Data Denoising and Derivative Estimation for Data-Driven Modeling of Nonlinear Dynamical Systems

2025-09-17

AI Summary

Observation noise in nonlinear dynamical systems severely distorts derivative estimation, leading to failure in system identification. Method: We propose RKTV-INR—a framework that jointly embeds implicit neural representations (INRs) with Runge–Kutta (RK) numerical integration structural priors and total variation (TV) regularization into the training objective, enabling smooth reconstruction of noisy state trajectories; high-fidelity derivatives are then obtained via automatic differentiation to drive sparse identification of nonlinear dynamics (SINDy). Contribution/Results: RKTV-INR is the first method to enforce RK integration schemes as hard constraints within INR training, while co-optimizing TV-based denoising and dynamical priors in an end-to-end fashion—simultaneously achieving denoising, accurate derivative estimation, and equation discovery. Experiments on chaotic and high-dimensional systems—including Lorenz and Kuramoto–Sivashinsky—demonstrate substantial improvements over state-of-the-art approaches, with superior robustness to strong noise and enhanced physical interpretability.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Data-driven modeling of nonlinear dynamical systems is often hampered by measurement noise. We propose a denoising framework, called Runge-Kutta and Total Variation Based Implicit Neural Representation (RKTV-INR), that represents the state trajectory with an implicit neural representation (INR) fitted directly to noisy observations. Runge-Kutta integration and total variation are imposed as constraints to ensure that the reconstructed state is a trajectory of a dynamical system that remains close to the original data. The trained INR yields a clean, continuous trajectory and provides accurate first-order derivatives via automatic differentiation. These denoised states and derivatives are then supplied to Sparse Identification of Nonlinear Dynamics (SINDy) to recover the governing equations. Experiments demonstrate effective noise suppression, precise derivative estimation, and reliable system identification.

Problem

• Denoising noisy measurements in nonlinear dynamical systems
• Accurately estimating derivatives from observed data
• Recovering governing equations through system identification

Innovation

• Implicit neural representation for trajectory denoising
• Runge-Kutta integration with total variation constraints
• SINDy method for governing equations recovery