srinivasan T. , Kalyani Desikan

Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:14 UTC · model grok-4.3

classification ✦ hep-ph

keywords neutrino oscillationsphysics-informed neural networkstwo-flavor approximationMSW effectreactor neutrinosatmospheric neutrinosdifferential equations

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The pith

Physics-informed neural networks solve two-flavor neutrino oscillation equations with errors of 10^{-3} to 10^{-4}.

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

The paper shows that physics-informed neural networks can solve the coupled differential equations for two-flavor neutrino oscillations in both vacuum and constant-density matter. The networks reproduce analytical results for low-energy reactor neutrinos and high-energy atmospheric neutrinos to mean squared errors around 10^{-3} to 10^{-4}. Traditional methods use meshes or reductions, while this approach embeds the governing equations directly into the loss function. A reader would care because it offers a flexible alternative for tracking flavor evolution without explicit discretization.

Core claim

Physics-informed neural networks trained on the vacuum mixing and MSW-effect equations produce oscillation probability solutions for reactor and atmospheric neutrinos that match known analytical expressions to mean squared errors of order 10^{-3} to 10^{-4} in both vacuum and constant-density matter cases.

What carries the argument

Physics-informed neural networks that incorporate the neutrino flavor evolution ODEs into the training loss to enforce physical constraints during solution of the oscillatory system.

If this is right

PINNs supply a mesh-free alternative to traditional solvers for neutrino propagation problems.
The same framework handles both vacuum and matter-induced effects at comparable precision.
The method demonstrates stability when solving the coupled ODEs that describe flavor oscillations.
It opens a route to treating variable-density matter profiles without new discretization schemes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach could incorporate real detector data directly into training for parameter inference.
Similar networks might address other two-state quantum oscillation systems such as neutral meson mixing.
Extension to three-flavor oscillations or time-dependent densities would test whether accuracy holds beyond the constant-density cases shown.

Load-bearing premise

The neural network optimization converges to the exact physical solution of the coupled oscillatory ODEs without phase errors or loss of accuracy.

What would settle it

Direct side-by-side comparison of PINN-computed oscillation probabilities against the exact analytical formulas over a range of baselines and energies, checking whether phase shifts or amplitude deviations exceed the reported error level.

Figures

Figures reproduced from arXiv: 2604.22862 by Kalyani Desikan, Srinivasan T..

**Figure 2.** Figure 2: FIG. 2. PINNs architecture for neutrino propagation in matter. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Survival and Appearance Probability of Reactor neutrino in vacuum case [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Survival and Appearance Probability of Reactor neutrino in constant matter case [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Survival and Appearance Probability of Atmospheric neutrino in vacuum case [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Survival and Appearance Probability of Atmospheric neutrino in constant matter case [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

Neutrino oscillations provide crucial insights into fundamental particle physics, with two-flavor approximations effectively describing reactor and atmospheric phenomena. This paper investigates the application of Physics-Informed Neural Networks (PINNs), which have several advantages over traditional solvers. Traditional methods typically depend on mesh-based techniques or dimensionality reduction approaches to solve the governing differential equations for neutrino evolution in vacuum and matter environments. We review the theoretical framework, including vacuum mixing and the Mikheyev-Smirnov-Wolfenstein (MSW) effect in matter, and demonstrate PINN implementations for vacuum and constant-density profiles. This Machine learning based approach for reactor (low-energy) and atmospheric (high-energy) neutrinos shows high precision similar to analytical solutions, with mean squared errors of the order of 10^{-3}~10^{-4}. We have also discussed the robustness of PINNs in solving coupled ODE systems, along with future extensions to three-flavor effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

PINNs solve the constant-density two-flavor neutrino cases with decent accuracy but the variable-density atmospheric results the title promises are not shown.

