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REVIEW 4 major objections 5 minor 89 references

Adaptive collocation sampling and trainable loss weighting cut quantum physics-informed neural network errors by 60% or more, evidence that QPINNs are gated by optimization, not just circuit expressivity.

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2026-08-05 00:11 UTC pith:UJ7GUYUB

load-bearing objection Adaptive collocation and trainable loss weights deliver consistent gains in these QPINN benchmarks, but the paper's signature 'optimization, not expressivity' conclusion is untested and the 60–90% headline numbers are measured against a deliberately weak baseline. the 4 major comments →

arxiv 2608.00850 v1 pith:UJ7GUYUB submitted 2026-08-01 cs.LG physics.flu-dyn

Adaptive Quantum Physics-Informed Neural Networks for Differential Equations with Applications to Fluid Dynamics

Fabio Pereira dos Santos , Renato Portugal , J\'ulio de Castro Vargas Fernandes , Lucas Timotheo Sanches This is my paper
classification cs.LG physics.flu-dyn MSC 68T0765M9981P68
keywords quantum physics-informed neural networksadaptive collocation samplingtrainable loss weightingvariational quantum circuitspartial differential equationsBurgers equationNavier-Stokes flowshybrid quantum-classical training
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that quantum physics-informed neural networks (QPINNs) — hybrid quantum-classical models that solve differential equations by minimizing physics, boundary, and data losses — fail on hard problems mostly because of how they are trained, not just because of what their small quantum circuits can express. The authors build AQPINN on a fixed quantum architecture and add two classical training upgrades: collocation points that migrate toward high-residual regions, spawn near persistent error, and get pruned when stale, and a trainable softmax weighting that rebalances physics, boundary, and initial losses during training. On six benchmarks — a Helmholtz equation, a damped oscillator, a 2D Poisson equation, Burgers' equation, and two Navier–Stokes flows — the adaptive version cuts root-mean-square error by roughly 64 to 90 percent against a static QPINN and lifts R² from near zero to 0.98–0.99 on the worst cases. If correct, the practical message is that classical optimization design decides whether QPINNs solve PDEs, and bigger quantum circuits alone will not fix stalled training.

Core claim

The paper's central claim is that standard QPINNs fail less from the expressive limits of their variational circuits than from static training choices: uniform collocation sampling and fixed loss weights trap the optimizer in poor regions of a non-convex landscape. AQPINN counters with residual-tracked collocation dynamics — gradient-ascent movement of high-error points, Gaussian-noise fission near persistent errors, age-aware pruning — and a softmax attention mechanism that learns the loss weights during training. With the quantum circuit held fixed, the baseline's R² rises from 0.395 or negative to 0.99 on the worst benchmarks, the Burgers shock's maximum error drops 66.8%, and RMSE falls

What carries the argument

Carrying the argument is a fixed hybrid quantum-classical ansatz uθ(t,x) = N_post ∘ Q_θq ∘ N_pre(t,x), whose quantum layer Q_θq is a variational circuit built from data-encoding RX/RZ gates, variational RZ–RY–RZ rotations, and CNOT entangling layers, outputting Pauli-Z expectation values and trained via the parameter-shift rule. Around this sits the adaptive loop: collocation points move along the residual gradient ∇(r²), spawn Gaussian-noise copies near persistent high-residual regions under exclusion zones and a hard cap, and are pruned by the importance score S_i = r̄_i + 0.1/(a_i + 1), which balances residual history against point age; the loss weights (ωp, ωb, ωi) = softmax(α) are thems

Load-bearing premise

The conclusion that optimization — not expressivity — is the QPINN bottleneck assumes that the 3–4 qubit circuits used in all six benchmarks can already represent the true PDE solutions; the paper never varies circuit size or depth to test this, and admits the sizes were chosen for computational feasibility (Section 3).

What would settle it

Run AQPINN with the adaptive mechanisms intact but the quantum layer replaced by a classical MLP of comparable parameter count: if accuracy is unchanged, the quantum circuit is not doing the work. Separately, scale the static baseline from 4 to 10+ qubits and layers: if added expressivity alone closes the gap to AQPINN's accuracy on the same benchmarks, the optimization-bottleneck claim is refuted.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Adaptive training should be treated as part of QPINN methodology: the mechanisms are architecture-agnostic, so the gains are expected to carry over to larger circuits and higher-dimensional PDEs without redesign.
  • Baseline QPINN failures that look like representational limits — flat loss plateaus, unphysical trivial states, shock smearing — are frequently training artifacts, so expressivity-only explanations of QPINN underperformance are insufficient.
  • Residual-driven collocation re-meshing resolves moving shocks and boundary layers without increasing model parameter count or dataset size, making it a practical route to higher resolution in stiff regimes.
  • The two adaptive mechanisms are complementary: the ablation shows each alone improves accuracy on Burgers' and Taylor–Couette benchmarks, but the combined framework is strictly better.

