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Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations

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Pith's one-line read A hybrid quantum-classical network solves fuzzy PDEs by jointly learning both alpha-cut endpoint functions, with a smaller error bound than the classical PIKAN whenever the quantum feature representation gain outweighs the extra…

desk verdict Honest incremental extension of QCPIKAN to fuzzy PDEs; the numerics are plausible but the proof of quantum advantage is a tautology and the experiments don't isolate the quantum circuit. read the letter →

arxiv 2608.08782 v1 pith:47CESTSD submitted 2026-08-09 cs.LG

Xiang Rao , Yuxuan Shen This is my paper · ORCID
classification cs.LG MSC 03E7235R1368T07
keywords fuzzypartialdifferentialequationsalpha-cutsphysics-informedKolmogorov-Arnoldnetworksquantum-classicalhybridcomputingparameterizedquantumcircuitsfuzzy-structureconstraintsChebyKANerroranalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a hybrid quantum-classical physics-informed Kolmogorov-Arnold network, QCPIKAN, for solving fuzzy partial differential equations written in $\alpha$-cut form. It treats the membership level as an extra input coordinate and trains one network to output the lower and upper endpoint functions of the solution interval for every $\alpha$, folding the governing equations, initial-boundary conditions, endpoint ordering, inter-level nesting, and endpoint coincidence at $\alpha=1$ into the loss. The paper also proves a conditional a priori error bound: QCPIKAN's endpoint-solution error bound is strictly smaller than PIKAN's whenever the representation gain from the quantum feature space exceeds the additional optimization, sampling, and fuzzy-structure errors. In ideal quantum-simulation experiments on elliptic, parabolic, and hyperbolic fuzzy equations, the classical PIKAN's mean relative $L^2$ error is roughly 1.1--2.7 times larger at most tested membership levels, and its wavefront-position error is about 1.77 times larger in the convection example. Both models still show local violations of the fuzzy-structure constraints near boundaries, high-gradient regions, and the wavefront.

What carries the argument

The load-bearing object is the $\alpha$-cut endpoint representation of a fuzzy PDE: the fuzzy solution is encoded as a nested family of intervals whose lower endpoint is nondecreasing and upper endpoint nonincreasing in the membership level $\alpha$. QCPIKAN maps the joint input (spatial coordinates, time, $\alpha$) through ChebyKAN pre- and post-processing layers—Kolmogorov-Arnold layers whose univariate functions are Chebyshev-polynomial expansions—and a four-qubit parameterized quantum circuit whose entangling two-qubit blocks are meant to generate interaction features among space, time, and $\alpha$. The identity that carries the theoretical argument is the a priori error decomposition $\mathcal{E}_M \le C_{\mathrm{stab}}(E_M^{\mathrm{app}}+\varepsilon_M)$ for $M\in\{\mathrm{PIKAN},\mathrm{QCPIKAN}\}$, with Theorem 1 stating that QCPIKAN's bound is strictly smaller when $\Delta^{\mathrm{app}}=E_P^{\mathrm{app}}-E_Q^{\mathrm{app}}$ exceeds the combined extra error terms in Eq. (43). The fuzzy-structure losses—endpoint ordering, inter-level nesting, and $\alpha=1$ coincidence—are what turn an ordinary function approximator into a solver for a fuzzy-valued solution rather than a merely interval-valued one.

What would settle it

Train a purely classical ChebyKAN (or a KAN with product and other interaction features) at the same parameter count, with the same training data, loss weights, and $\alpha$-cut constraints, on the same four fuzzy equations, and compare mean relative $L^2$ and wavefront-position errors at all five membership levels; if the classical model matches or beats QCPIKAN, the quantum entanglement features are not the cause of the reported gap. Alternatively, compute the best-approximation errors $E_P^{\mathrm{app}}$ and $E_Q^{\mathrm{app}}$ on the nonseparable component $U_{\mathrm{int}}$ directly; if $\Delta^{\mathrm{app}}$ does not exceed the combined optimization, sampling, and structure-constraint error differences in Eq. (44), the inequality in Theorem 1 fails.

