💻 computer science

Model discovery for dynamical systems with complex-valued product units

This paper introduces a data-driven model discovery method using complex-valued product-unit networks that learns governing equations directly from data without predefined function libraries, successfully recovering exact dynamics for chaotic systems and generating stable predictions for real-world human-gait signals.

Original authors: Martin Brückmann, Babette Dellen, Uwe Jaekel

Published 2026-05-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Martin Brückmann, Babette Dellen, Uwe Jaekel

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 a detective trying to figure out the secret recipe for a delicious, chaotic stew. You can't see the chef or the ingredients list, but you can watch the pot bubble and taste the soup at different moments. Your goal isn't just to guess what the soup will taste like next; you want to discover the exact mathematical formula that explains why it bubbles the way it does.

This paper presents a new "detective tool" called the Complex-Valued Product-Unit Network. Here is how it works, broken down into simple concepts:

1. The Old Way vs. The New Way

The Old Way (SINDy & Symbolic Regression): Imagine trying to find the recipe by testing every possible combination of ingredients from a giant, pre-written cookbook. You might try "salt + pepper," then "salt + pepper + sugar," then "salt + pepper + sugar + flour." If the real recipe uses a weird ingredient like "0.5 teaspoons of star anise" (a fractional exponent) or "negative salt" (negative exponents), your cookbook might not even have those options. You are limited by what you already know.
The New Way (This Paper): Instead of using a cookbook, this new tool is like a shape-shifting chef. It doesn't need a list of ingredients. It looks at the bubbling soup and learns to create its own ingredients on the fly. It can invent terms like $x^{0.5}$ (square root) or $x^{-2}$ (inverse) just by observing the data. It builds the formula from scratch, discovering the "monomials" (the building blocks of the math) directly from the observations.

2. How the "Shape-Shifting Chef" Works

The model uses special "units" that act like magic multipliers.

The Product Unit: Think of a standard calculator that just adds numbers. This model uses a special calculator that multiplies numbers together, but it can also raise them to any power it wants—even weird, fractional, or negative powers.
The Complex Domain: To make this math work smoothly (especially when dealing with negative numbers or roots), the model operates in the "complex number" world. Imagine a map where you have not just North/South (real numbers) but also East/West (imaginary numbers). The model uses this extra dimension to navigate tricky mathematical terrain that would trip up standard calculators.
The Result: The model spits out a list of terms (like $5x^2y$ or $-4z^{0.1}$ ) and adds them up to recreate the system's behavior.

3. The Test Kitchen: Benchmark Systems

The authors tested their detective tool on four famous "chaotic" systems (mathematical recipes known to be very unpredictable):

Lorenz63: The classic weather model.
Lorenz84: A more complex weather model.
Four-Wing Attractor: A system that looks like a butterfly with four wings.
Lorenz_Fract: A modified version of the first one that includes a "fractional" ingredient (a square root).

The Results:

When they fed the model enough data (at least 3,000 points), it successfully rediscovered the exact original recipes 90% of the time for the standard systems.
Even for the tricky "fractional" system, it got it right 70–90% of the time.
The "Crystal Ball" Test: Once the model found the recipe, they let it predict the future. If the recipe was perfect, the prediction stayed on track forever. If the recipe was slightly wrong, the prediction would eventually spiral out of control. The model was so accurate that its predictions stayed stable for a very long time.

4. The Real-World Challenge: Human Walking

Finally, they tried the tool on something messy and real: accelerometer data from a person walking.

The Problem: Unlike the clean math problems above, walking is noisy and high-dimensional (lots of variables). There is no known "recipe" for human walking to compare against.
The Approach: Instead of finding a differential equation (a recipe for change), the model learned to predict the next step of the walk based on the previous 50 steps.
The Outcome: The model learned a stable pattern. Even when asked to predict the walk for 30 seconds (three times longer than the time it was trained on), it didn't go crazy. The error stayed small and bounded, roughly 12–14% of the total movement range. It successfully captured the rhythm of walking without needing to know the physics beforehand.

5. The Catch (Limitations)

The paper admits a few things that aren't perfect yet:

Guessing the Size: You have to tell the model roughly how many "ingredients" (terms) to look for. If you guess too low, it can't find the recipe. If you guess too high, it gets messy.
Complexity: When applied to the walking data, the model became huge (thousands of parameters). While it worked, the resulting "recipe" was too complicated for a human to read and understand easily. It's a "black box" in this high-dimensional case, unlike the simple, readable formulas it found for the math benchmarks.
Noise: The walking data had to be cleaned up (filtered) before the model could learn from it. It hasn't been tested extensively on very noisy data yet.

Summary

This paper introduces a smart, flexible tool that can reverse-engineer the laws of nature (or at least complex systems) directly from data, without needing a pre-made list of possible formulas. It works brilliantly on mathematical puzzles and shows promise for real-world, messy data like human movement, provided you have enough data to feed it.

