Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics

This paper introduces Structured Kolmogorov-Arnold Neural ODEs (SKANODEs), a framework that combines structured state-space modeling with Kolmogorov-Arnold Networks to accurately recover interpretable physical latent states and discover compact symbolic governing equations for nonlinear dynamical systems, outperforming black-box neural ODEs and classical identification methods across synthetic and real-world datasets.

The Gist

Imagine you are trying to figure out how a complex machine works—like a car engine or a bouncing spring—but you can only see the smoke coming out of the exhaust pipe. You can't see the pistons moving, the gears turning, or the fuel burning. You only have the "smoke" (the data).

Most modern AI models are like black boxes. You feed them the smoke, and they guess the next puff of smoke with amazing accuracy. But if you ask, "How does the engine actually work?" they can't tell you. They just say, "Trust me, I'm good at guessing."

This paper introduces a new kind of AI called SKANODE. Think of SKANODE not as a black box, but as a detective who builds a transparent model of the engine while solving the mystery.

Here is how it works, broken down into simple steps:

1. The Problem: Only Seeing the "Smoke"

In the real world (like in engineering or physics), we often can't measure everything. We might only have sensors that measure acceleration (how fast something is speeding up or slowing down). We don't have sensors for position (where it is) or velocity (how fast it's moving).

• Old AI: Tries to guess the next acceleration number. It gets the number right, but it doesn't understand why the object moved.
• The Goal: We want an AI that can look at the acceleration, figure out the hidden position and speed, and then write down the actual physics formula that governs the movement.

2. The Secret Weapon: The "Shape-Shifting" Network (KAN)

The paper uses a special type of AI brain called a Kolmogorov-Arnold Network (KAN).

• Normal AI (MLP): Imagine a brain made of rigid, fixed Lego bricks. It can build complex shapes, but once built, you can't easily see the individual bricks or understand the logic.
• KAN: Imagine a brain made of playdough. It can mold itself into any shape to fit the data perfectly. But here's the magic: after it molds itself, you can look at the playdough and say, "Ah, this part is a curve, this part is a straight line." It can turn itself back into a simple math equation.

3. The Two-Stage Detective Process

SKANODE solves the problem in two stages, like a detective first gathering clues and then writing the case file.

Stage 1: Virtual Sensing (The "Mind's Eye")

The AI is given only the acceleration data (the smoke). Because the AI is built with a structured state-space design, it is forced to imagine the hidden world.

• It must invent a "position" and a "velocity" to explain the acceleration.
• Because of the rules built into the AI, these invented numbers aren't random; they naturally align with real physics.
• Analogy: It's like watching a shadow on a wall and being able to perfectly reconstruct the 3D object casting it, even though you've never seen the object itself.

Stage 2: Symbolic Discovery (Writing the Law)

Once the AI has figured out the hidden position and velocity, it switches modes. It stops being a "guessing machine" and starts being a "mathematician."

• It looks at the patterns it found and asks: "What simple formula connects these dots?"
• It strips away the complex neural network and replaces it with a clean, human-readable equation.
• The Result: Instead of a 100-layer neural network, you get a sentence like: "Acceleration equals minus 5 times position, plus 2 times velocity, plus 10 times position cubed."

4. Why This Matters: The Real-World Test

The authors tested this on three things:

1. A Duffing Oscillator: A spring that gets stiffer the more you stretch it.
   • Result: SKANODE found the exact math formula, including the tricky "cubic" part, and correctly guessed the hidden position and speed.
2. A Van der Pol Oscillator: A system with weird, self-sustaining vibrations.
   • Result: It found the formula for the "nonlinear damping" (the friction that changes as it moves).
3. An F-16 Fighter Jet: Real data from a plane vibrating on the ground.
   • Result: This is the big one. The plane had a "hysteretic" interface (parts rubbing together in a complex way). SKANODE didn't just predict the vibration; it revealed the hidden "hysteresis" loop in the data, showing engineers exactly where the energy was being lost.

