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arxiv: 2605.22235 · v1 · pith:T35BPZLPnew · submitted 2026-05-21 · 💻 cs.LG · math.DS

Holomorphic Neural ODEs with Kolmogorov-Arnold Networks for Interpretable Discovery of Complex Dynamics

Bhaskar Ranjan Karn , Dinesh Kumar This is my paper

Pith reviewed 2026-05-22 08:29 UTC · model grok-4.3

classification 💻 cs.LG math.DS
keywords holomorphic dynamicsKolmogorov-Arnold networksneural ordinary differential equationssymbolic regressionJulia setsCauchy-Riemann equationscomplex dynamical systemsinterpretable machine learning
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The pith

A Kolmogorov-Arnold network inside a neural ODE recovers the symbolic holomorphic maps that generate complex fractal dynamics from data.

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

The paper replaces the multilayer perceptron vector field inside a Neural ODE with a Kolmogorov-Arnold Network whose B-spline activations lie on the network edges. A differentiable penalty based on the Cauchy-Riemann equations is added during training to keep the learned dynamics holomorphic. Evaluated on six families of polynomial and transcendental complex maps, the resulting model reaches velocity-field R-squared above 0.95 with only 280 parameters, converts its learned splines directly into the correct symbolic governing equations, and reconstructs the associated Julia-set boundaries to 98 percent agreement. The same architecture shows markedly better noise tolerance and transfer performance than standard MLP baselines while supplying the governing equations that black-box models do not.

Core claim

Holomorphic KAN-ODE places learnable B-splines on the edges of a Kolmogorov-Arnold Network that serves as the right-hand side of a Neural ODE and augments the loss with a differentiable Cauchy-Riemann term. Across six holomorphic dynamical families the trained network produces velocity fields whose R-squared exceeds 0.95, converts the spline activations into the exact symbolic form of each governing map, and reconstructs the corresponding Julia-set fractal boundaries to within 98 percent agreement, all while using sixteen times fewer parameters than an MLP baseline.

What carries the argument

Kolmogorov-Arnold Network with edge-wise B-spline activations inside a Neural ODE, regularized by the Cauchy-Riemann equations, that permits direct conversion from learned splines to symbolic governing maps.

If this is right

  • Velocity-field R-squared exceeds 0.95 on all six tested holomorphic systems while using only 280 parameters.
  • Automatic conversion of learned B-splines yields the correct symbolic family for every one of the six dynamical systems.
  • Reconstructed Julia-set boundaries agree with the true fractals up to 98 percent.
  • Mean-squared error rises by only 4 percent when 10 percent observation noise is added.
  • Transfer from quadratic to cubic dynamics improves by 90.4 percent relative to MLP baselines.

Where Pith is reading between the lines

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

  • The same regularization approach could be tested on other geometric constraints such as conformality or periodicity to see whether it generalizes beyond holomorphicity.
  • If the spline-to-symbolic step proves reliable on new maps, it could replace separate symbolic-regression stages in other physics-informed models.
  • Applying the architecture to non-polynomial transcendental maps would test whether the holomorphic prior remains effective outside algebraic cases.

Load-bearing premise

Adding a differentiable Cauchy-Riemann penalty during training is sufficient to keep the network outputs truly holomorphic and that the automatic spline-to-formula conversion step then recovers the exact symbolic governing map without approximation or selection error.

What would settle it

Train the model on noisy trajectory data generated by a known holomorphic map such as z squared plus c, run the spline-to-formula extraction, and check whether the extracted closed-form expression exactly matches the original map rather than an approximate or different expression.

Figures

Figures reproduced from arXiv: 2605.22235 by Bhaskar Ranjan Karn, Dinesh Kumar.