read the letter

The one thing to know is that this paper takes Physics-Informed Neural Networks and applies them to the two-flavor neutrino oscillation problem in vacuum and constant matter density, getting MSE on the order of 10^{-3} to 10^{-4} that lines up with the known analytical solutions for reactor and atmospheric neutrinos in those conditions. What stands out is the clear presentation of the vacuum and MSW equations, followed by the PINN setup that avoids traditional mesh-based solvers. They show it works for both low-energy reactor and high-energy atmospheric cases under constant density, which is a useful check that the network can capture the oscillatory behavior in the coupled system. That's a concrete step forward for using PINNs in this area. The soft spot is the gap on variable density. Atmospheric neutrinos travel through Earth with changing density, so the Hamiltonian isn't constant, but the results stay with constant profiles where analytics exist. No loss curves or comparisons for the x-dependent case are given, so the broader claim depends on an assumption that the method generalizes without issues like phase drift. The abstract also skips over the network details and training process, which leaves the accuracy claim a bit thin on evidence. This paper is aimed at physicists or computational scientists who might use it as an example for applying PINNs to oscillation equations. Someone already familiar with neutrino phenomenology would see the value in the numerical approach, but it won't change how we think about the physics itself. It deserves a serious referee because the core results for the demonstrated cases are there and the method is grounded in the right equations. A reviewer could push for the missing variable-density tests and more implementation transparency. I would recommend sending it to peer review.

Referee Report

3 major / 2 minor

Summary. The paper applies Physics-Informed Neural Networks (PINNs) to solve the coupled ODEs for two-flavor neutrino oscillations in vacuum and constant-density matter, reviewing the vacuum mixing and MSW frameworks. It reports MSE values of order 10^{-3}–10^{-4} matching analytical solutions for the demonstrated cases and claims this precision extends to reactor (low-energy) and atmospheric (high-energy) neutrinos, while discussing robustness for coupled systems and future three-flavor extensions.

Significance. If the method can be shown to handle variable-density profiles without loss of accuracy or phase errors, it would provide a mesh-free alternative to standard integrators for neutrino propagation problems, particularly useful when density varies along the baseline. The reported agreement with analytics in constant-density cases indicates that the physics-informed loss can enforce the oscillation equations effectively in those regimes.

major comments (3)

[Abstract] Abstract: the claim that the approach 'shows high precision similar to analytical solutions' for atmospheric (high-energy) neutrinos is unsupported, as the text states that PINN implementations are demonstrated only for vacuum and constant-density profiles; no results are given for position-dependent density (e.g., PREM) that turns the MSW Hamiltonian into an x-dependent function.
[Results] Results section (demonstrations): the manuscript provides no architecture details, loss weighting, training procedure, or comparisons to independent numerical solvers beyond the quoted MSE values, so the central accuracy claim for the oscillatory system lacks the evidence needed to substantiate convergence to the true solution.
[Title/Abstract] Title and abstract scope: the headline applicability to atmospheric neutrinos rests on an untested generalization from constant-density to variable-density cases; the skeptic concern is confirmed by the absence of any loss curves, numerical results, or integrator comparisons for x-dependent coefficients.

minor comments (2)

[Introduction] Clarify in the introduction whether the reported MSE values apply only to the constant-density demonstrations or also to any variable-density tests that may exist in the full manuscript.
The notation for the two-flavor Hamiltonian and the PINN loss function should be cross-referenced to specific equations to improve readability for readers unfamiliar with PINN implementations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major point below and have revised the manuscript to clarify the demonstrated scope, supply missing technical details, and align all claims with the presented results.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the approach 'shows high precision similar to analytical solutions' for atmospheric (high-energy) neutrinos is unsupported, as the text states that PINN implementations are demonstrated only for vacuum and constant-density profiles; no results are given for position-dependent density (e.g., PREM) that turns the MSW Hamiltonian into an x-dependent function.

Authors: We agree that the abstract overstates applicability to atmospheric neutrinos. Our demonstrations and quantitative results are restricted to vacuum and constant-density matter. We will revise the abstract to state that high precision (MSE of order 10^{-3}–10^{-4}) is shown for the constant-density cases examined, and we will explicitly note that extension to position-dependent density profiles is a planned future direction rather than a current result. revision: yes
Referee: [Results] Results section (demonstrations): the manuscript provides no architecture details, loss weighting, training procedure, or comparisons to independent numerical solvers beyond the quoted MSE values, so the central accuracy claim for the oscillatory system lacks the evidence needed to substantiate convergence to the true solution.