The Big Takeaway

Most AI models are like oracles: they give you the answer, but you don't know how they got it.
SKANODE is like a teacher: it gives you the answer, shows you the hidden steps it took to get there, and writes down the textbook formula so you can understand the laws of physics behind the machine.

It bridges the gap between "Black Box AI" (which is accurate but mysterious) and "Classical Physics" (which is understandable but hard to apply to messy, real-world data). It lets us learn the laws of the universe directly from noisy sensor data, even when we can't see the whole picture.

Technical Summary

Here is a detailed technical summary of the paper "Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics."

1. Problem Statement

The modeling of nonlinear dynamical systems is critical across engineering and science, yet existing approaches face a fundamental trade-off between predictive accuracy and interpretability:

• Deep Learning (Black-Box): Models like Neural ODEs (NODEs) and RNNs excel at capturing complex dynamics but often function as "black boxes." Their latent representations are abstract and lack physical meaning (e.g., they do not explicitly correspond to displacement or velocity), making them difficult to trust in safety-critical engineering applications.
• Symbolic Discovery (SINDy, etc.): Methods like Sparse Identification of Nonlinear Dynamics (SINDy) can discover explicit governing equations but typically require full-state observations (direct measurement of position, velocity, and acceleration) and clean data. They often fail when only indirect, noisy measurements (like acceleration) are available, as numerical differentiation amplifies noise.
• The Gap: There is a lack of frameworks that can simultaneously learn accurate continuous-time dynamics from partial/indirect observations (e.g., accelerometers only) while recovering physically interpretable latent states and explicit symbolic governing equations.

2. Methodology: SKANODE

The authors propose Structured Kolmogorov-Arnold Neural ODEs (SKANODE), a unified framework that integrates structured state-space modeling with Kolmogorov-Arnold Networks (KANs).

Core Architecture

SKANODE operates within a Neural ODE framework but imposes strict physical inductive biases:

1. Structured State-Space Formulation:
   Instead of learning arbitrary latent variables, the latent state vector $\mathbf{z}(t)$ is explicitly defined as physical coordinates:
   $$\mathbf{z}(t) = [\mathbf{x}(t), \mathbf{v}(t)]^T$$
   where $\mathbf{x}$ is displacement and $\mathbf{v}$ is velocity. The system dynamics are modeled as:
   $$\frac{d}{dt} \begin{bmatrix} \mathbf{x} \\ \mathbf{v} \end{bmatrix} = \begin{bmatrix} \mathbf{v} \\ \mathbf{h}_\theta(\mathbf{x}, \mathbf{v}, \mathbf{u}) \end{bmatrix}$$
   The observation model is defined directly as acceleration:
   $$\mathbf{y}(t) = \mathbf{h}_\theta(\mathbf{x}, \mathbf{v}, \mathbf{u})$$
   This structure enforces that the latent states evolve as derivatives of each other and that the output is the second derivative of the position, ensuring physical consistency even when trained only on acceleration data.

2. Kolmogorov-Arnold Networks (KANs):
   SKANODE utilizes KANs, which replace fixed activation functions in Multi-Layer Perceptrons (MLPs) with learnable spline-based functions.

   • Universal Approximation: KANs act as powerful function approximators for the dynamics function $\mathbf{h}_\theta$.
   • Symbolic Extraction: The learnable spline functions can be parsed into compact symbolic expressions (e.g., polynomials, trigonometric functions), enabling the discovery of closed-form governing equations.
   • Sparsification: KANs include built-in $L_1$ and entropy regularization to prune unnecessary connections, promoting parsimonious and interpretable models.

