DAE-KAN: A Kolmogorov-Arnold Network Model for High-Index Differential-Algebraic Equations

Juan Tang; Kai Luo; Manqi Xie; Mingchao Cai; Ming Yan; Xiaoqing Zeng

REVIEW 4 major objections 5 minor 1 cited by

DAE-KAN: A Kolmogorov-Arnold Network Model for High-Index Differential-Algebraic Equations

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A KAN-based physics-informed neural network solves index-3 differential-algebraic equations directly, cutting errors by one to two orders of magnitude versus traditional PINNs.

desk verdict A sensible KAN-for-MLP swap for PINN-based DAE solvers with index-3 examples, but the evaluation needs a held-out grid and hidden-constraint checks before the broad claims can stand. read the letter →

arxiv 2504.15806 v2 pith:S7WZFCG3 submitted 2025-04-22 cs.LG cs.AI

Kai Luo , Juan Tang , Mingchao Cai , Xiaoqing Zeng , Manqi Xie , Ming Yan This is my paper · ORCID

classification cs.LGcs.AI MSC 65L8068T0734A09

keywords differential-algebraicequationshigh-indexDAEsphysics-informedneuralnetworksKolmogorov-Arnolddrift-offerrorindexreductionB-splineactivationDAEsolver

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 DAE-KAN, a physics-informed neural network that replaces the usual multilayer perceptron with a Kolmogorov-Arnold Network (KAN), whose trainable B-spline functions sit on the network edges. The framework uses two separate KANs, one for the differential variables and one for the algebraic variables, and trains them by minimizing only the DAE residual plus the initial condition. The authors claim that this lets them solve index-3 differential-algebraic equations directly (systems where three rounds of differentiating the constraints are needed before the equations behave like an ordinary differential equation), without index reduction, and that on two test problems (a particle on a circular orbit and a two-link robot arm) it cuts absolute errors of both differential and algebraic variables by one to two orders of magnitude compared with traditional MLP-based PINNs. They also report that both neural methods keep drift-off errors far smaller than classical numerical solvers such as DOPRI5, whose solutions to an index-reduced pendulum drift off the original constraints over time. If the claim holds, DAE-KAN is the first neural DAE solver that works at useful accuracy for index-3 systems.

What carries the argument

The load-bearing construction is the two-network DAE-KAN architecture. A Kolmogorov-Arnold Network represents a function as a composition of learnable univariate functions, here parametrized by B-splines on each edge rather than fixed nonlinearities at nodes. DAE-KAN uses one KAN to output all differential variables $u$ and a second KAN to output the algebraic variables $z$ (or the Lagrange multiplier $\lambda$), and the loss function is $MSE_F + MSE_i$: the mean squared residual of the DAE system (2.1) or (2.2) at 200 uniformly sampled collocation points plus the mean squared error at a single initial point. Automatic differentiation supplies $u'$ from the differential KAN. The paper credits the KAN's approximation power and the separation of variables to different networks for the accuracy gain, and credits the residual-based training for the drift-off control without explicitly enforcing differentiated hidden constraints.

What would settle it

On the circular-orbit index-3 system (4.2), train DAE-KAN exactly as described, then evaluate the hidden-constraint residuals $u_1 z_1 + u_2 z_2$ from (4.3) and $z_1^2+z_2^2+2u_1u_2(2-u_2^2-u_1^2)-\lambda(u_1^2+u_2^2)$ from (4.4) at 10,000 dense test points in $t\in[0,1]$ rather than the 200 training points. If the maximum residual on this dense grid is orders of magnitude above the reported absolute errors, or if it grows with $t$, then the solution does not actually satisfy the high-index constraints away from the training points, and the central claim fails.

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Extended reading notes

Core claim

The central discovery is that a KAN-based PINN with a differential network and a separate algebraic network can directly approximate solutions of high-index DAEs, including index-3 systems, and does so with consistently lower error than the same physics-informed setup built on MLPs. The paper demonstrates this on two index-3 systems: a constrained particle moving on a unit circle and a two-link planar robot arm, each solved in its index-1, index-2, and index-3 formulations. In every case DAE-KAN's relative errors are roughly one to three orders of magnitude smaller than PINNs' for the differential variables, and one to two orders smaller for the algebraic variable (the Lagrange multiplier $\lambda$ or its analogue). The paper further shows that when the trained solutions are substituted back into the constraint equations of the high-index formulations, the resulting drift-off errors stay small, unlike the drift-off produced by solving an index-reduced system with a classical integrator. The implicit point is that solving a lower-index form with a neural network can still respect the original high-index constraints, whereas classical numerical solutions of the reduced form cannot.