Authors: We accept this criticism. The original manuscript omitted these implementation details. In the revised version we will add a dedicated subsection describing the neural-network architecture (depth, width, activation functions), the precise form of the physics-informed loss and any weighting coefficients, the optimizer and training schedule, and direct side-by-side comparisons with solutions obtained from standard ODE integrators (e.g., SciPy’s odeint) to strengthen the evidence for convergence. revision: yes
Referee: [Title/Abstract] Title and abstract scope: the headline applicability to atmospheric neutrinos rests on an untested generalization from constant-density to variable-density cases; the skeptic concern is confirmed by the absence of any loss curves, numerical results, or integrator comparisons for x-dependent coefficients.

Authors: We concur that the title and abstract imply broader applicability than the results support. We will revise the title to focus on vacuum and constant-density matter environments and will update the abstract accordingly. We will also include training-loss curves and additional numerical-integrator benchmarks in the revised Results section to document performance on the x-independent cases actually solved. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard PINN application to known ODEs

full rationale

The paper inputs the standard two-flavor vacuum and MSW Hamiltonian equations into a PINN loss function and compares outputs to analytical solutions for vacuum and constant-density cases. This is a conventional numerical solver validation with no reduction of results to fitted parameters by construction, no self-definitional loops, and no load-bearing self-citations or imported uniqueness theorems. The derivation chain remains independent of the reported MSE values, which serve only as external benchmarks rather than redefinitions of the input physics.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard two-flavor oscillation framework and the assumption that PINN loss functions can enforce the governing ODEs accurately; no new physical entities are introduced.

free parameters (1)

PINN hyperparameters
Network depth, width, and loss-term weighting coefficients must be chosen or tuned to achieve the reported accuracy.

axioms (1)

domain assumption Neutrino flavor evolution obeys the standard two-flavor vacuum mixing and MSW matter equations.
Invoked as the physics constraints embedded in the PINN loss function.

pith-pipeline@v0.9.0 · 5462 in / 1188 out tokens · 32722 ms · 2026-05-14T22:14:07.605288+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We demonstrate PINN implementations for vacuum and constant-density profiles... mean squared errors of the order of 10^{-3}~10^{-4}.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

i d/dx ψ(x) − H(x)ψ(x) = 0 ... LODE = 1/Nf Σ ||i dψ/dx (xi) − Ĥψ (xi)||²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · 5 internal anchors

[1]

or an exponential solar profile, we use H(x) = Hvac + VCC(x), (34) with the electron density ne(x) given by the chosen model. The distinguishing feature for the matter case lies in the physics-informed loss function which forces the net- work to discover the non-linear amplitude enhancement and frequency shifts associated with the MSW effect. Figure 2 sho...

work page
[2]

Vacuum Propagation The PINN was trained to solve the vacuum evolution equation over a baseline length L suﬀicient to capture the oscillation cycles. Fig. 3 compares the PINN-predicted survival probability P (νe → νe) and the appearance probability P (νe → νµ) with the analytical vacuum for- mula, as given in equations (13) and (14) The generated graph dis...

work page
[3]

The PINN solved the modi- fied Schrödinger equation including the matter term

Constant Matter Propagation (MSW Effect) To investigate matter effects, we introduce a constant electron density potential V . The PINN solved the modi- fied Schrödinger equation including the matter term. De- spite the non-maximal vacuum mixing angle ( θ = 33.4◦), the matter potential modifies the effective mixing pa- rameters. The generated graph in Fig...

work page
[4]

Figure 5 illustrates the survival and appearance probabilities for the same

Vacuum Propagation In the atmospheric vacuum case, the oscillation is gov- erned by near-maximal mixing ( sin2 2θ ≈ 1). Figure 5 illustrates the survival and appearance probabilities for the same. The PINNs seemlessly produces the survival and ap- pearance probability graphs cleanly without any data points. The visualization closely mirrors the maximal am...

work page
[5]

Consequently, the matter effect is less pronounced than in the resonance region

Constant Matter Propagation In the case of matter, at E = 0 .2 GeV, the matter potential V is relatively small compared to the vacuum term ∆m2/2E. Consequently, the matter effect is less pronounced than in the resonance region. The PINN ef- fectively reproduces this behavior, generating a solution that deviates only slightly from the vacuum case, con- sis...

work page 2022
[6]

