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

TL;DR

LeARN introduces a learnable, adaptive library of basis functions for nonlinear system identification, removing the need for hand-crafted function libraries as required by SINDy. By applying a model-agnostic meta-learning framework (MAML), LeARN meta-trains the basis-function library and a feature-selection mechanism, enabling rapid adaptation to new dynamical regimes and varying noise conditions. The approach uses lightweight neural networks to parameterize the library and selection matrix, and enforces Lipschitz-based regularization during online adaptation. Empirical results on the Neural Fly dataset show LeARN achieving competitive dynamical error compared to SINDy, with notable gains in generalization as input dimensionality increases, demonstrating robust autonomous discovery of governing dynamics in nonlinear robotic systems. The work lays groundwork for autonomous, domain-agnostic modeling of complex dynamics in real-world environments and suggests future directions toward learning intrinsic residual dynamics and broader adaptive capabilities.

Abstract

System identification, the process of deriving mathematical models of dynamical systems from observed input-output data, has undergone a paradigm shift with the advent of learning-based methods. Addressing the intricate challenges of data-driven discovery in nonlinear dynamical systems, these methods have garnered significant attention. Among them, Sparse Identification of Nonlinear Dynamics (SINDy) has emerged as a transformative approach, distilling complex dynamical behaviors into interpretable linear combinations of basis functions. However, SINDy relies on domain-specific expertise to construct its foundational "library" of basis functions, which limits its adaptability and universality. In this work, we introduce a nonlinear system identification framework called LeARN that transcends the need for prior domain knowledge by learning the library of basis functions directly from data. To enhance adaptability to evolving system dynamics under varying noise conditions, we employ a novel meta-learning-based system identification approach that uses a lightweight deep neural network (DNN) to dynamically refine these basis functions. This not only captures intricate system behaviors but also adapts seamlessly to new dynamical regimes. We validate our framework on the Neural Fly dataset, showcasing its robust adaptation and generalization capabilities. Despite its simplicity, our LeARN achieves competitive dynamical error performance compared to SINDy. This work presents a step toward the autonomous discovery of dynamical systems, paving the way for a future where machine learning uncovers the governing principles of complex systems without requiring extensive domain-specific interventions.

Figures (2)

• Figure 1: For a concatenated feature vector $X\in\mathbb{R}^{1 \times (I+U)}$ of the form, $X = xu$, where $x\in\mathbb{R}^{1 \times I}$ is the state feature vector and $u\in\mathbb{R}^{1 \times U}$ is the control input corresponding to $x$, $\Theta(X;\psi) \in \mathbb{R}^{1\times P(I+U)}$ is the learned basis function library for a total of P basis functions, where $m_p$ represents the $p^{th}$ basis function and $\mathcal{E}(X;\phi) \in \mathbb{R}^{I\times P(I+U)}$ is the learned feature selection matrix, for a total of $n = I$ sets of coefficients, $e_k \in \mathbb{R}^{1 \times (P+U)}$ is coefficient for the $k^{th}$ state feature in $X$.
• Figure 2: Qualitative results of the full dynamics modeled by SINDy and LeARN across different wind conditions. Background represents the ground truth dynamics. Then, we have Adaptation performance , and the Generalization performance on unseen data. The horizontal axis runs across 2511 data points and the vertical axis represents the value.