Watch

Extended reading notes

Core claim

The central claim is that a single network in which classical ChebyKAN layers sandwich a parameterized quantum circuit can jointly approximate the lower and upper $\alpha$-cut endpoint functions of a fuzzy PDE solution, and that this hybrid representation can be more accurate than the classical PIKAN baseline. Theorem 1 decomposes the endpoint-solution error of each model into approximation, optimization, finite-sampling, fuzzy-structure, gradient, and hardware-noise components and shows that QCPIKAN has a strictly smaller a priori bound exactly when its representation gain $\Delta^{\mathrm{app}}$ dominates the sum of the extra computational errors. Under the ideal-simulation conditions used, this condition reduces to comparing the representation gain with the optimization, sampling, and structure-constraint error differences. Numerical experiments on a fuzzy Poisson equation, a fuzzy heat-conduction equation, a fuzzy reaction-diffusion equation, and a fuzzy convection equation support the claimed advantage: PIKAN's mean relative $L^2$ errors are about 1.1--2.7 times QCPIKAN's at most membership levels, and its mean wavefront-position error is about 1.77 times larger, while both models retain local endpoint-ordering and nesting violations near boundaries, high gradients, and the wavefront.

Load-bearing premise

The argument stands or falls on the assumption that, at a fixed model size, the QCPIKAN feature space can reproduce or beat the classical PIKAN representation on the nonseparable part of the fuzzy solution; the paper assumes this representation gain (A3) rather than measuring it independently of the experiments it is used to explain.

Editorial extensions

If this is right

  • One trained QCPIKAN returns the entire nested family of $\alpha$-cut interval solutions, with $\alpha$ as a continuous input, for elliptic, parabolic, and hyperbolic fuzzy PDEs.
  • On the four test equations, PIKAN's mean relative $L^2$ error is about 1.1--2.7 times QCPIKAN's at most tested membership levels, and its mean wavefront-position error in the convection example is about 1.77 times larger.
  • The theoretical comparison is conditional: QCPIKAN beats PIKAN only when the representation gain from the quantum feature space exceeds the combined extra optimization, sampling, fuzzy-structure, gradient, and hardware errors, so the advantage is not automatic.
  • Both models still violate fuzzy-structure constraints locally near boundaries, in high-gradient regions, and around the wavefront, so soft penalty enforcement does not guarantee a globally valid fuzzy solution.
  • Within the tested range, QCPIKAN accuracy improves with qubit count but varies nonmonotonically with circuit depth, consistent with growing optimization difficulty, including barren-plateau effects, at larger depth.

Reading between the lines

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

  • A direct extension not tested in the paper: the same ChebyKAN-plus-quantum-circuit architecture could serve as a general solver for parametric families of PDEs whose solutions depend nonseparably on the parameters, not only on the fuzzy membership level.
  • Because both models retain fuzzy-structure violations despite soft penalties, a natural next experiment is to enforce endpoint ordering and inter-level nesting with hard architectural constraints, such as monotonic output layers, to test whether the residual violations are a representation issue or a training issue.
  • The numerical comparisons use an ideal quantum simulator with analytic expectation values; on real hardware the finite-shot and gate-noise terms in Eq. (43) would enter, so the practical advantage should be re-measured under finite-shot conditions before drawing hardware conclusions.
  • To attribute the lower errors to quantum entanglement features, one would need a matched classical counterfactual with interaction features at the same parameter count, since the paper does not isolate $\Delta^{\mathrm{app}}$ independently of the experiments.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes QCPIKAN, a hybrid classical-quantum architecture that combines ChebyKAN layers with a four-qubit parameterized quantum circuit to solve fuzzy partial differential equations represented through alpha-cuts. The model treats the spatiotemporal coordinates and the membership level as joint inputs and outputs the lower and upper alpha-cut endpoint functions simultaneously. Its loss function includes the endpoint governing equations, initial and boundary conditions, endpoint ordering, inter-level nesting, and endpoint coincidence at alpha = 1. Section 2.3 presents an error decomposition and Theorem 1, which states that QCPIKAN has a smaller a priori error bound than PIKAN when the representation gain Delta_app exceeds the sum of the remaining computational-error differences. Numerical experiments cover a fuzzy Poisson equation, a fuzzy heat-conduction equation, a fuzzy reaction-diffusion equation, and a fuzzy convection equation, reporting that PIKAN's mean relative L2 error is approximately 1.1-2.7 times larger at most tested membership levels and that the wavefront-position error is about 1.77 times larger on average. The manuscript is candid that both models still exhibit local fuzzy-structure violations and that the experiments use an ideal quantum simulator.

