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

This paper introduces Holomorphic KAN-ODE, a parameter-efficient and interpretable framework that integrates Kolmogorov-Arnold Networks with Cauchy-Riemann regularization to accurately model complex dynamical systems, discover their symbolic governing equations, and robustly reconstruct fractal boundaries while outperforming traditional MLP-based Neural ODEs in noise resilience and transfer learning.

Original authors: Bhaskar Ranjan Karn, Dinesh Kumar

Published 2026-05-22

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Bhaskar Ranjan Karn, Dinesh Kumar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a computer to understand the chaotic, swirling patterns of a complex dance. In the world of mathematics, these dances are called complex dynamical systems. They are famous for creating beautiful, intricate fractal shapes (like the famous Mandelbrot set) where a tiny change in the starting step leads to a completely different ending.

The problem is that standard computer brains (called MLPs or Multi-Layer Perceptrons) are terrible at understanding the specific "rules of the dance" for these systems. They are like a student who memorizes the steps by rote but doesn't understand the rhythm or the physics, and they can't explain why they moved that way. They are also very fragile; if you add a little bit of static noise to the music, they get confused and fail.

The authors of this paper, Bhaskar Ranjan Karn and Dinesh Kumar, have built a new kind of computer brain called Holomorphic KAN-ODE. Here is how it works, explained simply:

1. The New Brain: KANs (The "Flexible Ropes")

Instead of using the standard rigid "black box" neurons, they used Kolmogorov–Arnold Networks (KANs).

The Analogy: Think of a standard neural network as a factory assembly line where every worker is fixed in place and does the same job. A KAN is more like a team of acrobats using flexible, stretchy ropes (called B-splines) to connect each other.
Why it matters: Because these ropes are flexible and learnable, the network can stretch itself to perfectly match the shape of the math it is trying to learn. Even better, once it learns the shape, it can look at the rope and say, "Ah, this rope is shaped exactly like a parabola" or "This one looks like a sine wave." This allows the computer to write down the actual math formula it discovered, rather than just guessing the answer.

2. The Rulebook: Cauchy–Riemann (The "Law of the Dance")

Complex math has a strict rulebook called the Cauchy–Riemann equations. If a function doesn't follow these rules, it's not a valid "complex" function.

The Analogy: Imagine teaching a dancer to waltz. If you don't tell them to keep their feet moving in a specific, coordinated way, they might trip or spin out of control.
The Solution: The authors added a "strict coach" (a regularization term) to the training. This coach constantly checks the dancer's moves. If the moves violate the Cauchy–Riemann rules, the coach gives a penalty. This forces the network to learn the correct mathematical structure, not just a messy approximation.

3. The Results: Small, Smart, and Strong

The authors tested this new system on six different types of complex dances (including polynomial and transcendental types). Here is what happened:

Efficiency: The new system learned the dances using only 280 parameters (tiny bits of memory). The old standard system needed 4,482 parameters (16 times more!) to get a similar level of accuracy on the raw numbers.
Interpretability: While the old system just gave a number, the new system looked at its own "ropes" and correctly identified the mathematical family of the dance (e.g., "This is a quadratic dance" or "This is an exponential dance"). It successfully wrote down the formulas.
Noise Resistance: This is the biggest win. When the researchers added "noise" (static) to the data:
- The old system's performance crashed by 15 times.
- The new system barely flinched, dropping only 4% in performance.
- Analogy: If you shout over a radio, the old system hears gibberish. The new system, because it knows the strict rules of the dance, ignores the shouting and keeps dancing correctly.
Fractals: They used the new system to redraw the famous fractal boundaries (Julia sets). The new system's drawings matched the real math with up to 98% accuracy.

4. What It Can't Do (The Limitations)

The paper is honest about what this tool cannot do yet:

It needs clean data: The system was trained on perfect, computer-generated data, not messy real-world measurements.
It struggles with "holes": It had trouble with a specific type of fluid flow that has a "singularity" (a mathematical hole or infinity point), suggesting it works best on smooth, continuous movements.
It's a specific tool: It is designed specifically for complex numbers and holomorphic functions, not for every type of physics problem.

The Bottom Line

The authors have created a tool that is smaller, smarter, and tougher than the current standard for learning complex mathematical dances. It doesn't just guess the answer; it learns the underlying rules, writes down the formula, and keeps dancing perfectly even when the music gets noisy. It proves that for this specific type of math, a specialized, rule-following network is far superior to a giant, generic black box.

Technical Summary: Holomorphic KAN-ODE for Interpretable Discovery of Complex Dynamics

1. Problem Statement

Complex dynamical systems governed by holomorphic maps (e.g., $z^2 + c$ ) exhibit fractal boundaries and extreme sensitivity to initial conditions. Accurately modeling these systems from data requires methods that respect the underlying complex-analytic geometry.

Current approaches face three primary limitations:

Lack of Structural Priors: Standard Neural Ordinary Differential Equations (Neural ODEs) typically employ Multi-Layer Perceptrons (MLPs) to parameterize velocity fields. These MLPs lack complex-analytic priors and fail to enforce the Cauchy–Riemann conditions necessary for holomorphic functions.
Opacity: MLPs function as black-box approximators, offering no pathway from learned weights to interpretable symbolic governing equations.
Real-Valued Constraints: Existing applications of Kolmogorov–Arnold Networks (KANs) to Neural ODEs have been restricted to real-valued state spaces, leaving complex-valued dynamics unaddressed.