Load-bearing premise

The entire reported accuracy and drift-off control depends on 200 uniformly spaced collocation points plus one initial point being enough for the KAN minimizer to satisfy the hidden constraints obtained by differentiating the algebraic equation, even though those hidden constraints are not part of the loss function.

Editorial extensions

If this is right

Index-3 DAEs can be solved directly by a neural network without first reducing them to index-1, so the algebraic constraints of the original system are the ones being approximated.
For the two tested systems, the KAN-based solver outperforms MLP-PINNs on every index formulation, with gains of one to three orders of magnitude in relative error depending on the variable.
Solutions obtained from any index formulation (index-1, index-2, or index-3) continue to nearly satisfy the original high-index constraints, so drift-off is controlled without projection or stabilization steps.
Prior neural DAE solvers that fail on index-3 systems are superseded, and the same dual-KAN recipe is proposed as a route toward partial differential-algebraic equations.

Reading between the lines

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

If the hidden constraints are indeed satisfied implicitly, the likely reason is that B-spline KANs produce smoother solutions, so minimizing the visible residual on 200 points also controls the differentiated constraints; this can be checked by comparing hidden-constraint residuals of KAN and MLP networks with identical loss and training budget.
An ablation that keeps the dual-network split but swaps KAN for an MLP would separate the contribution of the architecture from the contribution of the variable separation; the paper does not run this control.
The method's reliance on a fixed 200-point grid suggests a natural scaling test: as the time interval grows or the index increases, accuracy should degrade unless the number of collocation points and the KAN grid resolution grow; measuring that scaling would map the method's practical limits.
The same construction could be adapted to index-4 mechanical systems or to DAEs with inequality constraints, but nothing in the paper guarantees the hidden constraints will stay satisfied in those regimes.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: DAE-KAN is exactly what it sounds like—replace the MLP in a PINN with a KAN, run two KANs in parallel for differential and algebraic variables, and test on standard index-3 DAEs. The genuinely new part is that it takes KAN-based PINN solving into index-3 territory, where earlier neural DAE work stopped at index-1 or index-2. I believe the narrow claim, that KAN beats their own MLP-PINN on these two benchmark problems, is probably true. The numbers in Tables 1 and 2 are consistent with that, and the architecture is clean and easy to follow. The drift-off discussion is a useful motivation, and the authors are honest that they solve index-1, index-2, and index-3 forms separately rather than pretending the same network handles all cases.

The soft spots are mostly in the evaluation. Loss (3.10) contains only the DAE residual and the initial condition. For the index-3 problems, the hidden constraints—(4.3)–(4.4) for the circular particle, (4.16)–(4.17) for the robot arm—are not part of the loss. Nothing in the paper shows that a network trained on 200 residual points satisfies those constraints off the training grid. And the paper never states whether the AE/RE curves in Figures 5–6 and the numbers in the tables are computed on a held-out grid. If they are computed on the same 201 collocation points, those numbers are training fit, not solution error. That is a real gap. There are also no error bars or seeds, no code, and the classical baseline is used only to demonstrate drift-off, not as a serious accuracy comparison. The abstract's mention of 'generalization' is not backed by any extrapolation test, and the paper does not cite earlier KAN-PINN work, which is a miss but not a serious one.

Where does that leave us? The central idea is legitimate and likely correct in its narrow form. What is unverified is the broader claim that index-3 DAEs are now reliably solvable by neural networks at useful accuracy. A serious referee should engage with the paper, but it needs revision: a held-out test grid, hidden-constraint residuals reported, multiple seeds with error bars, and ideally code. I would not desk-reject it; I would send it to review and ask for those fixes.

Referee Report

4 major / 5 minor

Summary. The paper proposes DAE-KAN, a Physics-Informed Neural Network variant in which two separate Kolmogorov-Arnold Networks represent the differential and algebraic variables of a differential-algebraic equation, and the loss is the residual of the DAE plus the initial condition (Eq. 3.10). The method is tested on two index-3 benchmark systems, a particle on a circular orbit and a two-link planar robot arm, each formulated in index-1, index-2, and index-3 forms. The reported comparisons against an MLP-based PINN show one-to-two order-of-magnitude reductions in absolute and relative errors, and the paper claims that neural methods, especially DAE-KAN, control drift-off better than classical numerical methods such as DOPRI5 on a reduced-index ODE.