Significance. If the central claims were established, the paper would offer a useful computational framework for fuzzy PDEs that integrates KAN-based function representation, variational quantum circuits, and explicit fuzzy-structure constraints in a single physics-informed objective. The four numerical examples cover elliptic, parabolic, and hyperbolic problems, and the error metrics include both endpoint L2 errors and a wavefront-position error, which are appropriate for the considered problems. The paper also deserves credit for explicitly separating representation, optimization, sampling, fuzzy-structure, gradient, and hardware-noise errors, and for acknowledging that neither model strictly satisfies the fuzzy-structure constraints. However, the advertised theoretical result is essentially a restatement of its assumption, and the numerical experiments do not isolate the effect of the quantum feature map from additional classical capacity. As presented, the contribution is a plausible architecture plus a single-run numerical comparison whose central quantum-advantage claim is not yet established.

major comments (3)
  1. [Sec. 2.3, Theorem 1, Eqs. (43)-(45)] The theorem's sufficient condition is algebraically identical to its conclusion. Substituting Delta_app = E_app^P - E_app^Q into Eq. (43) gives E_app^Q + eps_Q < E_app^P + eps_P, and multiplying by C_stab yields exactly the claimed bound inequality Eq. (45). Thus Theorem 1 does not establish any nontrivial condition under which QCPIKAN beats PIKAN; it merely restates the target inequality after assuming, through Assumption A3, that QCPIKAN's best-approximation error is sufficiently smaller. The paper provides no independent estimate or bound for Delta_app, and the numerical experiments used to verify Eq. (44) are the same experiments that would need to supply Delta_app. Please reframe the theorem as an error decomposition with a stated condition, or provide an independent argument that Delta_app is positive and sufficiently large for the considered architecture.
  2. [Sec. 3 and Sec. 2.2.2] The empirical comparison does not isolate the quantum feature map from classical model capacity. The QCPIKAN pipeline is described as ChebyKAN 3->30->4, followed by a four-qubit circuit, followed by ChebyKAN 4->30->2, but the PIKAN architecture, its width, depth, and total parameter count are never specified. The reported 1.1-2.7x error ratios could therefore be caused by the extra ChebyKAN width, depth, or optimization capacity rather than by the entanglement features of the circuit. The paper should specify the PIKAN architecture exactly and include a counterfactual ablation in which the quantum circuit is replaced by a classical trainable nonlinear mixing layer of identical parameter count and comparable capacity.
  3. [Sec. 3, general experimental protocol] All numerical results appear to be single-run experiments without reported seeds, repeated trials, confidence intervals, or code. Because the reported performance ratios are as small as 1.1x, it is not possible to assess whether the observed differences are statistically meaningful or robust to initialization. The paper should report multi-seed statistics, specify all sampling sizes, and ideally provide code to make the comparison reproducible. This is load-bearing for the central empirical claim that PIKAN's mean relative L2 error is 1.1-2.7 times that of QCPIKAN.
minor comments (5)
  1. [Sec. 3.1] The text says QCPIKAN and PIKAN approximate Lu_alpha and Uu_alpha 'in Eq. (36)', but Eq. (36) is the generic error-bound inequality; the intended reference appears to be Eq. (51).
  2. [Eq. (52)] The Fourier sine-series reference solution contains garbled notation, including an apparent '?' character and an unclear summation range 'm,n odd, 1<=m,n<=49'. Please clean up the displayed formula and define all summation limits clearly.
  3. [Example 4, Eq. (60)] The initial condition for the convection equation is garbled in the displayed equation; the intended condition appears to be u(x,0)=u_0(x), but it should be written out explicitly.
  4. [Secs. 3.3 and 3.4] The manuscript honestly reports that in Example 3 QCPIKAN has more widely distributed inter-level nesting violations than PIKAN and that in Example 4 it does not show a consistent advantage in fuzzy-structure metrics. Since fuzzy-structure satisfaction is part of the model's stated objective, the paper should present quantitative structure-violation statistics and temper the conclusion that QCPIKAN provides a more accurate fuzzy solution family.
  5. [Sec. 2.2.3] The loss weights are chosen separately for each example and are not subjected to sensitivity analysis. A brief study of how the reported error ratios vary with these weights would strengthen the numerical comparison.