Fukuda, T

Y. Fukuda, T. Hayakawa, E. Ichihara, K. Inoue, K. Ishi- hara, H. Ishino, Y. Itow, T. Kajita, J. Kameda, S. Ka- suga, K. Kobayashi, Y. Kobayashi, Y. Koshio, M. Miura, M. Nakahata, S. Nakayama, A. Okada, K. Okumura, N. Sakurai, M. Shiozawa, Y. Suzuki, Y. Takeuchi, Y. Totsuka, S. Yamada, M. Earl, A. Habig, E. Kearns, M. D. Messier, K. Scholberg, J. L. Stone,...

work page 1998
[7]

Q. R. Ahmad, R. C. Allen, T. C. Andersen, J. D.Anglin, J. C. Barton, E. W. Beier, M. Bercovitch, J. Bigu, S. D. Biller, R. A. Black, I. Blevis, R. J. Boardman, J. Boger, E. Bonvin, M. G. Boulay, M. G. Bowler, T. J. Bowles, S. J. Brice, M. C. Browne, T. V. Bullard, G. Bühler, J. Cameron, Y. D. Chan, H. H. Chen, M. Chen, X. Chen, B. T. Cleveland, E. T. H. C...

work page 2002
[8]

Giunti and C

C. Giunti and C. W. Kim, Fundamentals of Neu- trino Physics and Astrophysics (Oxford University Press, 2007)

work page 2007
[9]

Eguchi, S

K. Eguchi, S. Enomoto, K. Furuno, J. Goldman, H. Hanada, H. Ikeda, K. Ikeda, K. Inoue, K. Ishihara, W. Itoh, T. Iwamoto, T. Kawaguchi, T. Kawashima, H. Kinoshita, Y. Kishimoto, M. Koga, Y. Koseki, T. Maeda, T. Mitsui, M. Motoki, K. Nakajima, M. Naka- jima, T. Nakajima, H. Ogawa, K. Owada, T. Sak- abe, I. Shimizu, J. Shirai, F. Suekane, A. Suzuki, K. Tada,...

work page 2003
[10]

K. Abe, N. Abgrall, Y. Ajima, H. Aihara, J. B. Al- bert, C. Andreopoulos, B. Andrieu, S. Aoki, O. Araoka, J. Argyriades, A. Ariga, T. Ariga, S. Assylbekov, D. Au- tiero, A. Badertscher, M. Barbi, G. J. Barker, G. Barr, M. Bass, F. Bay, S. Bentham, V. Berardi, B. E. Berger, I. Bertram, M. Besnier, J. Beucher, D. Beznosko, S. Bhadra, F. d. M. Blaszczyk, A. ...

work page 2011
[11]

F. P. An, J. Z. Bai, A. B. Balantekin, H. R. Band, D. Beavis, W. Beriguete, M. Bishai, S. Blyth, K. Boddy, R. L. Brown, B. Cai, G. F. Cao, J. Cao, R. Carr, W. T. Chan, J. F. Chang, Y. Chang, C. Chasman, H. S. Chen, H. Y. Chen, S. J. Chen, S. M. Chen, X. C. Chen, X. H. Chen, X. S. Chen, Y. Chen, Y. X. Chen, J. J. Cher- winka, M. C. Chu, J. P. Cummings, Z. ...

work page 2012
[12]

P. A. Zyla et al. (Particle Data Group), Review of Par- ticle Physics, PTEP 2020, 083C01 (2020)

work page 2020
[13]

Esteban, M.C

I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09, 178, arXiv:2007.14792 [hep-ph]

work page arXiv 2007
[14]

P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez- Miravé, O. Mena, C. A. Ternes, M. Tórtola, and J. W. F. Valle, 2020 global reassessment of the neutrino oscillation picture, Journal of High Energy Physics 2021, 71 (2021), arXiv:2006.11237 [hep-ph]

work page arXiv 2020
[15]

T. A. Mueller, D. Lhuillier, M. Fallot, A. Letourneau, S. Cormon, M. Fechner, L. Giot, T. Lasserre, J. Mar- tino, G. Mention, A. Porta, and F. Yermia, Improved predictions of reactor antineutrino spectra, Phys. Rev. C 83, 054615 (2011)

work page 2011
[16]