Significance. If the central claims hold, DAE-KAN would be among the first neural solvers capable of directly handling index-3 DAEs at useful accuracy, which is relevant for constrained multibody dynamics and other high-index applications. The paper has concrete strengths: the dual-network separation of differential and algebraic variables is sensible; two nontrivial benchmarks with known exact solutions are used; and the exact solutions appear only as external benchmarks, not in the loss, so the accuracy comparison is not circular. However, the empirical evidence as presented is not yet sufficient: the evaluation appears to be performed on the training collocation points rather than a held-out grid, the loss omits the hidden constraints that are essential for high-index DAE solutions, and the classical baseline is weak. These issues are fixable with additional experiments and reporting, so the contribution is promising but not yet established.

major comments (4)

[Sec. 3.2, Eq. (3.10); Sec. 4.1, Eqs. (4.3)-(4.4); Sec. 4.2, Eqs. (4.16)-(4.17)] The loss (3.10) penalizes only the original DAE residual and the initial condition at N_F=200 collocation points. For the index-3 direct formulations, the hidden constraints obtained by differentiating the algebraic constraints, namely (4.3)-(4.4) for the particle example and (4.16)-(4.17) for the robot arm, are not included in the loss. Satisfying the algebraic position constraint at isolated collocation points does not enforce its derivative, so the hidden velocity and multiplier constraints may be violated at those points and between them. A B-spline KAN can fit the residual at 201 discrete points while oscillating off-grid. Consequently, the central claim that DAE-KAN 'directly solves' index-3 DAEs and controls drift-off is not supported by the reported loss. Please add the hidden-constraint residuals to the loss, or at minimum verify constraint satisfaction and drift-off on a dense held-out grid that is not used for training.
[Sec. 4.1, Sec. 4.2, Eq. (4.1), Figures 5-9, Tables 1-2] No held-out test split is described in either experiment. The text states that N_F=200 residual points are uniformly distributed over t in [0,1], and the AE and RE values in Tables 1-2 and Figures 5-9 appear to be computed on the same 201-point training set. Eq. (4.1) refers to 'test points' but does not state that they differ from the training collocation points. If the reported errors are evaluated at the training nodes, they measure training fit rather than solution quality or generalization. Please evaluate on a separate, preferably finer grid and state the evaluation protocol explicitly.
[Sec. 4.1, Sec. 4.2, Algorithm 1] All numerical results come from a single training run with no seeds, no repeated initializations, and no variance information. The KAN grid size and spline order, which determine the expressive capacity of the B-spline basis, are also not reported. The claim of one-to-two-order accuracy improvement over MLP-PINN would be much more convincing with multiple random seeds and reported means/standard deviations or min/max ranges. At minimum, the missing KAN hyperparameters must be provided for reproducibility.
[Sec. 2.2, Figures 1, 6, 9] The comparison with classical numerical methods uses DOPRI5 applied to an index-reduced ODE, which is a generic explicit ODE solver rather than a numerical method designed for DAEs. The claim that neural network methods outperform classical methods in controlling drift-off therefore rests on a weak baseline. A stronger comparison would use a modern DAE solver, such as a projected implicit Runge-Kutta method, BDF with projection, or an index-aware solver applied to the original or index-reduced systems.

minor comments (5)

[Sec. 2.1, Definition 1] The sentence defining the differential index is incomplete: it reads 'The differential index ν is the smallest integer such that the system.' and should be completed or rewritten.
[Sec. 4.1, Figure 5 and Sec. 4.2, Figure 8] The figure captions state that the green line represents DAE-KAN and the blue line represents PINNs, while the body text describes a blue curve for PINNs and a red curve for DAE-KAN. The color labels are inconsistent and should be aligned.
[Sec. 4.2, Eq. (4.13)] The mass matrix in Eq. (4.13) is malformed: the second row appears to have missing entries. It should likely read [[5+3 cos(u2), 1+1.5 cos(u2)], [1+1.5 cos(u2), 1]].
[Sec. 4.2, paragraph after Eq. (4.18)] The text says that the traditional PINNs model predicts absolute errors for 'the algebraic variable v' when the algebraic variable in this system is the Lagrange multiplier lambda. This appears to be a typographical error.
[General] The abstract and conclusion mention 'generalization', but the experiments only cover two closed-form benchmark problems on a fixed time interval and no unseen-initial-condition or extrapolation tests are conducted. Please either add such experiments or soften the generalization claim.