2. Methodology: Holomorphic KAN-ODE

The authors propose Holomorphic KAN-ODE, a framework that integrates Kolmogorov–Arnold Networks with Neural ODEs, constrained by Cauchy–Riemann regularization to ensure the learned dynamics are holomorphic.

2.1 Architecture

The framework replaces the fixed activation functions of standard MLPs with learnable univariate B-spline functions residing on the network edges.

Input/Output: The network takes the real and imaginary parts of the complex state $z = x + iy$ as input and outputs the real and imaginary parts of the velocity field $f(z) = u + iv$.
Parameterization: Each edge activation $\phi_{i,j}(x)$ is defined as a linear combination of B-spline basis functions $B_m(x)$ with learnable coefficients.
Efficiency: The proposed architecture uses a width of $[2, 5, 2]$ , resulting in only 280 trainable parameters, compared to 4,482 in the MLP baseline.

2.2 Cauchy–Riemann Regularization

To enforce holomorphicity, the authors introduce a differentiable regularization term derived from the Cauchy–Riemann equations:
$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
The loss function includes a term $L_{CR}$ that penalizes deviations from these conditions. To prevent training collapse caused by enforcing this constraint immediately, a linear warmup schedule is employed, gradually increasing the regularization weight $\lambda_{CR}$ from zero over the initial training steps.

2.3 Symbolic Extraction

A key feature of KANs is their ability to perform automatic symbolic regression. After training, the learned spline activations on each edge are fitted against a candidate library of basis functions (e.g., polynomials, exponentials, trigonometric terms). The dominant basis functions across the network edges are used to identify the symbolic family of the governing equation (e.g., identifying a system as belonging to the $z^2 + c$ family).

3. Key Contributions

Holomorphic KAN Architecture: The first framework to combine KANs with Neural ODEs under Cauchy–Riemann regularization for complex dynamics.
Comprehensive Evaluation: Testing across six families of complex dynamical systems, including polynomial ( $z^2+c, z^3+c$ ) and transcendental classes ( $e^z+c, \sin(z)+c, \cos(z)+c, z \cdot e^z+c$ ), as well as a physical potential flow application.
Interpretability and Robustness: Demonstrating that the model can recover symbolic formulas, enforce holomorphic structure, and maintain superior noise resilience compared to MLPs.

4. Experimental Results

The framework was evaluated on six pure complex-analytic systems and one physical system (potential flow around a cylinder).

4.1 Accuracy and Efficiency

Velocity Field Approximation: The Holomorphic KAN achieved $R^2 > 0.95$ on all six pure systems using only 280 parameters.
Comparison with MLP: While the MLP baseline (4,482 parameters) achieved lower pointwise Mean Squared Error (MSE) due to its larger capacity, it lacked interpretability and structural enforcement.
Fractal Reconstruction: The model reconstructed Julia set fractal boundaries with up to 98.0% agreement (e.g., for the sine system), validating that the learned velocity fields preserve fine-grained geometric details.

4.2 Symbolic Discovery

The model correctly identified the governing symbolic family for all six systems through automatic spline-to-formula fitting (100% accuracy).

Polynomial systems were identified as polynomial.
Transcendental systems were correctly fitted to exponential, trigonometric, or mixed exponential forms.

4.3 Stability and Noise Resilience

Lyapunov Analysis: The model correctly classified the stability of the systems, identifying polynomial systems as chaotic and transcendental systems as predominantly stable.
Noise Robustness: Under 10% observation noise, the KAN exhibited only a 4% degradation in MSE, whereas the MLP degraded by a factor of 15.2×. This highlights the inductive bias provided by the Cauchy–Riemann constraint and spline parameterization.
Transfer Learning: Fine-tuning a KAN trained on quadratic dynamics ( $z^2+c$ ) to learn cubic dynamics ( $z^3+c$ ) resulted in a 90.4% improvement in final MSE compared to training from scratch, demonstrating the transferability of holomorphic priors.

4.4 Limitations Observed

Rational/Singular Systems: In the potential flow application ( $f(z) = Uz + Ua^2/z$ ), the model achieved a lower $R^2$ (0.333), indicating that B-spline parameterizations optimized for smooth functions struggle with rational structures and singularities.
Gradient Challenges: Systems with steep higher-order gradients (e.g., $z^3+c$ ) showed slightly lower $R^2$ values compared to smoother systems.

5. Significance and Claims

The paper positions Holomorphic KAN-ODE as a parameter-efficient, interpretable alternative to MLPs for the physics-informed discovery of holomorphic dynamics. The authors claim that while MLPs may offer superior pointwise reconstruction accuracy due to higher capacity, they fail to provide the three critical capabilities essential for scientific discovery in this domain:

Symbolic Formula Recovery: The ability to extract closed-form governing equations.
Enforced Holomorphic Structure: Guaranteeing that the learned dynamics satisfy complex-analytic constraints.
Superior Noise Resilience: Maintaining performance in noisy observational settings where black-box models fail.

The work establishes KANs as a viable tool for bridging complex analysis, dynamical systems theory, and scientific machine learning, specifically for systems where structural priors and interpretability are paramount.