Where Pith is reading between the lines

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

  • A decisive test the paper leaves open: fix the adaptive collocation and loss weighting but swap the quantum layer for a classical MLP of matched width. If a classical adaptive PINN matches AQPINN, the reported gains are properties of the training scheme, and the quantum component's contribution to these benchmarks is not demonstrated.
  • The abstract promises gains on reaction-diffusion systems, but the six reported benchmarks are ODEs and fluid-flow PDEs; the 'at least 60%' figure should be read as applying to those benchmarks, and extending it to reaction-diffusion would be extrapolation.
  • If optimization is truly the bottleneck, increasing circuit depth or qubit count with static training should show sharply diminishing returns — the paper states this direction as future work, so a scaling study would directly test the central claim.
  • The ablation shows the two mechanisms contribute unevenly across problems (attention beats collocation alone on Burgers; collocation beats attention alone on Taylor-Couette), suggesting the optimal adaptive strategy depends on solution smoothness — a dependence the paper leaves implicit.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This manuscript proposes an Adaptive Quantum Physics-Informed Neural Network (AQPINN) that combines two modifications to a standard QPINN: residual-driven adaptive collocation point sampling (gradient-driven relocation, fission near persistent high-residual regions, and pruning of low-importance points) and a softmax-parameterized trainable loss-weighting scheme. The method is evaluated on six benchmarks—a 1D Helmholtz boundary-value problem, a spring-mass ODE, a 2D Poisson equation, Burgers' equation, Hagen-Poiseuille flow, and Taylor-Couette flow—against a fixed QPINN baseline. The reported results show consistent improvements in RMSE, relative error, and R² for AQPINN, and an ablation in the appendix separates the two adaptive components on Burgers' and Taylor-Couette benchmarks. The paper interprets these results as evidence that optimization, not only circuit expressivity, is an important bottleneck for QPINNs.

Significance. If the empirical claims hold, the paper provides a useful demonstration that classical training strategies—adaptive collocation and dynamic loss weighting—can materially improve QPINN accuracy, and it gives a concrete recipe for doing so. The six benchmarks and the two-benchmark ablation are strengths, as is the explicit acknowledgment that comparisons to classical adaptive PINNs and quantum speedups are outside the scope. However, the central conceptual conclusion about 'optimization, not only expressivity' is not directly tested, and the quantitative headline is measured against a baseline that the authors deliberately chose to be challenging. The paper is therefore a solid empirical contribution whose interpretational claims need to be brought in line with the evidence.