J. K. Ahn, S. Chebotaryov, J. H. Choi, S. Choi, W. Choi, Y. Choi, H. I. Jang, J. S. Jang, E. J. Jeon, I. S. Jeong, K. K. Joo, B. R. Kim, B. C. Kim, H. S. Kim, J. Y. Kim, S. B. Kim, S. H. Kim, S. Y. Kim, W. Kim, Y. D. Kim, J. Lee, J. K. Lee, I. T. Lim, K. J. Ma, M. Y. Pac, I. G. Park, J. S. Park, K. S. Park, J. W. Shin, K. Siyeon, B. S. Yang, I. S. Yeo, S....

work page 2012
[17]

F. P. An, A. B. Balantekin, H. R. Band, M. Bishai, S. Blyth, I. Butorov, D. Cao, G. F. Cao, J. Cao, W. R. Cen, Y. L. Chan, J. F. Chang, L. C. Chang, Y. Chang, H. S. Chen, Q. Y. Chen, S. M. Chen, Y. X. Chen, Y. Chen, J. H. Cheng, J. Cheng, Y. P. Cheng, J. J. Cher- winka, M. C. Chu, J. P. Cummings, J. de Arcos, Z. Y. Deng, X. F. Ding, Y. Y. Ding, M. V. Diwa...

work page 2016
[18]

M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado, and T. Schwetz, Global fit to three neutrino mixing: critical look at present precision, JHEP 12, 123, arXiv:1209.3023 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv
[19]

T. K. Gaisser and M. Honda, Flux of atmospheric neutri- nos, Ann. Rev. Nucl. Part. Sci. 52, 153 (2002), arXiv:hep- ph/0203272

work page arXiv 2002
[20]

KAJITA, Atmospheric neutrinos and discovery of neu- trino oscillations, Proceedings of the Japan Academy, Se- ries B 86, 303 (2010)

T. KAJITA, Atmospheric neutrinos and discovery of neu- trino oscillations, Proceedings of the Japan Academy, Se- ries B 86, 303 (2010)

work page 2010
[21]

E. K. Akhmedov, Neutrino Physics, in Particle Physics , edited by G. Senjanović and A. Y. Smirnov (2000) p. 103, arXiv:hep-ph/0001264 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[22]

K. Abe, C. Bronner, Y. Haga, Y. Hayato, M. Ikeda, K. Iyogi, J. Kameda, Y. Kato, Y. Kishimoto, L. Marti, M. Miura, S. Moriyama, M. Nakahata, T. Naka- jima, Y. Nakano, S. Nakayama, Y. Okajima, A. Orii, G. Pronost, H. Sekiya, M. Shiozawa, Y. Sonoda, A. Takeda, A. Takenaka, H. Tanaka, S. Tasaka, T. To- mura, R. Akutsu, T. Irvine, T. Kajita, I. Kametani, K. Ka...

work page 2018
[23]

M. A. Acero, P. Adamson, L. Aliaga, T. Alion, V. Al- lakhverdian, S. Altakarli, N. Anfimov, A. Antoshkin, A. Aurisano, A. Back, C. Backhouse, M. Baird, N. Bal- ashov, P. Baldi, B. A. Bambah, S. Bashar, K. Bays, S. Bending, R. Bernstein, V. Bhatnagar, B. Bhuyan, J. Bian, T. Blackburn, J. Blair, A. C. Booth, P. Bour, C. Bromberg, N. Buchanan, A. Butkevich, ...

work page 2019
[24]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Com- puting (Cambridge University Press, 2007)

work page 2007
[25]

A. M. Dziewonski and D. L. Anderson, Preliminary refer- ence earth model, Phys. Earth Planet. Interiors 25, 297 (1981)

work page 1981
[26]

Kopp, Phenomenology of Three-Flavour Neutrino Oscillations, diploma, Technische Universität München, München (2006)

J. Kopp, Phenomenology of Three-Flavour Neutrino Oscillations, diploma, Technische Universität München, München (2006)

work page 2006
[27]

Morales, G

G. Morales, G. Lehaut, A. Vacheret, F. Jurie, and J. Fadili, Neutrino Oscillation Parameter Estimation Us- ing Structured Hierarchical Transformers, arXiv e-prints , arXiv:2603.22342 (2026), arXiv:2603.22342 [hep-ph]

work page arXiv 2026
[28]

Raissi, P

M. Raissi, P. Perdikaris, and G. 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 (2019)

work page 2019
[29]

Carleo, I

G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Machine learning and the physical sciences, Reviews of Modern Physics 91, 045002 (2019)

work page 2019
[30]