Two-Stage Learning Pipeline

The training proceeds in two distinct stages:

• Stage 1: Virtual Sensing (KAN$_{approx}$): A fully trainable KAN approximates the acceleration dynamics $\mathbf{h}_\theta$. The model is trained solely on indirect acceleration measurements. Due to the structured ODE constraints, the latent states $\mathbf{x}$ and $\mathbf{v}$ naturally evolve into physically meaningful displacement and velocity trajectories ("virtual sensing").
• Stage 2: Symbolic Discovery & Calibration (KAN$_{symbolic}$): Once the latent dynamics are learned, the spline functions are analyzed to extract a symbolic expression. This symbolic form replaces the neural network in the ODE. The affine coefficients of the symbolic equation are then fine-tuned (calibrated) within the ODE framework to maximize predictive accuracy while maintaining the interpretable structure.

3. Key Contributions

• End-to-End Symbolic Discovery from Indirect Data: Unlike SINDy, SKANODE does not require full-state measurements or numerical differentiation. It recovers governing equations directly from noisy, partial observations (e.g., accelerations).
• Physically Interpretable Latent States: By enforcing a structured state-space representation, SKANODE ensures that latent variables correspond to real physical quantities (displacement/velocity), solving the "black-box" opacity of standard NODEs.
• Theoretical Identifiability: The paper provides a formal proposition proving that under sufficient data richness and structural assumptions, SKANODE can uniquely recover the true governing dynamics and latent states from acceleration measurements alone.
• Superior Robustness: The framework demonstrates high robustness to noise, outperforming black-box NODE variants and classical system identification methods (ARX/NARX).

4. Experimental Results

The authors evaluated SKANODE on three benchmarks:

A. Duffing Oscillator (Synthetic)

• Task: Recover cubic stiffness nonlinearity from acceleration data.
• Results: SKANODE successfully identified the symbolic equation containing the $x^3$ term. It achieved significantly lower Mean Squared Error (MSE) in recovering latent displacement and velocity compared to ANODE and SONODE.
• Downstream Application: The extracted symbolic equation was used in a Feedback Linearization (FBL) controller. It achieved control performance comparable to Newton-based inversion but with a 7,400x reduction in online computation time, demonstrating practical utility for real-time control.

B. Van der Pol Oscillator (Synthetic)

• Task: Recover nonlinear damping ($\mu(1-x^2)\dot{x}$).
• Results: SKANODE correctly identified the quadratic coupling between displacement and velocity. It outperformed all baselines (including numerical SINDy) in both prediction accuracy (MSE) and structural similarity (SSIM), while ANODE failed to converge under noise.

C. F-16 Aircraft Ground Vibration (Real-World)

• Task: Model nonlinear interactions at the wing-payload interface using real sensor data.
• Results:
  • SKANODE accurately predicted acceleration responses where baselines failed or diverged.
  • Hysteretic Discovery: The inferred latent phase portraits revealed closed-loop patterns characteristic of hysteresis, a complex nonlinear phenomenon.
  • Symbolic Model: The discovered equation included a latent hysteretic state, confirming the presence of energy dissipation mechanisms that purely numerical methods (SINDy) failed to capture, leading to unstable predictions.

5. Significance

SKANODE represents a significant advancement in Physics-Encoded Deep Learning. Its significance lies in:

1. Bridging the Gap: It unifies the high predictive power of deep learning with the transparency of symbolic physics, making AI models trustworthy for engineering diagnostics.
2. Practical Applicability: It solves the "partial observability" problem common in structural health monitoring, where only accelerometers are available, yet engineers need to understand underlying displacements and forces.
3. Real-Time Control: The ability to distill complex learned dynamics into compact symbolic equations enables efficient, model-based control strategies that are computationally feasible for real-time deployment.
4. Scientific Discovery: It provides a robust tool for discovering governing equations in complex systems where traditional physics-based modeling is difficult due to unknown nonlinearities or unmeasured states.

In conclusion, SKANODE demonstrates that by embedding physical structure into the neural architecture and leveraging the symbolic capabilities of KANs, it is possible to learn accurate, robust, and interpretable models of nonlinear dynamics from limited sensor data.