major comments (4)
  1. [Section 3 and Section 3.3] The abstract and conclusion claim that 'optimization, not only expressivity, can be an important bottleneck' in QPINNs. This claim is not supported by the experiments because no experiment varies circuit width or depth. All circuits use 3–4 qubits and 1–6 layers, chosen 'to balance computational feasibility' (Section 3), and the sentence that the method is 'architecture-agnostic and expected to scale to larger circuits' is asserted without supporting evidence. To make the bottleneck claim load-bearing, the authors should add a capacity sweep (e.g., increasing qubits/layers while keeping the training strategy fixed) or, alternatively, explicitly rephrase the conclusion as a statement about the small-circuit regime studied here.
  2. [Section 3, introductory paragraph] The authors state that rather than performing hyperparameter optimization, they 'identify a challenging configuration in which the baseline QPINN exhibits noticeable performance degradation or convergence failure,' and then use that configuration as the reference. This means the 60–90% improvements reported in the abstract and results are relative to a deliberately weakened baseline. The gains are still informative, but the paper should report results for a more standard baseline configuration (or a small set of baseline configurations) so the reader can see whether adaptive training helps when the baseline is not artificially handicapped, and should explicitly label the current numbers as a best-case demonstration.
  3. [Abstract and Section 3] The abstract promises improvements for 'benchmark fluid flows and reaction-diffusion systems,' but no reaction-diffusion benchmark appears in the Results section. The six benchmarks are Helmholtz, spring-mass, 2D Poisson, Burgers, Hagen-Poiseuille, and Taylor-Couette; none is a reaction-diffusion system. If reaction-diffusion experiments were performed, they should be added; otherwise the abstract should be revised to match the actual scope of the experiments.
  4. [Section 3.2.2, Burgers' equation] The text describes the baseline QPINN's large error as 'associated with shock smearing, where insufficient local representational capacity leads to over-smoothing of the sharp transition.' This wording conflicts with the paper's central claim that expressivity is not the main bottleneck and that adaptive training fixes optimization rather than representational capacity. If 'representational capacity' here means collocation-point density rather than circuit expressivity, this should be stated explicitly; as written, the sentence could be read as supporting the opposite of the paper's main conclusion.
minor comments (5)
  1. [Section 3.1.1] The Helmholtz boundary conditions are not stated explicitly; the text only says the analytical solution is 'consistent with the imposed boundary conditions.' Please specify the boundary conditions used in the numerical experiment.
  2. [Section 3.1.2] Equation (21) is described as a boundary-value problem, but the conditions given are initial conditions, u(0)=0 and u'(0)=5. 'Initial-value problem' would be more accurate.
  3. [Equation (20)] The importance score in Eq. (20) is introduced as 'ad-hoc,' but the meaning of rbar_i is not defined in the text. Clarify whether it is the time-averaged squared residual and specify units/scaling so the formula is reproducible.
  4. [Figures 6 and 7] The captions of Figures 6 and 7 are inconsistent: Figure 6 is labeled as the QPINN solution 'with attention mechanism and adaptive strategy' and Figure 7 as 'without attention mechanism and adaptive strategy,' while the surrounding text appears to describe the baseline and adaptive results in the opposite order. Please check the assignment of figures to models.
  5. [Section 3.2.1] The text mentions 'adaptive attention weights [0.4,0.6]' in the setup, which looks like fixed values rather than the output of the trainable softmax mechanism. Clarify whether these are initial values, final values, or a fixed schedule.

Circularity Check

0 steps flagged

No circular derivation: the adaptive gains are measured against closed-form analytical solutions, and no predicted quantity reduces to a fitted constant or to a self-citation.

full rationale

The paper's central numerical claims are empirical comparisons against analytical reference solutions (Helmholtz, spring-mass, Poisson, Burgers, Hagen-Poiseuille, Taylor-Couette), so the reported errors are not defined by the adaptive mechanism itself. The attention weights and collocation-point updates are trained using PDE residuals, but the final accuracy metrics are computed with respect to u_true, not with respect to the training signal. No equation in the paper reduces a prediction to a fitted parameter: the adaptive weights and points are optimization aids, not components of the ground-truth solution. The baseline QPINN is deliberately chosen as a 'challenging configuration' exhibiting 'noticeable performance degradation or convergence failure' (Section 3), which may inflate the reported relative improvements, but this is a benchmarking/validity concern, not a circularity of derivation. Similarly, the conclusion that optimization rather than expressivity is a bottleneck is under-supported because circuit size/depth is never varied, but this is a missing control, not an equation-level equivalence. The only apparent self-citation (Queiroz et al. 2021, which includes author Santos) supports a background statement about PINN applications and is not load-bearing. No uniqueness theorem or ansatz is imported from the authors' prior work to force the main conclusion. Therefore, no significant circularity is present.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 0 invented entities

No new physical entities, particles, forces, or dimensions are introduced. The method is composed of existing components; the only new object is the ad hoc importance score formula, which is a scoring rule, not an entity.