Circularity Check

2 steps flagged · score 5.0 of 10

Theorem 1's conclusion is a restatement of its main assumption: Eq. (43) is algebraically identical to Eq. (45), and the experiments cited as 'verification of Eq. (44)' never isolate the quantum feature-map advantage.

  1. self definitional [Sec. 2.3, Theorem 1 and proof, Eqs. (41)-(48)]
    "Second, Eq. (43) is equivalent to (E_Q^app + ε_Q) - (E_P^app + ε_P) < 0. ... Because the stability constant C_stab > 0, multiplying both sides of Eq. (47) by C_stab does not change the direction of the inequality. Therefore, C_stab(E_Q^app + ε_Q) < C_stab(E_P^app + ε_P). Thus, E_Q < E_P and Eq. (45) holds, completing the proof."

    Two paragraphs earlier the paper defines Δ_app = E_P^app − E_Q^app (Eq. (41)). Substituting that definition into condition (43) yields exactly E_Q^app + ε_Q < E_P^app + ε_P, which is Eq. (47). Multiplying by C_stab produces E_Q < E_P, the theorem's conclusion. So Theorem 1 does not derive the quantum advantage from independent premises; it restates, in decomposed notation, the assumption that QCPIKAN's approximation advantage outweighs its computational-error disadvantage. Assumption A3 already assumes a strict best-approximation advantage at fixed model size, and no independent estimate of Δ_app is supplied.

  2. fitted input called prediction [Sec. 2.3, final paragraph; Sec. 3 numerical experiments]
    "The numerical results presented here should therefore be interpreted as empirical verification of Eq. (44) under ideal conditions."

    Eq. (44) is the theorem's sufficient-condition inequality, which is algebraically equivalent to the claimed bound comparison E_Q < E_P. The experiments report final mean relative L2 errors, wavefront-position errors, and loss curves — the same quantities the bound comparison concerns — but never measure E_Q^app, E_P^app, or the individual optimization, sampling, and fuzzy-structure error terms at a fixed PIKAN parameter count. PIKAN's architecture and total parameter count are not specified, and QCPIKAN adds ChebyKAN layers 3→30→4 before and 4→30→2 after the circuit. Thus 'empirical verification of Eq.

full rationale

The paper contains no load-bearing self-citation: [65] and [66] are contextual references, and the fuzzy-structure losses and ChebyKAN-plus-circuit architecture are described directly. The numerical benchmarks are also self-contained against analytic eigenfunction series, exact heat-kernel solutions, IMEX finite differences, and method-of-characteristics solutions, so the empirical comparisons have independent content. The central theoretical result, however, is tautological. With Δ_app defined as E_P^app − E_Q^app, inequality (43) is exactly E_Q^app + ε_Q < E_P^app + ε_P; multiplying by C_stab gives E_Q < E_P, the theorem's conclusion. Assumption A3 already assumes the strict approximation advantage, and the paper offers no independent estimate of Δ_app. The experiments do not close this gap: PIKAN's architecture and parameter count are unspecified, and QCPIKAN's extra ChebyKAN layers precede and follow the quantum circuit, so the observed 1.1–2.7x error ratios cannot be uniquely attributed to quantum entanglement features. Calling the numerical results 'verification of Eq. (44)' is circular in the sense that Eq. (44) is the same algebraic comparison as the claimed bound advantage. Overall, the error decomposition and benchmarks have independent value, but the advertised a priori quantum-advantage theorem reduces by construction to its own assumption, giving a partial-circularity score of 5.

