🤖 machine learning

LeARN: Learnable and Adaptive Representations for Nonlinear Dynamics in System Identification

The paper introduces LeARN, a novel system identification framework that leverages meta-learning and a lightweight deep neural network to automatically learn and adapt basis functions from data, thereby overcoming the domain-specific knowledge limitations of traditional SINDy methods while achieving competitive performance in modeling nonlinear dynamical systems.

Original authors: Arunabh Singh, Joyjit Mukherjee

Published 2026-06-02

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Arunabh Singh, Joyjit Mukherjee

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 robot how to fly a drone through a windy day. To do this safely, the robot needs a "rulebook" that explains exactly how the wind, the motors, and the drone's weight interact to move the drone.

The Old Way: The "Pre-Written Dictionary" Problem

For a long time, scientists used a method called SINDy to write this rulebook. Think of SINDy as a student trying to solve a math problem using a pre-written dictionary of words (like "sine," "cosine," or "polynomials").

The Good News: If the problem is simple, this dictionary works great. The student can find the right words, write a clear sentence, and explain exactly why the drone moves the way it does. It's very transparent.
The Bad News: The student needs a human expert to pick the right dictionary beforehand. If the wind behaves in a weird, new way that isn't in the dictionary, the student is stuck. They can't invent new words; they can only use the ones they were given. This makes the system rigid and hard to adapt to new, messy real-world situations.

The New Way: LeARN (The "Inventive Student")

The authors of this paper created a new system called LeARN (Learnable and Adaptive Representations for Nonlinear Dynamics).

Instead of giving the student a fixed dictionary, LeARN gives them a blank notebook and a super-smart, flexible pen.

Learning from Scratch: LeARN doesn't need a human to tell it which words to use. It looks at the data (the drone's flight logs) and learns to write its own words (mathematical functions) that perfectly describe what's happening.
The "Meta-Learning" Trick: To make sure this student is ready for any wind condition (not just the ones they practiced on), the authors used a technique called Meta-Learning.
- Analogy: Imagine training an athlete not just to run on a track, but to learn how to learn how to run on sand, ice, or mud. The athlete practices on many different surfaces so that when they face a new surface they've never seen before, they can instantly adjust their running style without starting from zero.
- In the paper, LeARN practiced on various wind speeds (from calm to very strong) so it could instantly adapt when the drone flew in a completely new wind pattern.

How It Works (The "Two-Part Engine")

LeARN breaks the problem into two parts, like a two-person team:

The Creator (The DNN): A lightweight neural network that invents the "words" (basis functions) needed to describe the physics. It's flexible and can create complex shapes to fit the data.
The Editor (The Selection Matrix): Another network that decides which of those invented words are actually important. It keeps the rulebook simple by ignoring the noise, ensuring the final model is still easy to understand (interpretable).

The Results: Does It Work?

The team tested LeARN on a famous dataset called Neural Fly, which involves a drone flying in a figure-eight pattern under different wind conditions.

The Challenge: The drone's movements were complex and oscillatory (wiggly), which is hard to model.
The Comparison: They compared LeARN against the old SINDy method.
- SINDy struggled when the wind got very strong or changed unexpectedly because its "dictionary" wasn't built for those specific conditions.
- LeARN adapted quickly. As the problem got more complex (adding more variables like rotation and speed together), LeARN actually started to perform better than SINDy.
The Verdict: LeARN achieved error rates very close to, and sometimes better than, the traditional method, but without needing a human expert to pre-select the math formulas.

Summary

In short, LeARN is a system that teaches a robot to write its own physics rulebook on the fly. Instead of relying on a human to guess the right math formulas, it learns the formulas directly from the data and uses a special "learning-to-learn" technique to stay flexible when the environment changes. It bridges the gap between the flexibility of modern AI and the clarity of traditional math, making robots better at handling the unpredictable chaos of the real world.