Circularity Check

1 steps flagged · score 2.0 of 10

Central DAE-KAN accuracy claims are not circular; one drift-off diagnostic re-reports the training residual.

fitted input called prediction [Section 3.2 Eq. (3.10)-(3.12) and Section 4.1 Fig. 6 caption]
"Loss(θ)=MSE_F + MSE_i ... MSE_F = 1/N_F Σ |F(u'_KAN(t_n^F), u_KAN(t_n^F), t_n^F)|^2. ... Fig. 6: The predicted solutions in the form of (4.2) index-1, index-2, and index-3 obtained by PINNs and DAE-KAN are respectively inserted into the constraint equations in the form of (4.2) index-1, index-2, and index-3. That is, the (4.4), (4.3), and the constraint equations of (4.2), resulting in the drift-off error curve."

For each index formulation solved with loss (3.10), the algebraic constraint used to define the drift-off curve (e.g., (4.4), (4.3), or the final equation of (4.2)) is exactly one of the residual components already summed in MSE_F through F in (2.1)/(2.2). Thus the same-index drift-off curves report the training residual that was minimized, not an independent validation. The off-diagonal substitutions (a solution of one index checked against another index's constraint) are not in the loss and give independent content, so this circularity is partial and does not affect the main AE/RE accuracy comparisons against exact solutions.

full rationale

The main derivation is self-contained: the loss (3.10) contains only the DAE residual and the initial condition; the exact solutions (4.6) and the robot-arm solution are used exclusively as external error benchmarks and do not enter the loss. No fitted parameter is renamed as a prediction, and no author-uniqueness theorem is imported. The omission of hidden constraints (4.3)-(4.4) and (4.16)-(4.17) from the index-3 loss is a soundness/generalization concern, not an Eq-X-equals-Eq-Y circularity. Self-citations (refs. [10], [20]) are contextual literature claims and are not load-bearing for the derivation. The only self-referential element is the diagonal drift-off diagnostic in Fig. 6, which re-evaluates a term already minimized in MSE_F; since this is secondary to the accuracy claim, the overall circularity score is low.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central accuracy claim rests on hand-chosen KAN hyperparameters and on assumptions about smooth solvability and optimizer success. No new physical entities are introduced. The largest external load is the assumption that LBFGS finds the global minimizer of a nonconvex residual loss, which is not proved.

free parameters (5)

KAN architecture widths = [1,5,5,4] and [1,5,5,1] for particle; [1,4,4,4] and [1,2,2,1] for robot
Hand-chosen per problem; no search or sensitivity analysis is reported.
Number of collocation points N_F = 200
Uniform over [0,1]; no convergence study in N_F is shown.
Training epochs = 24,000 (particle), 20,000 (robot)
Stopping rule is not given; no loss or variance curves are shown.
Optimizer and initialization = LBFGS; B-spline coefficients N(0,0.1^2), Xavier weights
No learning rate or optimizer settings are reported; initialization choices are not varied.
KAN grid size and spline order = not reported
A defining hyperparameter for KAN accuracy is omitted, so the results may not reproduce.

assumptions (4)

standard math Kolmogorov-Arnold representation theorem supports the KAN approximation structure.
Used in Section 3.2, eq. (3.3), as the theoretical basis for the network architecture.
domain assumption The DAE systems considered are smooth and have unique solutions on [0,1].
The residual loss and the known exact solutions in Sections 4.1 and 4.2 require solvability and smoothness.
domain assumption Consistent initial conditions for the benchmark DAEs are known and used.
Initial conditions in eqs. (4.5) and (4.18) are derived from the known exact solutions and are not learned.
ad hoc to paper LBFGS converges to a global minimizer of the nonconvex residual loss.
Eq. (3.13) assumes the argmin is achieved; no convergence proof is offered for the nonconvex KAN loss.

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Cite this review

Pith. "Pith review of DAE-KAN: A Kolmogorov-Arnold Network Model for High-Index Differential-Algebraic Equations." pith.science (2026). https://pith.science/paper/S7WZFCG3

@misc{pith2026250415806,
  author       = {Pith},
  title        = {Pith review of: DAE-KAN: A Kolmogorov-Arnold Network Model for High-Index Differential-Algebraic Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S7WZFCG3}},
  note         = {Machine review of arXiv:2504.15806}
}

read the original abstract

Kolmogorov-Arnold Networks (KANs) have emerged as a promising alternative to Multi-layer Perceptrons (MLPs) due to their superior function-fitting abilities in data-driven modeling. In this paper, we propose a novel framework, DAE-KAN, for solving high-index differential-algebraic equations (DAEs) by integrating KANs with Physics-Informed Neural Networks (PINNs). This framework not only preserves the ability of traditional PINNs to model complex systems governed by physical laws but also enhances their performance by leveraging the function-fitting strengths of KANs. Numerical experiments demonstrate that for DAE systems ranging from index-1 to index-3, DAE-KAN reduces the absolute errors of both differential and algebraic variables by 1 to 2 orders of magnitude compared to traditional PINNs. To assess the effectiveness of this approach, we analyze the drift-off error and find that both PINNs and DAE-KAN outperform classical numerical methods in controlling this phenomenon. Our results highlight the potential of neural network methods, particularly DAE-KAN, in solving high-index DAEs with substantial computational accuracy and generalization, offering a promising solution for challenging partial differential-algebraic equations.