Figure 1
Figure 1. Figure 1: Architecture of the Holomorphic KAN-ODE. Inputs [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The training dynamics for z 2 +c. Left: The MSE convergence comparison shows that HoloKAN (280 parameters) achieves optimal performance earlier than MLP (4,482 parameters) due to the inductive bias of B-spline edge activations. Right: The Cauchy–Riemann residual LCR (solid) decreases as the linear warmup weight λCR(t) (dashed) increases over 100 steps. 3.5 Symbolic Extraction KANs are unique in their abili… view at source ↗
Figure 3
Figure 3. Figure 3: Velocity field MSE comparison across all systems (log scale). While the MLP achieves [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Julia set fractal boundary for f(z) = z 2 + c with c = −0.4 + 0.6i, visualized by escape-time iteration count. Boundary agreement between KAN-generated and ground-truth fractal reaches 95.5% ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Noise robustness comparison. The KAN maintains near-constant performance under [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

Complex dynamical systems governed by holomorphic maps such as $z^2 + c$ exhibit fractal boundaries with extreme sensitivity to initial conditions. Accurately modelling these structures from data requires methods that respect the underlying complex-analytic geometry, yet Multi-Layer Perceptrons (MLPs) within Neural Ordinary Differential Equations (Neural ODEs) lack complex-analytic priors, violate the Cauchy--Riemann conditions, and function as opaque approximators incapable of yielding governing equations. We introduce Holomorphic KAN-ODE, a framework that replaces the MLP with a Kolmogorov-Arnold Network (KAN) whose learnable B-spline activations reside on network edges, and incorporates Cauchy--Riemann equations as a differentiable regularization to preserve holomorphic structure. We evaluate on six families of complex dynamical systems spanning polynomial and transcendental classes. With only 280 parameters ($16\times$ fewer than the MLP baseline), the network achieves velocity-field $R^2 > 0.95$ on all six systems, correctly identifies all six governing symbolic families through automatic spline-to-formula fitting, and reconstructs Julia set fractal boundaries with up to 98.0\% agreement. Crucially, the model exhibits only 4\% MSE degradation under 10\% observation noise versus $15.2\times$ for MLPs, and achieves 90.4\% improvement in transfer learning from quadratic to cubic dynamics. While the MLP attains lower pointwise reconstruction error due to its larger capacity, the KAN uniquely provides interpretable symbolic equations, enforced holomorphic structure, and superior noise resilience, capabilities that are entirely absent in black-box architectures. These results establish KANs as a parameter-efficient, interpretable alternative to MLPs for physics-informed discovery of holomorphic dynamics.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. The manuscript introduces Holomorphic KAN-ODE, a Neural ODE variant that replaces the MLP with a Kolmogorov-Arnold Network whose B-spline activations are regularized by a differentiable Cauchy-Riemann penalty to enforce holomorphic structure. Evaluated on six polynomial and transcendental complex dynamical systems, the model with 280 parameters reports velocity-field R² > 0.95, automatic recovery of all six governing symbolic families via spline-to-formula conversion, Julia-set boundary reconstruction up to 98 % agreement, 4 % MSE degradation under 10 % noise, and strong transfer-learning gains from quadratic to cubic dynamics, while claiming interpretability and noise resilience absent in MLP baselines.

Significance. If the central claims hold, the work would offer a parameter-efficient, interpretable route to symbolic discovery of holomorphic flows that standard Neural ODEs cannot provide. The explicit complex-analytic prior and automatic symbolic step address a genuine gap in physics-informed learning for systems with fractal sensitivity. Reproducible code and explicit parameter counts are positive features, but the absence of derivation details for the regularization and fitting steps limits immediate assessment of whether the reported identification is truly parameter-free and independent of post-training approximation.

major comments (3)
  1. [§3.2] §3.2 (Loss function and CR regularization): the Cauchy-Riemann term is introduced as a soft penalty with finite coefficient λ. Because the constraint is not hard, residual violations of the CR equations can remain; any such residuals allow the B-spline activations to encode non-holomorphic corrections that still fit the training trajectories, undermining the claim that the subsequent spline-to-formula step recovers the exact holomorphic governing map.
  2. [§4.3] §4.3 (Symbolic recovery procedure): the automatic conversion from learned splines to closed-form expressions is presented as identifying all six families without post-hoc selection. No explicit separation between training data and the fitting procedure, nor verification that the recovered symbols reproduce the true flow near fractal boundaries, is provided; this leaves open the possibility that the reported identification is a post-training fit rather than an independent derivation.
  3. [Table 2] Table 2 (velocity-field R² and symbolic accuracy): while R² > 0.95 is reported for all six systems, the table does not include error bars or the precise methodology used to compute them, nor does it show the dynamical divergence of the recovered symbolic equations versus the ground-truth holomorphic flow; without these, the cross-system claim of exact family identification cannot be fully evaluated.
minor comments (2)
  1. [§3] Notation for complex variables (z = x + iy) is introduced but not consistently carried through the equations in §3; explicit component-wise CR equations would improve clarity.
  2. [Figure 4] Figure 4 caption states 'up to 98.0 % agreement' but does not specify the exact metric (pixel overlap, Hausdorff distance, etc.) or the number of initial conditions used for the Julia-set comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below, indicating revisions to the manuscript where appropriate.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Loss function and CR regularization): the Cauchy-Riemann term is introduced as a soft penalty with finite coefficient λ. Because the constraint is not hard, residual violations of the CR equations can remain; any such residuals allow the B-spline activations to encode non-holomorphic corrections that still fit the training trajectories, undermining the claim that the subsequent spline-to-formula step recovers the exact holomorphic governing map.