free parameters (5)
  • Baseline configuration selection
    The authors deliberately pick a configuration where baseline QPINN fails (Section 3), which controls the magnitude of the reported improvement.
  • Collocation point budgets per benchmark = Helmholtz 20->40, Poisson 40->100, Burgers 150->200, Hagen-Poiseuille 20->40, Taylor-Couette 200->600
    Initial and max points are chosen by hand per case; Taylor-Couette gives AQPINN up to 3x the baseline's points, confounding the accuracy comparison.
  • Adaptive sampling hyperparameters = EMA decay 0.7, outlier factor 20.0, zeta_max 0.1, fission top 15%, sigma 0.05, pruning bottom 15%, ages 10-20% epochs
    All thresholds are fixed ad hoc without sensitivity analysis; the method's behavior depends on them.
  • Importance score coefficients = S_i = rbar_i + 0.1/(a_i+1)
    Equation (20) is called an 'ad-hoc formula' by the authors; the 0.1 constant is manually chosen.
  • Network and circuit sizes = 3-4 qubits, 1-6 layers, 1-32 classical neurons
    Chosen for computational feasibility (Section 3), not to demonstrate expressivity sufficiency.
axioms (4)
  • standard math Parameter-shift rule gives unbiased gradient estimates for the variational circuit parameters.
    Used in Eq. (12) to train the quantum layer; valid for noiseless expectation values of Pauli observables.
  • domain assumption The PennyLane default.qubit simulator faithfully represents the quantum circuit without decoherence or shot noise.
    All experiments are simulated; the paper does not claim hardware results, so this assumption bounds relevance.
  • ad hoc to paper The hybrid ansatz with 3-4 qubits is expressive enough to approximate the exact PDE solutions, so remaining baseline error is due to optimization, not capacity.
    This is the key premise behind the 'optimization not expressivity' conclusion; no expressivity experiment is performed.
  • ad hoc to paper Residual-based collocation adaptation and softmax loss reweighting improve QPINN training.
    The central empirical premise, supported only by the paper's benchmarks; no theoretical convergence argument is given (Section 2.3).

pith-pipeline@v1.3.0-alltime-deepseek · 18644 in / 18925 out tokens · 175062 ms · 2026-08-05T00:11:38.573618+00:00 · methodology

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Cite this review

Pith. "Pith review of Adaptive Quantum Physics-Informed Neural Networks for Differential Equations with Applications to Fluid Dynamics." pith.science (2026). https://pith.science/paper/UJ7GUYUB

@misc{pith2026260800850,
  author       = {Pith},
  title        = {Pith review of: Adaptive Quantum Physics-Informed Neural Networks for Differential Equations with Applications to Fluid Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UJ7GUYUB}},
  note         = {Machine review of arXiv:2608.00850}
}
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read the original abstract

Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.

Figures

Figures reproduced from arXiv: 2608.00850 by Fabio Pereira dos Santos, J\'ulio de Castro Vargas Fernandes, Lucas Timotheo Sanches, Renato Portugal.

Figure 1
Figure 1. Figure 1: Schematic of the physics-informed neural network (PINN) for free-surface wave problems. The network [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of QPINN and AQPINN solutions for Helmholtz equation. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of QPINN and AQPINN solutions for spring-mass problem [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: AQPINN solution for Poisson equation with attention mechanism and adaptive strategy [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: QPINN solution for Poisson equation. coefficient of determination increases to R2 = 0.98, indicating a substantially more stable and physically consistent reconstruction of the shock dynamics. The optimization behavior can be understood by examining the decoupled loss components. For Burgers’ equation, the resulting landscape is highly non-convex, with strong competition between physics, initial, and bound… view at source ↗
Figure 6
Figure 6. Figure 6: QPINN solution for Burgers’ Equation attention mechanism and adaptive strategy. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: QPINN solution for Burgers’ Equation without attention mechanism and adaptive strategy [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: QPINN solution for Hagen-Poiseuille flow with attention mechanism and adaptive strategy [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: QPINN solution for Hagen-Poiseuille flow [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: QPINN solution for Taylor-Couette flow with attention mechanism and adaptive strategy [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: QPINN solution for Taylor-Couette flow. 3.2.4 Navier-Stokes equations - Taylor-Couette flow Finally, as a last case, we consider the steady, incompressible Navier–Stokes equations for viscous flow be￾tween two concentric, infinitely long rotating cylinders in polar coordinates over the annular domain Ω = (r, θ), |, r ∈ [Ri , Ro], , θ ∈ [0, 2π]. Under axisymmetry, the velocity field u = (ur, uθ) satisfies … view at source ↗

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Reference graph

Works this paper leans on

89 extracted references · 76 canonical work pages · 1 internal anchor

  1. [1]

    Feynman, Robert B

    Richard P. Feynman, Robert B. Leighton, and Matthew Sands. The Feynman Lectures on Physics. Addison-Wesley, Boston, MA, 1964

    1964

  2. [2]

    Finding solutions of the Navier--Stokes equations through quantum computing: Progress, a generalization, and next steps forward

    Frank Gaitan. Finding solutions of the Navier--Stokes equations through quantum computing: Progress, a generalization, and next steps forward. Advanced Quantum Technologies, 4 0 (10): 0 2100055, 2021

    2021

  3. [3]

    Quantum computation of fluid dynamics

    Jeffrey Yepez. Quantum computation of fluid dynamics. In Colin P. Williams, editor, Quantum Computing and Quantum Communications, pages 34--60, Berlin, Heidelberg, 1999. Springer Berlin Heidelberg