Figures

Figures reproduced from arXiv: 2504.15806 by the authors.

**Figure 1.** Error in the constraints for DOPRI5. 3. Proposed framework: DAE-KAN In this section, we provide a concise overview of the PINN method and introduce the fundamental concepts of Kolmogorov-Arnold networks. Additionally, we conclude this section by presenting our DAE-KAN framework. 3.1. The brief review of PINNs Physics-Informed Neural Networks (PINNs) [15] represent an innovative approach initially proposed for solvin… view at source ↗

**Figure 2.** Conventional physical information neural networks with MLP. and incorporated the physical information contained in equations into the neural network model to construct a neural network approximation of the solution: 𝑢 PINN(𝑡, 𝜽) ≈ 𝑢(𝑡), (3.1) where 𝜃 is the neural network parameter and 𝑢(𝑡) denote the exact solution of the equation at time 𝑡, 𝑢 PINN(𝑡, 𝜽) is the predicted solution of the neural network at time 𝑡, wh… view at source ↗

**Figure 3.** Illustration of a 3-layer KAN network of the shape [2,3,2,1]. 𝜙𝑙,𝑖,𝑗(𝐭 𝑙,𝑖) = 𝑤𝑙,𝑖,𝑗 ⋅ (𝑏(𝐭 𝑙,𝑖) + 𝑠𝑝𝑙𝑖𝑛𝑒(𝐭 𝑙,𝑖)), (3.7) where 𝑤𝑙,𝑖,𝑗 is the weight factor controlling the overall amplitude of the activation function, and 𝑏(𝑡) is the basis function, which can be set as: 𝑏(𝐭 𝑙,𝑖) = 𝑠𝑖𝑙𝑢(𝐭 𝑙,𝑖) = 𝐭 𝑙,𝑖 1 + 𝑒 −𝐭 𝑙,𝑖 , (3.8) the 𝑠𝑝𝑙𝑖𝑛𝑒(𝐭 𝑙,𝑖) is a spline function, which can be parameterized as a linear combination of B-s… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Diagram of the DAE-KAN network structure. differential neural network predicts the differential variables, and the algebraic neural network predicts the algebraic variables. Using two KAN networks to separately predict the differential and algebraic variables helps the…

**Figure 5.** Figure 5: The AE curves of the predicted solution and the exact solution obtained by using the traditional PINNs and DAE-KAN to solve the corresponding index-1, index-2, and index-3 forms of equation (4.2), respectively. The green line represents DAE-KAN, and the blue line repre…

**Figure 6.** Figure 6: The predicted solutions in the form of (4.2) index-1, index-2, and index-3 obtained by PINNs and DAE-KAN are respectively inserted into the constraint equations in the form of (4.2) index-1, index-2, and index-3. That is, the (4.4), (4.3), and the constraint equations …

**Figure 7.** Figure 7: Schematic diagram of a robot arm. The coordinates of the intermediate joint of the robot arm (𝑥1 , 𝑦1 ) are given by: {𝑥1 = 𝑙1 cos 𝜃1 , 𝑦1 = 𝑙1 sin 𝜃1 , (4.7) The coordinates of the end effector of the robot arm (𝑥2 , 𝑦2 ) are given by: {𝑥2 = 𝑥1 + 𝑙2 cos(𝜃1 + 𝜃2 ), 𝑦2 …

**Figure 8.** Figure 8: The AE curves of the predicted solution and the exact solution obtained by using the traditional PINNs and DAEKAN to solve the corresponding index-1, index-2, and index-3 forms of (2.2), respectively. The green line represents DAE-KAN, and the blue line represents PIN…

**Figure 9.** Figure 9: The predicted solutions in the form of (2.2) index-1 index-2 and index-3 obtained by PINNs and DAE-KAN are respectively put into the constraint equations in the form of (2.2) index-1 index-2 and index-3. That is, the Eqs (4.17) (4.16) and (4.15), and finally the drift-…

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jaxdae: A JAX-native Differentiable Solver for Differential-Algebraic Equations in Coupled Multi-physics
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