Technical Summary: Model Discovery for Dynamical Systems with Complex-Valued Product Units

Problem Statement
The recovery of governing equations from observed trajectories offers deeper structural insight into dynamical systems than mere state prediction. However, identifying the relevant monomials—particularly those involving fractional or negative exponents—in high-dimensional systems is challenging due to the potentially infinite number of candidate terms. Established methods like SINDy (Sparse Identification of Nonlinear Dynamics) and symbolic regression rely on predefined libraries of candidate functions. This restriction limits the representation of nonlinear function classes and renders exhaustive search intractable as input dimensionality increases, as the number of candidate terms grows exponentially.

Methodology
The authors propose a data-driven approach using Complex-Valued Product-Unit Networks to discover governing equations without a predefined candidate library.

Architecture: The model utilizes product units where each unit computes a product of inputs raised to complex-valued powers ( $x_i^{w_i}$ ). These are implemented by concatenating summation units with logarithmic and exponential activation functions defined in the complex domain: $\prod x_i^{w_i} = \exp(\sum w_i \log x_i)$ .
Formulation: The network output represents the right-hand side (RHS) of ordinary differential equations (ODEs) as a sparse linear combination of these complex monomials:
$\hat{f}(x_1, \dots, x_n) = \sum_{k=1}^m c_k \prod_{i=1}^n x_i^{w_{k,i}}$
where $c_k$ and $w_{k,i}$ are complex coefficients and exponents learned directly from data.
Training: The network is trained via gradient descent (Adam optimizer) using a complex-valued mean squared error loss. For benchmark systems, the model learns to predict time derivatives ( $\dot{x}$ ) given state variables. For real-world gait data, where derivatives are unavailable, the model employs time-delay embedding to predict the next state directly from a sequence of past observations.
Post-Processing: To handle numerical noise and ensure stability, terms with absolute coefficients below $10^{-3}$ are excluded. Similar terms (where exponents differ by $\le 0.1$ ) are merged by averaging exponents and summing coefficients.

Key Contributions

Library-Free Discovery: Unlike SINDy or symbolic regression, this approach does not require a preset set of candidate functions. It learns relevant monomials, including those with fractional or negative exponents, directly from the data.
Complex-Valued Domain: By operating fully in the complex domain (inputs, weights, and outputs), the model naturally handles negative inputs and fractional powers, which are problematic for standard real-valued product units.
Sparse Representation: The method aims to recover sparse algebraic structures driving the dynamics, facilitating interpretability.

Results
The method was evaluated on four chaotic benchmark systems and real-world human gait accelerometer data.

Benchmark Systems (Lorenz63, Lorenz84, Four-Wing, Lorenz_Fract):
- Using at least 3,000 training points, the model recovered the exact governing equations in 90% of trials for the integer-exponent systems (Lorenz63, Lorenz84, Four-Wing).
- For the fractional variant (Lorenz_Fract, containing a $z^{0.5}$ term), exact recovery was achieved in 70–90% of trials with 3,000–5,000 points.
- When all leading terms were correctly identified, the Effective Prediction Time (EPT) was effectively infinite (indistinguishable from the true system). If terms were missed, the EPT dropped significantly (below one Lyapunov time).
- The study noted that spurious imaginary components in the output often served as early indicators of trajectory divergence (prediction uncertainty).
Real-World Application (Human Gait):
- Applied to 150-dimensional gait signals (3 axes $\times$ 50 time lags), the model was trained to predict future states from past observations.
- The model produced stable trajectories over a test horizon three times longer than the training interval.
- Prediction errors remained bounded, with a Root Mean Square Error (RMSE) of approximately 12–14% of the signal amplitude range.
- The model successfully captured the quasi-periodic nature of gait without accumulating substantial drift.

Significance and Claims
The paper claims that complex-valued product-unit networks offer a viable alternative to library-based methods for discovering dynamical systems, particularly when the functional form is unknown or involves non-integer powers. The approach demonstrates that:

Exact governing equations can be recovered from data without prior knowledge of the candidate function set.
The method extends naturally to fractional power laws, which are difficult to handle with standard sparse regression libraries.
The model can generate stable, bounded predictions for high-dimensional, noisy real-world systems (like human gait) even when analytic equations are unavailable.

Limitations and Future Work
The authors acknowledge several constraints:

Hyperparameter Sensitivity: The number of product units must be specified in advance and must be at least equal to the number of dominant terms; while training loss can hint at model adequacy, automatic determination remains an open challenge.
Interpretability vs. Dimensionality: In high-dimensional applications (e.g., gait analysis with ~46,000 parameters), the sparse, human-readable nature of the solution degrades.
Noise Robustness: The method required low-pass filtering for gait data, and systematic evaluation of robustness to measurement noise is needed.

Future work suggested by the authors includes developing adaptive methods to automatically determine the number of product units, extending the framework to partial differential equations, and conducting systematic evaluations for biomedical applications such as fall detection or gait anomaly identification.