Technical Summary: LeARN – Learnable and Adaptive Representations for Nonlinear Dynamics

Problem Statement
System identification, the derivation of mathematical models from input-output data, faces significant challenges in nonlinear dynamical systems, particularly for robotics operating in unstructured environments. While traditional physics-based models offer interpretability, they struggle with complex nonlinearities, unknown disturbances, and high-dimensional inputs. Data-driven methods like Sparse Identification of Nonlinear Dynamics (SINDy) have emerged as a solution, distilling dynamics into interpretable linear combinations of basis functions. However, SINDy relies heavily on domain expertise to construct a pre-defined library of basis functions (e.g., polynomials, trigonometric terms). This dependency limits adaptability when encountering unknown or evolving dynamics. Conversely, deep learning approaches (e.g., universal approximators, PINNs) lack the structural interpretability required for safety-critical applications. Existing attempts to learn latent coordinate transformations often sacrifice the interpretability of the resulting dynamical equations.

Methodology
The authors propose LeARN (Learnable and Adaptive Representations for Nonlinear Dynamics), a framework designed to bridge the gap between the expressiveness of deep learning and the interpretability of SINDy without requiring prior domain knowledge.

Learnable Basis Library: Unlike SINDy, which uses a fixed library $\Theta(\cdot)$ , LeARN parameterizes the basis function library using a lightweight Deep Neural Network (DNN), denoted as $\Theta(X; \psi)$ . This network learns the functional form of the basis functions directly from the data in the physical state space.
Feature Selection: The framework decomposes the system dynamics $\dot{x} \approx \Theta(X; \psi)E(X; \phi)^T$ , where $E(X; \phi)$ is a second DNN acting as a feature selection matrix. This structure mimics the sparse regression of SINDy, allowing the model to identify which input features contribute dominantly to the dynamics, thereby retaining structural interpretability regarding feature importance, even if the functional form of the basis functions is learned.
Meta-Learning for Adaptation: To handle varying noise conditions and evolving system dynamics, LeARN employs Model-Agnostic Meta-Learning (MAML). The meta-training process optimizes the initial parameters ( $\psi, \phi$ ) across a distribution of tasks (e.g., different wind conditions). This ensures the model possesses a robust initialization that allows for rapid adaptation to unseen environments via a few gradient steps during online deployment.
Online Adaptation: During deployment, the model adapts to new tasks (e.g., new wind speeds) using an adaptation loss that includes a regularizer to enforce Lipschitz continuity on the state inputs, ensuring stability.

Key Contributions

Elimination of Pre-defined Libraries: LeARN removes the requirement for a pre-defined function library and domain expertise, learning the basis functions directly from data.
Adaptability via Meta-Learning: By meta-learning the library of basis functions, the framework enables rapid adaptation to changing dynamics and noisy conditions.
Balanced Interpretability: The use of lightweight DNNs allows for modeling complex nonlinearities while maintaining interpretability through the sparsity structure of the learned feature selection matrix, identifying dominant input features.

Experimental Results
The framework was validated on the Neural Fly dataset, a challenging benchmark involving quadrotor flight under various wind conditions. The authors compared LeARN against SINDy across three formulations of quadrotor dynamics:

Partial translational dynamics.
Partial attitude dynamics.
Full system dynamics (combining translation and attitude).

Performance Findings:

Generalization: LeARN demonstrated competitive generalization performance compared to SINDy.
Dimensionality Scaling: A key finding is that the performance gap between LeARN and SINDy narrows as the dimensionality of the input feature space increases. For full-system dynamics (higher dimension), LeARN achieved a generalization error difference of only 0.001811 compared to SINDy, suggesting superior scalability for complex, high-dimensional systems.
Qualitative Comparison: Visual comparisons of the full dynamics across different wind conditions (e.g., 35wind, 70wind, 100wind) showed that LeARN effectively captures the dynamical evolution, matching the ground truth and SINDy's performance.

Significance and Claims
The paper positions LeARN as a step toward the autonomous discovery of dynamical systems. Its primary significance lies in its ability to uncover the governing principles of complex systems without extensive domain-specific interventions. The authors claim that LeARN offers a pathway to robust, versatile, and adaptive robotic systems capable of operating safely in real-world, unstructured environments. By learning the basis functions directly, the framework addresses the scalability and flexibility limitations of SINDy while preserving the interpretability necessary for safety-critical applications. The work suggests future potential in health monitoring of systems via data-driven identification and modeling of intrinsic residual dynamics.