{"id":"d420554f-ecce-467d-a2fc-e2aa97924082","arxiv_id":"2504.15806","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"DAE-KAN, a dual-network KAN-PINN, is reported to solve index-1 through index-3 differential-algebraic equations with one to two orders of magnitude lower error than MLP-based PINNs on two benchmark problems.","lead":"DAE-KAN combines Kolmogorov-Arnold networks with physics-informed training to solve differential-algebraic equations, including index-3 systems, and reports one to two orders of magnitude lower error than standard MLP-based PINNs on two benchmark problems. A generalist might read it to gauge whether KAN-based neural solvers are becoming practical for constrained dynamics and circuit-style equations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Loss (3.10) omits the hidden constraints (4.3)-(4.4), so the 200-point residual does not uniquely determine the index-3 solution, and the paper does not report evaluation on a distinct grid, leaving constraint satisfaction and generalization unverified.","rationale":"The paper's headline is that high-index DAEs can be solved directly and more accurately than traditional PINNs. The most load-bearing step is the move from a 201-point residual fit to a claim about the true solution on [0,1]. For an index-3 DAE, the algebraic constraint is only part of the solution manifold; the hidden constraints obtained by differentiation are needed to determine the velocities and multiplier. Since (3.10) contains no hidden-constraint term, the optimization problem is underdetermined at the continuous level: the residual can be zero while (4.3) or (4.4) is violated away from the training points. The numerical section does not report a separate evaluation grid, so the reported AE/RE and drift-off values could reflect only the training nodes. This does not make the paper wrong, but it makes the central claim currently unsupported. The proposed check, dense held-out evaluation plus a control with hidden constraints added to the loss, would settle it. I keep the reader's CONDITIONAL verdict because the concern is not yet demonstrated, but if the check fails, the claim should be rejected.","tokens_in":14183,"tokens_out":6753,"duration_ms":63925,"concrete_test":"Re-run the Section 4.1 experiment with the published loss (3.10) and the stated 200 collocation points, then evaluate the trained DAE-KAN on 10,000 random points in [0,1] distinct from the training nodes. Compute the maximum residual of hidden constraints (4.3)-(4.4) and the maximum absolute error over all five variables. If either exceeds roughly 1e-3 or is more than 10x the corresponding Table 1 entries, the direct index-3 claim is unsupported. As a control, train the same networks with (4.3)-(4.4) added to the loss and compare; large differences would show the published loss is underdetermined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"For the index-3 problems, equation (3.10) penalizes only the original DAE residual and the initial condition. For the circular-particle example, the solution must also satisfy the hidden constraints (4.3) u1*z1 + u2*z2 = 0 and (4.4) z1^2 + z2^2 + 2*u1*u2*(2 - u2^2 - u1^2) - lambda*(u1^2 + u2^2) = 0. These are not included in the loss. With NF = 200 and Ni = 1, a B-spline KAN can fit the residual at the training nodes while violating these constraints between nodes or on a finer grid, because the constraints restrict only the position manifold, not the velocity and multiplier variables, except at isolated points. The paper does not state that Figures 5-6 and Tables 1-2 are computed on a held-out grid; if the AE/RE and drift-off curves are evaluated on the same 200 collocation points, they measure training fit rather than solution quality. The central claim of direct high-index solution therefore rests on an unverified assumption that 201 points are sufficient to pin down the unique index-3 solution and that the reported errors transfer off-grid. The same issue applies to the robot-arm example, where the hidden constraints are (4.16) and (4.17).