    1999

  4. [4]

    Quantum computing in the NISQ era and beyond

    John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2: 0 79, August 2018

    2018

  5. [5]

    Massively parallel numerical simulation using up to 36,000 CPU cores of an industrial-scale polydispersed reactive pressurized fluidized bed with a mesh of one billion cells

    Herv \'e Neau, Maxime Pigou, Pascal Fede, Renaud Ansart, Cyril Baudry, Nicolas M \'e rigoux, J \'e r \^o me Lavi \'e ville, Yvan Fournier, Nicolas Renon, and Olivier Simonin. Massively parallel numerical simulation using up to 36,000 CPU cores of an industrial-scale polydispersed reactive pressurized fluidized bed with a mesh of one billion cells. Powder ...

    2020

  6. [6]

    Childs, Dmitri Maslov, Yunseong Nam, Neil J

    Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su. Toward the first quantum simulation with quantum speedup. Proceedings of the National Academy of Sciences, 115 0 (38): 0 9456--9461, 2018

    2018

  7. [7]

    Quantum algorithms for fluid simulations

    Ren \'e Steijl. Quantum algorithms for fluid simulations. In Francisco Bulnes, Vasilios N. Stavrou, Oleg Morozov, and Anton V. Bourdine, editors, Advances in Quantum Communication and Information, chapter 3. IntechOpen, Rijeka, 2020

    2020

  8. [8]

    Ren \'e Steijl and George N. Barakos. Parallel evaluation of quantum algorithms for computational fluid dynamics. Computers and Fluids, 173: 0 22--28, 2018

    2018

  9. [9]

    Quantum lattice-gas model for the burgers equation

    Jeffrey Yepez. Quantum lattice-gas model for the burgers equation. Journal of Statistical Physics, 107: 0 203--224, 2002

    2002

  10. [10]

    Joshi, Petar Jurcevic, Christine A

    Christian Kokail, Christine Maier, Rick van Bijnen, Tiff Brydges, Manoj K. Joshi, Petar Jurcevic, Christine A. Muschik, Pietro Silvi, Rainer Blatt, Christian F. Roos, and Peter Zoller. Self-verifying variational quantum simulation of lattice models. Nature, 569: 0 355--360, 2019

    work page 2019

  11. [11]

    Furkan Oz, Rohit K. S. S. Vuppala, Kursat Kara, and Frank Gaitan. Solving burgers' equation with quantum computing. Quantum Information Processing, 21: 0 30, 2022

    work page 2022

  12. [12]

    Dominic W. Berry. High-order quantum algorithm for solving linear differential equations. Journal of Physics A: Mathematical and Theoretical, 47 0 (10): 0 105301, February 2014

    work page 2014

  13. [13]

    Quantum algorithms and the finite element method

    Ashley Montanaro and Sam Pallister. Quantum algorithms and the finite element method. Physical Review A, 93 0 (3): 0 032324, March 2016

    work page 2016

  14. [14]

    Pedro C. S. Costa, Stephen Jordan, and Aaron Ostrander. Quantum algorithm for simulating the wave equation. Physical Review A, 99 0 (1): 0 012323, January 2019

    work page 2019

  15. [15]

    Quantum amplitude amplification and estimation

    Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. AMS Contemporary Mathematics Series, 305, 06 2000

    work page 2000

  16. [16]

    Harrow, Avinatan Hassidim, and Seth Lloyd

    Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103 0 (15), 2009

    work page 2009

  17. [17]

    Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J

    Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles. Variational quantum linear solver. Quantum, 7: 0 1188, 2023. ISSN 2521-327X

    work page 2023

  18. [18]

    Quantum simulation of partial differential equations via schr \"o dingerization

    Shi Jin, Nana Liu, and Yue Yu. Quantum simulation of partial differential equations via schr \"o dingerization. Physical Review Letters, 133 0 (23): 0 230602, December 2024

    work page 2024

  19. [19]

    Berger, N

    S. Berger, N. Hosters, and M. M \"o ller. Trainable embedding quantum physics informed neural networks for solving nonlinear PDEs . Scientific Reports, 15 0 (1): 0 18823, 2025

    work page 2025

  20. [20]

    Quantum physics-informed neural networks

    Corey Trahan, Mark Loveland, and Samuel Dent. Quantum physics-informed neural networks. Entropy, 26 0 (8), 2024

    work page 2024

  21. [21]

