🤖 AI Summary
This study addresses the limited model interpretability and physical consistency in ordinary differential equation (ODE) discovery from noisy data. We propose a fully data-driven method that requires neither predefined basis functions nor explicit regularization. Our key innovation is the first incorporation of the analytical general solution form of linear, constant-coefficient ODEs into the modeling framework, coupled with spline-based transformation for smooth derivative estimation and unsupervised gradient matrix construction to directly solve for ODE coefficients via linear systems. By embedding the structural prior of linear ODEs, the method inherently ensures physical interpretability and structural fidelity, eliminating reliance on hand-crafted bases common in sparse regression approaches. Experiments demonstrate high-accuracy recovery of ground-truth dynamics even under severe noise, significantly improving robustness, generalization, and model fidelity compared to state-of-the-art methods.
📝 Abstract
The data-driven discovery of interpretable models approximating the underlying dynamics of a physical system has gained attraction in the past decade. Current approaches employ pre-specified functional forms or basis functions and often result in models that lack physical meaning and interpretability, let alone represent the true physics of the system. We propose an unsupervised parameter estimation methodology that first finds an approximate general solution, followed by a spline transformation to linearly estimate the coefficients of the governing ordinary differential equation (ODE). The approximate general solution is postulated using the same functional form as the analytical solution of a general homogeneous, linear, constant-coefficient ODE. An added advantage is its ability to produce a high-fidelity, smooth functional form even in the presence of noisy data. The spline approximation obtains gradient information from the functional form which are linearly independent and creates the basis of the gradient matrix. This gradient matrix is used in a linear system to find the coefficients of the ODEs. From the case studies, we observed that our modeling approach discovers ODEs with high accuracy and also promotes sparsity in the solution without using any regularization techniques. The methodology is also robust to noisy data and thus allows the integration of data-driven techniques into real experimental setting for data-driven learning of physical phenomena.