Karniadakis, Y

G. Karniadakis, Y. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Physics-informed machine learn- ing, Nature Reviews Physics , 1 (2021)

work page 2021
[31]

Cuomo, V

S. Cuomo, V. Schiano di Cola, F. Giampaolo, G. Rozza, M. Raissi, and F. Piccialli, Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What’s next, arXiv e-prints , arXiv:2201.05624 (2022), arXiv:2201.05624 [cs.LG]

work page arXiv 2022
[32]

Raissi, A

M. Raissi, A. Yazdani, and G. E. Karniadakis, Hid- den fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science 367, 1026 (2020), https://www.science.org/doi/pdf/10.1126/science.aaw4741

work page doi:10.1126/science.aaw4741 2020
[33]

X. Jin, S. Cai, H. Li, and G. E. Karniadakis, NSFnets (Navier-Stokes flow nets): Physics-informed neural net- works for the incompressible Navier-Stokes equations, Journal of Computational Physics 426, 109951 (2021), arXiv:2003.06496 [physics.comp-ph]

work page arXiv 2021
[34]

I. E. Lagaris, A. Likas, and D. I. Fotiadis, Artificial Neu- ral Networks for Solving Ordinary and Partial Differen- tial Equations, arXiv e-prints , physics/9705023 (1997), arXiv:physics/9705023 [physics.comp-ph]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[35]

L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis, DeepXDE: A deep learning library for solving differen- tial equations, arXiv e-prints , arXiv:1907.04502 (2019), arXiv:1907.04502 [cs.LG]

work page arXiv 1907
[36]

Z. Maki, M. Nakagawa, and S. Sakata, Remarks on the Unified Model of Elementary Particles, Progress of The- oretical Physics 28, 870 (1962)

work page 1962
[37]

Wolfenstein, Neutrino oscillations in matter, Phys

L. Wolfenstein, Neutrino oscillations in matter, Phys. Rev. D 17, 2369 (1978)

work page 1978
[38]

J. N. Bahcall, NEUTRINO ASTROPHYSICS (Cam- bridge University Press, 1989)

work page 1989
[39]

S. Wang, Y. Teng, and P. Perdikaris, Understanding and Mitigating Gradient Flow Pathologies in Physics- Informed Neural Networks, SIAM Journal on Scientific Computing 43, A3055 (2021), arXiv:2001.04536 [cs.LG]

work page arXiv 2021
[40]

D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014)

work page internal anchor Pith review Pith/arXiv arXiv 2014
[41]

Rathore, W

P. Rathore, W. Lei, Z. Frangella, L. Lu, and M. Udell, Challenges in Training PINNs: A Loss Landscape Perspective, arXiv e-prints , arXiv:2402.01868 (2024), arXiv:2402.01868 [cs.LG]

work page arXiv 2024
[42]

Hannan Mustajab, H

A. Hannan Mustajab, H. Lyu, Z. Rizvi, and F. Wuttke, Physics-Informed Neural Networks for High-Frequency and Multi-Scale Problems using Transfer Learning, arXiv e-prints , arXiv:2401.02810 (2024), arXiv:2401.02810 [cs.LG]

work page arXiv 2024
[43]

Sitzmann, J

V. Sitzmann, J. N. P. Martel, A. W. Bergman, D. B. Lindell, and G. Wetzstein, Implicit Neural Representa- tions with Periodic Activation Functions, arXiv e-prints , arXiv:2006.09661 (2020), arXiv:2006.09661 [cs.CV]

work page arXiv 2006
[44]

A. D. Jagtap, K. Kawaguchi, and G. E. Karniadakis, Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, Jour- nal of Computational Physics 404, 109136 (2020), arXiv:1906.01170 [physics.comp-ph]

work page arXiv 2020
[45]

Paszke, S

A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al. , Pytorch: An imperative style, high-performance deep learning library, Advances in neural information processing systems 32 (2019)

work page 2019
[46]

Automatic differentiation in machine learning: a survey

A. Gunes Baydin, B. A. Pearlmutter, A. Andreye- vich Radul, and J. M. Siskind, Automatic differenti- ation in machine learning: a survey, arXiv e-prints , arXiv:1502.05767 (2015), arXiv:1502.05767 [cs.SC]

work page internal anchor Pith review Pith/arXiv arXiv 2015