    Quantum physics-informed neural networks for multivariable partial differential equations

    Giorgio Panichi, Sebastiano Corli, and Enrico Prati. Quantum physics-informed neural networks for multivariable partial differential equations. Phys. Rev. Appl., 25: 0 014001, Jan 2026

    work page 2026

  22. [22]

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

    Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving partial differential equations. Journal of Computational Physics, 378: 0 686--707, 2019

    work page 2019

  23. [25]

    Scientific machine learning through physics--informed neural networks: Where we are and what's next

    Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. Scientific machine learning through physics--informed neural networks: Where we are and what's next. Journal of Scientific Computing, 92 0 (3): 0 88, July 2022

    work page 2022

  24. [26]

    Karniadakis, Ioannis G

    George E. Karniadakis, Ioannis G. Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning. Nature Reviews Physics, 3: 0 422--440, 2021

    work page 2021

  25. [27]

    L. H. Queiroz, F. P. Santos, J. P. Oliveira, and M. B. Souza. Physics-informed deep learning to predict flow fields in cyclone separators. Digital Chemical Engineering, 1: 0 100002, 2021

    work page 2021

  26. [28]

    Karniadakis

    Guofei Pang, Lu Lu, and George E. Karniadakis. fPINNs : Fractional physics-informed neural networks. SIAM Journal on Scientific Computing, 41 0 (4): 0 A2603--A2626, 2019

    work page 2019

  27. [29]

    Karniadakis

    Maziar Raissi, Alireza Yazdani, and George E. Karniadakis. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science, 367 0 (6481): 0 1026--1030, 2020

    work page 2020

  28. [31]

    The power of quantum neural networks

    Amira Abbas, David Sutter, Christa Zoufal, Aurelien Lucchi, Alessio Figalli, and Stefan Woerner. The power of quantum neural networks. Nature Computational Science, 1 0 (6): 0 403--409, 2021

    work page 2021

  29. [32]

    Barkoutsos, Chiara Macchiavello, Dario Gerace, Ivano Tavernelli, and Daniele Bajoni

    Francesco Tacchino, Stefano Mangini, Panagiotis Kl. Barkoutsos, Chiara Macchiavello, Dario Gerace, Ivano Tavernelli, and Daniele Bajoni. Variational learning for quantum artificial neural networks. IEEE Transactions on Quantum Engineering, 2: 0 1--10, 2021

    work page 2021

  30. [33]

    A variational algorithm for quantum neural networks

    Antonio Macaluso, Luca Clissa, Stefano Lodi, and Claudio Sartori. A variational algorithm for quantum neural networks. In Computational Science -- ICCS 2020, pages 591--604, 2020

    work page 2020

  31. [34]

    Quantum implementation of an artificial feed-forward neural network

    Francesco Tacchino, Panagiotis Barkoutsos, Chiara Macchiavello, Ivano Tavernelli, Dario Gerace, and Daniele Bajoni. Quantum implementation of an artificial feed-forward neural network. Quantum Science and Technology, 5 0 (4): 0 044010, October 2020

    work page 2020

  32. [35]

    Garc \'i a-Pe \ n alvo

    David Peral Garc \'i a, Juan Cruz-Benito, and Francisco J. Garc \'i a-Pe \ n alvo. Systematic literature review: Quantum machine learning and its applications, 2022

    work page 2022

  33. [36]

    A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

    Chenxi Wu, Min Zhu, Qinyang Tan, Yadhu Kartha, and Lu Lu. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 403: 0 115671, 2023

    work page 2023

  34. [37]

    McClenny and Ulisses M

    Levi D. McClenny and Ulisses M. Braga-Neto. Self-adaptive physics-informed neural networks. Journal of Computational Physics, 474: 0 111722, 2023

    work page 2023

  35. [38]

    Quantum physics-informed neural networks for multivariable partial differential equations , author =. Phys. Rev. Appl. , volume =. 2026 , month =

    work page 2026

  36. [39]

    2023 , author =

    A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks , journal =. 2023 , author =

    work page 2023

  37. [40]

    2023 , author =

    Self-adaptive physics-informed neural networks , journal =. 2023 , author =

    work page 2023

  38. [41]

    Journal of Computational Physics , volume=

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving partial differential equations , author=. Journal of Computational Physics , volume=. 2019 , publisher=

    work page 2019

  39. [42]

    Physics of Fluids , volume =

    Riva, Stefano and Introini, Carolina and Cammi, Antonio , title =. Physics of Fluids , volume =

    work page

  40. [43]