    Authors: We agree that the Cauchy-Riemann regularization is a soft penalty and therefore does not enforce a hard constraint. In the current experiments the final CR residual is kept below 5×10^{-4} for all six systems, which is small enough that the dominant spline terms remain holomorphic and permit exact symbolic recovery. To make this explicit we will add a supplementary table reporting the per-system CR residual norms at convergence together with a short derivation of the differentiable penalty term. revision: yes

  2. Referee: [§4.3] §4.3 (Symbolic recovery procedure): the automatic conversion from learned splines to closed-form expressions is presented as identifying all six families without post-hoc selection. No explicit separation between training data and the fitting procedure, nor verification that the recovered symbols reproduce the true flow near fractal boundaries, is provided; this leaves open the possibility that the reported identification is a post-training fit rather than an independent derivation.

    Authors: The spline-to-formula conversion is a deterministic post-training procedure that operates solely on the learned B-spline coefficients and knot vectors; it does not re-optimize or refit to the original training trajectories. We therefore view it as independent of the data-fitting stage. We nevertheless accept that explicit verification near fractal boundaries is missing and will add a new experiment that integrates both the neural and the recovered symbolic vector fields from points on the Julia-set boundary and reports the resulting trajectory divergence. revision: partial

  3. Referee: [Table 2] Table 2 (velocity-field R² and symbolic accuracy): while R² > 0.95 is reported for all six systems, the table does not include error bars or the precise methodology used to compute them, nor does it show the dynamical divergence of the recovered symbolic equations versus the ground-truth holomorphic flow; without these, the cross-system claim of exact family identification cannot be fully evaluated.

    Authors: The reported R² values are coefficients of determination evaluated on a held-out test set of 2000 points per system. We will revise Table 2 to include standard deviations computed over five independent runs with different random seeds and will add a supplementary table that quantifies dynamical divergence (integrated squared error of 100-step trajectories) for both the KAN-ODE and the recovered symbolic equations against the ground-truth flow. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results are empirical evaluations on external benchmarks

full rationale

The paper introduces Holomorphic KAN-ODE by combining Kolmogorov-Arnold Networks with Neural ODEs and adding a differentiable Cauchy-Riemann regularization term. Claims rest on experimental outcomes: velocity-field R² > 0.95 across six systems, Julia-set reconstruction agreement up to 98%, 4% MSE degradation under noise, and transfer-learning gains. The symbolic identification step is explicitly described as post-training automatic spline-to-formula fitting rather than a first-principles derivation or a quantity forced by construction from the training inputs. No equations reduce the output to the inputs by definition, no load-bearing self-citation chains appear, and no fitted parameters are relabeled as independent predictions. The framework is therefore self-contained against the reported external benchmarks and does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the target systems are exactly holomorphic and that B-spline activations plus a differentiable penalty can both learn the dynamics and permit exact symbolic recovery; no new physical entities are postulated.

free parameters (2)
  • B-spline knot coefficients and grid size
    Learnable parameters on network edges that are fitted to velocity-field data for each dynamical system.
  • Cauchy-Riemann regularization strength
    Hyper-parameter controlling the weight of the holomorphic constraint during training.
axioms (1)
  • domain assumption The underlying maps satisfy the Cauchy-Riemann equations everywhere in the domain of interest.
    Invoked to justify both the problem setup and the choice of regularization term.

pith-pipeline@v0.9.0 · 5848 in / 1448 out tokens · 38343 ms · 2026-05-22T08:29:56.731005+00:00 · methodology

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