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14489,"tokens_out":9145,"duration_ms":85990,"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":[{"comment":"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.","section":"Sec. 3.2, Eq. (3.10); Sec. 4.1, Eqs. (4.3)-(4.4); Sec. 4.2, Eqs. (4.16)-(4.17)"},{"comment":"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.","section":"Sec. 4.1, Sec. 4.2, Eq. (4.1), Figures 5-9, Tables 1-2"},{"comment":"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.","section":"Sec. 4.1, Sec. 4.2, Algorithm 1"},{"comment":"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.","section":"Sec. 2.2, Figures 1, 6, 9"}],"minor_comments":[{"comment":"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.","section":"Sec. 2.1, Definition 1"},{"comment":"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.","section":"Sec. 4.1, Figure 5 and Sec. 4.2, Figure 8"},{"comment":"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]].","section":"Sec. 4.2, Eq. (4.13)"},{"comment":"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.","section":"Sec. 4.2, paragraph after Eq. (4.18)"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main concerns are empirical verification issues rather than fundamental design flaws. The authors can address them by adding a held-out evaluation grid, incorporating hidden constraints into the loss or at least verifying them off-grid, reporting multiple seeds and KAN hyperparameters, and strengthening the classical baseline. If the revised evidence supports the claims, I see no reason to reject the paper. I do not see a novelty or citation-practice concern beyond the scope of the report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nWhere 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.","headline":"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.","tokens_in":15060,"tokens_out":2594,"would_cite":true,"duration_ms":25470,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65L80","68T07","34A09"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["differential-algebraic equations","high-index DAEs","physics-informed neural networks","Kolmogorov-Arnold networks","drift-off error","index reduction","B-spline activation","neural DAE solver"],"falsifier":"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.","tokens_in":13966,"feed_emoji":"🧮","tokens_out":8666,"duration_ms":71324,"temperature":0.7,"pith_summary":"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.","feed_headline":"KAN-based network solves index-3 DAEs directly","feed_subtitle":"Dual KAN nets for differential and algebraic variables beat MLP-based PINNs and tame drift-off error.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the definition of DAE index and the drift-off phenomenon that motivates direct high-index solving.","marker":"[1]"},{"why":"Provides the PINN methodology whose loss structure DAE-KAN inherits and the MLP baseline it is compared against.","marker":"[15]"},{"why":"Introduces Kolmogorov-Arnold Networks with B-spline activations, the function approximator at the core of DAE-KAN.","marker":"[22]"},{"why":"Is the previous DAE-PINN architecture, limited to index-1 systems, which DAE-KAN extends.","marker":"[19]"},{"why":"Is Radau-PINN, which handles index-2 but not index-3 DAEs, defining the accuracy ceiling DAE-KAN claims to break.","marker":"[20]"},{"why":"Is the DOPRI5 integrator whose drift-off error on the index-reduced pendulum illustrates why classical methods need index reduction.","marker":"[26]"},{"why":"Provides the circular-orbit index-3 DAE test problem with its exact solution.","marker":"[29]"},{"why":"Provides the two-link robot arm index-3 multibody test problem with its exact solution.","marker":"[30]"}],"fun_headline_variants":["KAN net beats MLP PINNs on high-index DAEs","Index-3 DAEs tamed by dual KAN networks","KAN-PINN cuts error on algebraic constraints","High-index DAEs solved with KAN-based PINN"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["KAN net beats MLP PINNs on high-index DAEs","Index-3 DAEs tamed by dual KAN networks","KAN-PINN cuts error on algebraic constraints","High-index DAEs solved with KAN-based PINN"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1311,"prompt_tokens":991,"completion_tokens":320,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":251}},"tokens_in":607,"tokens_out":320,"duration_ms":3365,"temperature":1.0,"reasoning_tokens":251,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:17:23.878711+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the definition of DAE index and the drift-off phenomenon that motivates direct high-index solving."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the previous DAE-PINN architecture, limited to index-1 systems, which DAE-KAN extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is Radau-PINN, which handles index-2 but not index-3 DAEs, defining the accuracy ceiling DAE-KAN claims to break."},{"cited_title":"Benhammouda, Solution of nonlinear higher-index hessenberg DAEs by adomian polynomials and differential transform method, SpringerPlus 4 (2015) 648","cited_arxiv_id":null,"evidence_quote":"Provides the circular-orbit index-3 DAE test problem with its exact solution."},{"cited_title":"Ascher, P","cited_arxiv_id":null,"evidence_quote":"Provides the two-link robot arm index-3 multibody test problem with its exact solution."}],"review_version":1}