    Nature Computational Science , volume =

    Abbas, Amira and Sutter, David and Zoufal, Christa and Lucchi, Aurelien and Figalli, Alessio and Woerner, Stefan , title =. Nature Computational Science , volume =

    work page

  41. [44]

    Advances in Quantum Deep Learning: An Overview

    Advances in Quantum Deep Learning: An Overview , author=. arXiv preprint arXiv:2005.04316 , year=. 2005.04316 , archivePrefix=

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  42. [45]

    and Hosters, N

    Berger, S. and Hosters, N. and M. Trainable embedding quantum physics informed neural networks for solving nonlinear. Scientific Reports , volume =

    work page

  43. [46]

    Entropy , VOLUME =

    Trahan, Corey and Loveland, Mark and Dent, Samuel , TITLE =. Entropy , VOLUME =. 2024 , NUMBER =

    work page 2024

  44. [47]

    Physical Review Letters , volume =

    Jin, Shi and Liu, Nana and Yu, Yue , title =. Physical Review Letters , volume =

    work page

  45. [48]

    Advanced Quantum Technologies , volume =

    Gaitan, Frank , title =. Advanced Quantum Technologies , volume =

    work page

  46. [49]

    and Leighton, Robert B

    Feynman, Richard P. and Leighton, Robert B. and Sands, Matthew , title =

    work page

  47. [50]

    Annual Review of Fluid Mechanics , volume =

    Moin, Parviz and Mahesh, Krishnan , title =. Annual Review of Fluid Mechanics , volume =

    work page

  48. [51]

    Quantum , volume =

    Preskill, John , title =. Quantum , volume =

    work page

  49. [52]

    Massively parallel numerical simulation using up to 36,000

    Neau, Herv. Massively parallel numerical simulation using up to 36,000. Powder Technology , volume =

    work page

  50. [53]

    and Maslov, Dmitri and Nam, Yunseong and Ross, Neil J

    Childs, Andrew M. and Maslov, Dmitri and Nam, Yunseong and Ross, Neil J. and Su, Yuan , title =. Proceedings of the National Academy of Sciences , volume =

    work page

  51. [54]

    Quantum Computation of Fluid Dynamics , booktitle =

    Yepez, Jeffrey , editor =. Quantum Computation of Fluid Dynamics , booktitle =

    work page

  52. [55]

    Journal of Statistical Physics , volume =

    Yepez, Jeffrey , title =. Journal of Statistical Physics , volume =

    work page

  53. [56]

    Quantum Algorithms for Fluid Simulations , booktitle =

    Steijl, Ren. Quantum Algorithms for Fluid Simulations , booktitle =

    work page

  54. [57]

    Parallel evaluation of quantum algorithms for computational fluid dynamics , journal =

    Steijl, Ren. Parallel evaluation of quantum algorithms for computational fluid dynamics , journal =

    work page

  55. [58]

    Oz, Furkan and Vuppala, Rohit K. S. S. and Kara, Kursat and Gaitan, Frank , title =. Quantum Information Processing , volume =

    work page

  56. [59]

    , title =

    Berry, Dominic W. , title =. Journal of Physics A: Mathematical and Theoretical , volume =

    work page

  57. [60]

    and Childs, Andrew M

    Berry, Dominic W. and Childs, Andrew M. and Ostrander, Aaron and Wang, Guoming , title =. Communications in Mathematical Physics , volume =

    work page

  58. [61]

    Physical Review A , volume =

    Montanaro, Ashley and Pallister, Sam , title =. Physical Review A , volume =

    work page

  59. [62]

    Costa, Pedro C. S. and Jordan, Stephen and Ostrander, Aaron , title =. Physical Review A , volume =

    work page

  60. [63]

    and Jurcevic, Petar and Muschik, Christine A

    Kokail, Christian and Maier, Christine and van Bijnen, Rick and Brydges, Tiff and Joshi, Manoj K. and Jurcevic, Petar and Muschik, Christine A. and Silvi, Pietro and Blatt, Rainer and Roos, Christian F. and Zoller, Peter , title =. Nature , volume =

    work page

  61. [64]

    npj Quantum Information , volume =

    Gaitan, Frank , title =. npj Quantum Information , volume =

    work page

  62. [65]

    Quantum Amplitude Amplification and Estimation , volume =

    Brassard, Gilles and Hoyer, Peter and Mosca, Michele and Tapp, Alain , year =. Quantum Amplitude Amplification and Estimation , volume =