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

The central claim rests on a long chain of hand-chosen loss weights, an argued but unverified representation-gain assumption, and an ideal-simulator idealization that removes noise and gradient errors. No new physical entities are introduced. The number of free parameters is moderate for a PINN-style paper, but the theoretical advantage theorem is conditional on the existence of the very gain it tries to explain.

free parameters (5)
  • Loss weights, Example 1 (lambda_res, lambda_BC, lambda_nest, lambda_order, lambda_{alpha=1}) = 1, 1, 0.1, 0.2, 1
    Chosen by hand in Sec. 3.1; the reported accuracy comparison depends on these weights.
  • Loss weights, Example 2 (lambda_res, lambda_IC, lambda_BC, lambda_nest, lambda_order, lambda_{alpha=1}) = 1, 1, 1, 0.5, 0.2, 1
    Chosen by hand in Sec. 3.2; the comparison is tied to these values.
  • Loss weights, Example 3 (lambda_res, lambda_IC, lambda_BC, lambda_nest, lambda_order, lambda_{alpha=1}) = 1, 1, 1, 0.1, 0.2, 1
    Chosen by hand in Sec. 3.3; the nonlinear Fisher-KPP comparison uses these weights.
  • Loss weights, Example 4 (lambda_res, lambda_IC, lambda_BC, lambda_nest, lambda_order, lambda_{alpha=1}) = 5, 10, 5, 0.5, 0.15, 0.15
    Chosen by hand in Sec. 3.4; the wavefront-error comparison is affected by these choices.
  • Qubit count and circuit depth = n_q = 4, 1 variational layer
    Default architecture in Sec. 2.2.2; Sec. 3.1 Fig. 7 shows error varies nonmonotonically with depth, so the reported gain is tied to this choice.
assumptions (5)
  • domain assumption Residual-stability estimate Eq. (35): ||U - V||_H <= C_stab ||R(V) - R(U)||_Y
    Invoked to convert training loss into solution error bound Eq. (36) and Theorem 1; only plausible for well-posed problems and not verified for the four examples.
  • domain assumption Endpoint problems are well-posed (A1)
    Needed to make Eq. (35) hold; no analysis of well-posedness for the fuzzy endpoint systems is supplied.
  • domain assumption Monotonic dependence of solution endpoints on fuzzy parameters in Examples 2-4 (e.g., Eq. (55) uses a_U for the lower endpoint, Eq. (59) uses (a_U, b_L), Eq. (62) uses c_U)
    The endpoint equations are assigned by assuming solutions decrease with diffusion coefficient, increase with reaction coefficient, and decrease with convection velocity; if this monotonicity fails, the alpha-cut endpoint system is wrong.
  • domain assumption Ideal quantum simulation with analytic expectation values and no noise
    Paper sets epsilon_noise_Q = 0 and epsilon_grad = 0 in Eq. (43); the theoretical and empirical advantage may not persist on real hardware, as the paper concedes.
  • standard math Triangular fuzzy inputs with singleton core imply endpoint coincidence at alpha = 1 (Eq. (9))
    Standard property of triangular fuzzy numbers; used to define loss Eq. (31) and the reference solutions.

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Cite this review

Pith. "Pith review of Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations." pith.science (2026). https://pith.science/paper/47CESTSD

@misc{pith2026260808782,
  author       = {Pith},
  title        = {Pith review of: Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/47CESTSD}},
  note         = {Machine review of arXiv:2608.08782}
}
read the original abstract

