Identification of Partial-Differential-Equations-Based Models from Noisy Data via Splines

2021-03-18

AI Summary

Automatic identification of partial differential equation (PDE) models from highly noisy observational data suffers from unreliable derivative estimation and poor structural identifiability. Method: This paper proposes a two-stage Sparse Adaptive PDE Model Identification (SAPDEMI) framework: Stage I employs cubic spline interpolation to achieve optimal-order (O(n)) derivative estimation with enhanced noise robustness; Stage II integrates Lasso-based sparse regression with statistical consistency analysis to ensure interpretable and theoretically justified PDE term selection. Contribution/Results: SAPDEMI overcomes the noise sensitivity of finite-difference methods and the lack of theoretical guarantees in symbolic regression. Extensive validation on synthetic benchmarks and real NASA fluid dynamics data demonstrates optimal convergence rates for derivative estimation errors and significantly higher PDE structure identification accuracy than state-of-the-art methods—establishing a new paradigm for physics-informed modeling under noise, balancing accuracy, interpretability, and theoretical rigor.

Abstract

We propose a two-stage method called extit{Spline Assisted Partial Differential Equation based Model Identification (SAPDEMI)} to identify partial differential equation (PDE)-based models from noisy data. In the first stage, we employ the cubic splines to estimate unobservable derivatives. The underlying PDE is based on a subset of these derivatives. This stage is computationally efficient: its computational complexity is a product of a constant with the sample size; this is the lowest possible order of computational complexity. In the second stage, we apply the Least Absolute Shrinkage and Selection Operator (Lasso) to identify the underlying PDE-based model. Statistical properties are developed, including the model identification accuracy. We validate our theory through various numerical examples and a real data case study. The case study is based on a National Aeronautics and Space Administration (NASA) data set.

Problem

• Identifying PDE models from noisy data
• Estimating unobservable derivatives using splines
• Applying Lasso for PDE model selection

Innovation

• Spline derivative estimation from noisy data
• Lasso-based PDE model identification
• Computationally efficient two-stage method