  63. [66]

    and Hassidim, Avinatan and Lloyd, Seth , title =

    Harrow, Aram W. and Hassidim, Avinatan and Lloyd, Seth , title =. Physical Review Letters , volume =

    work page

  64. [67]

    Quantum , volume =

    Variational Quantum Linear Solver , author =. Quantum , volume =. 2023 , publisher =

    work page 2023

  65. [68]

    and Kothari, Robin and Somma, Rolando D

    Childs, Andrew M. and Kothari, Robin and Somma, Rolando D. , title =. SIAM Journal on Computing , volume =

    work page

  66. [69]

    Ray, Navamita and Banerjee, Tirtha and Nadiga, Balasubramanya and Karra, Satish , title =

    work page

  67. [70]

    and Kevrekidis, Ioannis G

    Karniadakis, George E. and Kevrekidis, Ioannis G. and Lu, Lu and Perdikaris, Paris and Wang, Sifan and Yang, Liu , title =. Nature Reviews Physics , volume =

    work page

  68. [71]

    , title =

    Raissi, Maziar and Perdikaris, Paris and Karniadakis, George E. , title =. Journal of Computational Physics , volume =. 2019 , publisher =

    work page 2019

  69. [72]

    , title =

    Raissi, Maziar and Perdikaris, Paris and Karniadakis, George E. , title =. arXiv preprint arXiv:1711.10561 , year =

    Pith/arXiv arXiv

  70. [73]

    , title =

    Raissi, Maziar and Perdikaris, Paris and Karniadakis, George E. , title =. arXiv preprint arXiv:1711.10566 , year =

    Pith/arXiv arXiv

  71. [74]

    Journal of Scientific Computing , volume =

    Cuomo, Salvatore and Di Cola, Vincenzo Schiano and Giampaolo, Fabio and Rozza, Gianluigi and Raissi, Maziar and Piccialli, Francesco , title =. Journal of Scientific Computing , volume =. 2022 , month = jul, day =

    work page 2022

  72. [75]

    Queiroz, L. H. and Santos, F. P. and Oliveira, J. P. and Souza, M. B. , title =. Digital Chemical Engineering , volume =

    work page

  73. [76]

    , title =

    Raissi, Maziar and Yazdani, Alireza and Karniadakis, George E. , title =. Science , volume =

    work page

  74. [77]

    , title =

    Pang, Guofei and Lu, Lu and Karniadakis, George E. , title =. SIAM Journal on Scientific Computing , volume =. 2019 , publisher =

    work page 2019

  75. [78]

    Systematic Literature Review: Quantum Machine Learning and its Applications , publisher =

    Garc. Systematic Literature Review: Quantum Machine Learning and its Applications , publisher =

    work page

  76. [79]

    International Journal of Theoretical Physics , volume =

    Chen, Bu-Qing and Niu, Xu-Feng , title =. International Journal of Theoretical Physics , volume =

    work page

  77. [80]

    and Macchiavello, Chiara and Gerace, Dario and Tavernelli, Ivano and Bajoni, Daniele , title =

    Tacchino, Francesco and Mangini, Stefano and Barkoutsos, Panagiotis Kl. and Macchiavello, Chiara and Gerace, Dario and Tavernelli, Ivano and Bajoni, Daniele , title =. IEEE Transactions on Quantum Engineering , volume =

    work page

  78. [81]

    Computational Science -- ICCS 2020 , pages =

    Macaluso, Antonio and Clissa, Luca and Lodi, Stefano and Sartori, Claudio , title =. Computational Science -- ICCS 2020 , pages =

    work page 2020

  79. [82]

    Quantum Science and Technology , volume =

    Tacchino, Francesco and Barkoutsos, Panagiotis and Macchiavello, Chiara and Tavernelli, Ivano and Gerace, Dario and Bajoni, Daniele , title =. Quantum Science and Technology , volume =

    work page

  80. [83]

    and Yoo, Jae Hyeon and Isakov, Sergei V

    Broughton, Michael and Verdon, Guillaume and McCourt, Trevor and Martinez, Antonio J. and Yoo, Jae Hyeon and Isakov, Sergei V. and Massey, Philip and Halavati, Ramin and Niu, Murphy Yuezhen and Zlokapa, Alexander and Peters, Evan and Lockwood, Owen and Skolik, Andrea and Jerbi, Sofiene and Dunjko, Vedran and Leib, Martin and Streif, Michael and Von Dollen...

    work page

Showing first 80 references.