In this study, we propose a quantum-classical physics-informed Kolmogorov-Arnold network (QCPIKAN) dedicated to the solution of fuzzy differential equations. The network takes the spatiotemporal coordinates and membership level as joint inputs and employs ChebyKAN modules and a parameterized quantum circuit to construct a hybrid function approximator. It simultaneously approximates the lower and upper endpoint functions associated with the {\alpha}-cuts and incorporates the governing equations, initial-boundary conditions, and fuzzy-structural constraints into the training objective. Theoretically, a unified error-analysis framework is established for QCPIKAN and PIKAN, in which the endpoint-solution error is decomposed into approximation, sampling, optimization, and fuzzy-structure constraint errors. Under the assumptions of well-posedness and residual stability, it is proved that QCPIKAN has a smaller a priori error bound when the representational gain introduced by quantum entanglement features exceeds the additional computational error. Numerical experiments are conducted for elliptic, parabolic, and hyperbolic equations in an ideal quantum-simulation environment. The results show that QCPIKAN captures the overall contraction of the solution interval as increases. At most tested membership levels, the mean relative L2 error of PIKAN is approximately 1.1-2.7 times that of QCPIKAN. In the fuzzy convection example, the mean wavefront-position error of PIKAN is approximately 1.77 times that of QCPIKAN. Nevertheless, both models still exhibit local fuzzy-structure violations near boundaries, in high-gradient regions, and around the wavefront. These results indicate that QCPIKAN provides a quantum-classical hybrid physics-informed computational framework with comparatively high predictive accuracy for solving fuzzy partial differential equations represented by {\alpha}-cuts.

Figures

Figures reproduced from arXiv: 2608.08782 by the authors.

Figure 1
Figure 1. Schematic of the QCPIKAN computational procedure for fuzzy differential equations. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Comparison of training losses in Example 1. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Spatial distributions of the lower- and upper-endpoint solutions in Example 1 at  = 0.50. (a) QCPIKAN (b) PIKAN [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: Spatial interval width and inter-level nesting violations along the  direction at  = 0.5 in Example 1. (a) Absolute error of QCPIKAN lower bound (b) Absolute error of QCPIKAN upper bound (c) Absolute error of PIKAN lower bound (d) Absolute error of PIKAN upper bound …
Figure 5
Figure 5. Figure 5: Absolute errors of the lower- and upper-endpoint solutions in Example 1 at  = 0.50 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Prediction errors and comparison with the reference solution at different [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Effects of the number of qubits and parameterized quantum [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Comparison of training losses in Example 2. [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Temporal evolution of the lower and upper solution endpoints at [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Spatial interval widths at different  levels and inter-level nesting violations along the  direction at t  0.49 in Example 2. (a) Absolute error of QCPIKAN lower bound (b) Absolute error of QCPIKAN upper bound [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Absolute errors of the lower- and upper-endpoint solutions in Example 2 at  = 0.50. (a) Variation of the predicted and reference endpoints with  at a fixed spatiotemporal location (b) Mean relative 𝐿2 error [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Prediction errors and comparison with the reference solution at different [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Comparison of training losses in Example 3. [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: Temporal evolution of the lower and upper solution endpoints at [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]
Figure 15
Figure 15. Figure 15: Spatial interval widths at different  levels and inter-level nesting violations along the  direction at t  0.49 in Example 3. (a) Absolute error of QCPIKAN lower bound (b) Absolute error of QCPIKAN upper bound (c) Absolute error of PIKAN lower bound (d) Absolute er…
Figure 16
Figure 16. Figure 16: Absolute errors of the lower- and upper-endpoint solutions in Example 3 at  = 0.50 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: Prediction errors and comparison with the reference solution at different [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 18
Figure 18. Figure 18: Comparison of training losses in Example 4. [PITH_FULL_IMAGE:figures/full_fig_p024_18.png]
Figure 19
Figure 19. Figure 19: Space-time distributions of the lower- and upper-endpoint solutions in Example 4 at  = 0.50 [PITH_FULL_IMAGE:figures/full_fig_p024_19.png]
Figure 20
Figure 20. Figure 20: Space-time interval width and inter-level nesting violations along the  direction at  = 0.5 in Example 4. (a) At t  0.24, predicted wavefronts and reference solutions for  = 0, 0.50, and 1.00 (b) Mean wavefront-position errors at different  levels [PITH_FULL_IMA…
Figure 21
Figure 21. Figure 21: Wavefront-profile predictions and mean position errors in Example 4. 4 Conclusions This study constructs a QCPIKAN framework for solving fuzzy partial differential equations represented by  -cuts. ChebyKAN modules and a parameterized quantum circuit are used to joint…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.